ASPECTS OF PART VS WHOLE RELATIONSHIPS IN QUANTUM INFORMATION PROCESSING A Thesis Submitted to the Faculty in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics and Astronomy by Peter Douglas Johnson DARTMOUTH COLLEGE Hanover, New Hampshire October 20, 2016 Examining Committee: Lorenza Viola, Chair Chandrasekhar Ramanathan Miles Blencowe F. Jon Kull, Ph.D. Dean of Graduate and Advanced Studies Fernando G.S.L. Brandão
Transcript
ASPECTS OF PART VS WHOLE RELATIONSHIPS IN
QUANTUM INFORMATION PROCESSING
A Thesis
Submitted to the Faculty
in partial fulfillment of the requirements for the
degree of
Doctor of Philosophy
in
Physics and Astronomy
by
Peter Douglas Johnson
DARTMOUTH COLLEGE
Hanover, New Hampshire
October 20, 2016
Examining Committee:
Lorenza Viola, Chair
Chandrasekhar Ramanathan
Miles Blencowe
F. Jon Kull, Ph.D.Dean of Graduate and AdvancedStudies
Fernando G.S.L. Brandão
Abstract
Since its inception in quantum theory, the phenomenon of quantum entanglement hasevolved from Einstein’s enigmatic “spooky action at a distance” to a crucial resourcefor quantum information processing. Recent technological advances geared towardscontrolling quantum systems and harnessing quantum entanglement have borne newperspectives and challenges. One major challenge is the development of a completetheory of multi-partite entanglement. Quantum theory places highly non-trivial con-straints on how entanglement may be distributed among the parts of a whole com-posite quantum system. In the simplest example, the more entangled system A iswith system B, the less entangled system B can be with system C. This principle,known as the “monogamy of entanglement”, is a uniquely quantum feature enabling,in particular, secure quantum key distribution protocols and having ramifications forcontrol of many-body quantum systems.
In the first half of this thesis, I describe our contributions toward understandingthe principles governing the distribution of multipartite entanglement. In particular,we elucidate surprising connections between the underlying kinematic constraints andthe dynamical constraints stemming from the “no-cloning” principle and the uncer-tainty principle for incompatible quantum observables.
In the second half of the thesis, I describe our contributions towards developingmethods to create and control multipartite entanglement under realistic resource con-straints. Thanks to a number of recent experimental realizations, dissipative controlof quantum systems is garnering increasing attention, alongside traditional unitaryapproaches. We investigate the use of dissipative control for driving a quantum sys-tem towards a target entangled state independently of initialization, a task known as“stabilization” – subject to the constraint that control resources be “quasi-local”. Inparticular, we develop mathematical tools for discovering hidden structures amongthe parts of a multi-partite entangled state which enable their stabilization.
ii
Preface
I cannot imagine a more ideal setting for balancing work and play than DartmouthCollege in Hanover, New Hampshire. I am grateful to have spent such formative yearsin an environment that combines intense intellectual stimulation with striking naturalbeauty. I will truly miss the Norwich hills cycling, Connecticut river swimming,and granite peaks hiking. Certainly, this sense of connection to place is rooted inconnections to friends. I am sincerely thankful to all of you who have helped mewrite such a vibrant chapter of my life.
First and foremost, I thank my PhD adviser and mentor Lorenza Viola. You haveshaped my development as a scientist in ways that I have yet to fully appreciate andhave taught me lessons that extend far beyond physics. For the remainder of mycareer I will be drawing inspiration from your tireless demand of quality.
I thank my local thesis committee Chandrasekhar Ramanathan and Miles Blencowefor guidance and encouragement throughout my PhD. I also thank my outside ex-aminer Fernando Brandão for participating in my defense and encouraging me as ascientist.
I owe thanks to two other members of my Italian academic family. Thanks to myacademic older brother, Francesco Ticozzi, for friendship, for expanding my mathe-matical tool set, and for supporting me in many ways. Thanks to my academic uncle,Roberto Onofrio, for thoughtful, timely guidance over the past six years.
My working days were made brighter by the companionship of two wonderfulfriends Abhijeet Alase and Salini Karuvade. I look forward to the evolution of ourfriendship and collaboration for years to come.
A number of physicists have been responsible for steering my course at variouspoints over the past six years. Thanks to: Sandu Popescu for a conversation overbilliards that blossomed into an undying pursuit; Stephon Alexander for helping meto see some some beautiful connections between my passions of physics and music;Carlton Caves for some memorable anecdotes, your encouragement, and for being arole model. Finally, I want to give special thanks to Ben Schumacher for setting mycourse and to Bill Wootters for sustaining it.
Outside of academia, I thank my Hanover friends Dan Reeves, Ian Adelstein, BillyBraasch, and Mana Francisquez for pulling me out of the office for adventures and
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then still talking shop along the way. Thanks to Sam, Russ, Andrew, and Ben formaking deep thinking a fond passtime. Lastly, I want to especially thank my parents,brothers, and Ariana for your love and for putting up with (and even encouraging!)my pursuit of physics.
Quantum information processing provides a radically different means of comput-ing that shows promise for solving previously intractable problems. From a physicalstandpoint, one of the most compelling applications is to the simulation of quan-tum systems [1, 2]. A very practical and promising use for quantum simulation is,for instance, the calculation of molecular energies [3]. With a classical computer,such calculations are feasible for small systems, but they soon become too resource-intensive as larger molecules are considered. Quantum algorithms have been devel-oped which are believed to solve this problem by requiring exponentially fewer stepsthan the best known classical algorithms. A few months before the time of writing,Google Inc. and collaborators published work that demonstrated the first example ofa scalable quantum simulation calculating molecular energies [4].
Besides its technological applications, quantum information science also offers adiverse set of mathematical tools and concepts. Many of these are actively beingdeveloped or, even, created. These tools and concepts are proving useful in areas ofphysics as diverse as condensed matter physics [5, 6], quantum field theory [7, 8], andquantum gravity [9, 10, 11].
Even with its technological potential and powerful mathematical toolbox, quan-tum information science has captivated many researchers, including the author, fora different reason. Some of the seminal contributions to the field were initiated byphysicists asking foundational questions about quantum mechanics [12]. Quantuminformation science has given us deep insights into the workings of Nature. Certainly,more insights lie just beyond the horizon.
In the most common setting, quantum information is processed with quantumbits, or qubits as opposed to bits. In principle, any two-level quantum system mayserve as a qubit. Examples include the spin of an electron, the polarization of aphoton, or the two lowest energy levels in a superconducting Josephson junction [13].No single system has yet to triumph as the universally used qubit. In practice, certainsystems (e.g. trapped ions or superconducting qubits) are more accurately controlledthan others, and therefore, show more promise for becoming the standard qubit.
Quantum bits obey more subtle rules than classical bits do. In particular, multiplequantum systems can exhibit a uniquely quantum type of correlation known as quan-tum entanglement. Although we do not yet fully understand its ramifications, it isthought that quantum entanglement plays a crucial role in providing the advantagesof quantum computing over classical computing. Also, not all classical informationprocesses translate to the quantum case. For instance, there is no quantum processwhich can perfectly copy, or clone, an unknown quantum bit [14]. This mentioning ofquantum entanglement and the no-cloning principle anticipate some of the nuancesof quantum information processing that we address throughout the thesis.
The field of quantum information science, which gained traction in the 1990s, isnow a firmly established research field [15]. Many of the most pressing challenges in
2
Introduction
quantum information science demand improvements in hardware engineering. Muchof the theoretical groundwork is, by now, relatively well-established. The importantconcepts from classical information theory have been appropriately repurposed for usein quantum information theory [16]. There are a number of well-established modelsof quantum computing such as the circuit model [17], measurement-based quantumcomputing [18], and, towards solving certain optimization problems, quantum an-nealing [19]. The theory of quantum error correction, necessary for building scalable,fault-tolerant quantum computers, is relatively well-understood [20].
Despite these advances, many theoretical challenges remain and are being activelypursued. We still lack a complete understanding of the principles ensuring “quantumsupremacy” over classical information processing. And, furthermore, we cannot antic-ipate the breakthroughs in understanding and unforeseen applications that are likelyto result from a deeper inquiry into the quantum.
The theory of bipartite entanglement, quantum correlations between two systems,is well-understood, including mixed-state entanglement. However, even in the sim-plest setting where subsystems are distinguishable, multipartite entanglement de-mands a much richer tool-kit of mathematical concepts, many of which are currentlybeing developed. Some of these tools are already finding diverse applications, such astowards exploring space-time as an emergent phenomenon based on quantum entan-glement [11]. Nevertheless, questions remain as for how best to describe multipartiteentanglement, to characterize its features, and to use it as a resource.
In the circuit model approach to quantum information processing, one applies asequence of unitary quantum gates to some fiducial input state, seeking to maintainsufficient coherent control of the relevant parts of the system. However, even withinthe quantum circuit model, access to suitable non-unitary control is crucial for properinitialization or “entropy removal” in fault-tolerant architectures, as well as for uni-versal “digital” simulation of open-system dynamics [21]. Control theoretic advancesin open quantum systems have further revealed new avenues for utilizing incoherentcontrol by means of engineered dissipation. Such non-unitary control resources havebeen shown to be beneficial for the tasks of robust preparation of resource states [22],rapid quantum state purification [23], and engineered dissipative quantum memories[24]. The last few years, in particular, have seen a surge of interest in proposals fordissipatively preparing strongly-correlated and topological phases of matter. Recentexperimental advances have used engineered dissipation to autonomously drive a two-qubit system towards an entangled resource state. This feat has been achieved withsuperconducting qubits [25] as well as with ion traps [26]. With these advances comesa demand for developing mathematical tools which can fully describe the capabilitiesand limitations of controlled dissipation.
This thesis contributes to the problems identified above. The unifying theme tyingthis work together is the quantum part vs whole relationship. Although the content
3
Introduction
Figure 1.1: Three quantum systems ABC with delineated subsystems AB and BC.
is varied in motivation and mathematical techniques, nearly all of the problems weconsider admit a simple example on three subsystems as pictured in Fig. 1.1. Manyquestions we consider involve confronting the interaction, or “overlap”, of subsystems(e.g. subsystem AB overlaps with subsystem BC). Like discussing the weather,the ubiquitous difficulty of “the interacting case” gives even physicists with differingbackgrounds a topic with which to mutually relate. It may be useful for the readerto keep this picture in mind throughout.
The thesis naturally divides into two parts. The first half explores a concept thatwe developed known as quantum joinability. The notion of joinability addresses thequestion of whether or not there exists a description of the whole (e.g. ABC) whichis consistent with constraints on the parts (e.g AB and BC); equivalently, we ask ifthe constrained parts can be “joined” into some physically allowed whole. Due to thephenomenon of entanglement, quantum theory places non-trivial constraints on theways that the parts of a quantum system may be correlated among one another. Asan example, the more entangled qubit A is with B, the less entangled qubit B canbe with C. This phenomenon, known as the monogamy of entanglement, is uniquelyquantum in that classical correlations are not limited by such a constraint. As thenumber of systems increases, the complexity of such constraints grows drastically.The study of such constraints is often referred to as the quantum marginal problem[27]. This problem originated from the field of quantum chemistry [28], motivated bythe attempt to simplify calculations of atomic and molecular ground state energies.More recently, this problem has gained attention from the quantum information com-munity since it addresses the nature of entanglement distribution in a multipartitequantum system. Researchers have drawn on and developed diverse mathematicaltools towards tackling this very difficult problem. We have contributed to this effortby analyzing, in detail, a number of tractable cases which bear insight on the moregeneral problem. Furthermore, we have established rigorous connections among theconcepts of monogamy of entanglement, the no-cloning principle, and incompatiblemeasurements. These concepts are ubiquitous in quantum information and are find-ing application in other areas of physics such as black-hole thermodynamics [29]. Wedevelop the framework of quantum joinability in order to put these notions on equalfooting and elucidate their common origin.
4
Introduction
• Chapter 2 defines the problems of quantum sharability and quantum joinabilityfor bipartite quantum states. We investigate some basic scenarios in which agiven set of bipartite quantum states may consistently arise as the set of re-duced states of a global N -partite quantum state. We restrict the discussionto bipartite reduced states that belong to the paradigmatic classes of Wernerand isotropic states in d dimensions, and focus on two specific versions of thequantum marginal problem which we find to be tractable. The first is Alice-Bob, Alice-Charlie joining, with both pairs being in a Werner or isotropic state.The second is m-n sharability of a Werner state across N subsystems, whichmay be seen as a variant of the N -representability problem to the case wheresubsystems are partitioned into two groupings of m and n parties, respectively.By exploiting the symmetry properties that each class of states enjoys, we de-termine necessary and sufficient conditions for three-party joinability and 1-nsharability for arbitrary d. Our results explicitly show that although entangle-ment is required for sharing limitations to emerge, correlations beyond entan-glement generally suffice to restrict joinability, and not all unentangled statesnecessarily obey the same limitations. The relationship between joinability andquantum cloning as well as implications for the joinability of arbitrary bipartitestates are discussed. In particular, the observations regarding quantum cloninglead naturally into the investigation of the subsequent chapter.
• Chapter 3 develops the framework of quantum joinability which unifies seem-ingly different joinability problems for bipartite quantum states and channels.This includes well known problems such as optimal quantum cloning and quan-tum marginal problems as special instances. Central to our generalization isa variant of the Choi-Jamiolkowski isomorphism between bipartite states anddynamical maps which we term the “homocorrelation map”: while the formeremphasizes the preservation of the positivity constraint, the latter is designedto preserve statistical correlations, allowing more direct contact with entan-glement. In particular, we define and analyze state-joining, channel-joining,and local-positive joining problems in three-party settings exhibiting collectiveU ⊗ U ⊗ U symmetry, obtaining exact analytical characterizations in low di-mension. Suggestively, we find that bipartite quantum states are limited inthe degree to which their measurement outcomes may agree, while quantumchannels are limited in the degree to which their measurement outcomes maydisagree. Loosely speaking, quantum mechanics enforces an upper bound on theextent of positive correlation across two subsystems at a single time, as well ason the extent of negative correlation between the state of a single system acrosstwo instants of time. We argue that these general statistical bounds inform thequantum joinability limitations, and show that they are in fact sufficient for thethree-party U ⊗ U ⊗ U -invariant setting.
5
Introduction
• Chapter 4 delves deeper into underpinnings of quantum joinability. We adopt adifferent approach to the joinability problem, ensuring positive-semidefinitenessor complete positivity from the outset. With this, we formalize the “compositionlaw” of correlations from the previous two chapters. We incorporate the notionof incompatibility of observables as another example of quantum joinability.Furthermore, we draw parallels between the quantum and classical problems,finding a intuitive common cause for their respective joinability constraints.The mathematical techniques developed here are intended to elucidate somequantum peculiarities such as the distinction between causal and acausal quan-tum relationships and the origin of monogamy of entanglement, no-cloning, andmeasurement incompatibility.
The second half of this thesis explores engineering dissipation to stabilize quantumstates both asymptotically and in finite time. Towards preparing the quantum re-source of many-body entanglement in a realistic setting, one must address the issue ofconstrained control capabilities. This becomes increasingly relevant as the number ofparts in the systems is scaled up, as is needed for quantum information processing todeliver its full potential. In particular, a possible realistic constraint on one’s controlof the system is that only parts of the system can be addressed at a time. Much likea logical circuit, we may assume that manipulations of the whole are achieved by aseries of manipulations of the parts. Such an implementation is said to be quasi-local.Thus, in quasi-local stabilization, the part-whole relationship features, in that we seeka preparation of the whole (e.g. ABC) by addressing only its parts (e.g. AB andBC).
The task of engineering quasi-local dissipation to drive a quantum system, all-to-one, into a target quantum state has been initiated and explored in [30, 31, 32, 33,34]. These works consider a continuous-time dynamics generated by an engineeredquasi-local Markovian master equation, giving examples and exploring conditions forpreparing, or “stabilizing”, a target pure state. In practice, pure quantum statesare never available and natural dissipative dynamics exhibit mixed steady states.Accordingly, we contribute to this line of research by exploring quasi-local stabilizationin the case of mixed target states. Some implementations of engineered dissipationare best modeled by discrete-time dynamics [35]. Constraining the dynamical mapsto act quasi-locally, we can view the sequence of maps as a dissipative quantumcircuit. An advantage in this case, compared to that of continuous-time, is that atarget state may be exactly stabilized in a finite time. We contribute to the fieldof dissipative quantum control by determining conditions for finite-time stabilizationof a target state. We describe how well-known resource states, such as graph statesfor measurement based quantum computing, may be stabilized with these schemes.To conclude, we present some preliminary work geared towards achieving quantumencoding using quasi-local resources.
6
Introduction
• Chapter 5 builds off of previous work by L. Viola and F. Ticozzi to investi-gate the engineering of dissipative continuous-time dynamics to render a targetmixed quantum state as the unique global attractor of the dynamics. In par-ticular, we determine necessary and sufficient conditions for whether or not agiven target state can be the unique steady state of frustration-free quasi-localcontinuous-time Markovian dynamics. We investigate under which conditionsa mixed state on a finite-dimensional multipartite quantum system may bethe unique, globally stable fixed point of frustration-free semigroup dynamicssubject to specified quasi-locality constraints. Our central result is a linear-algebraic necessary and sufficient condition for a generic (full-rank) target stateto be frustration-free quasi-locally stabilizable, along with an explicit procedurefor constructing Markovian dynamics that achieve stabilization. If the targetstate is not full-rank, we establish sufficiency under an additional condition,which is naturally motivated by consistency with pure-state stabilization re-sults yet provably not necessary in general. Several applications are discussed,of relevance to both dissipative quantum engineering and information process-ing, and non-equilibrium quantum statistical mechanics. In particular, we showthat a large class of graph product states (including arbitrary thermal graphstates) as well as Gibbs states of commuting Hamiltonians are frustration-freestabilizable relative to natural quasi-locality constraints. Likewise, we provideexplicit examples of non-commuting Gibbs states and non-trivially entangledmixed states that are stabilizable despite the lack of an underlying commutingstructure, albeit scalability to arbitrary system size remains in this case an openquestion.
• Chapter 6 complements the work of Chapter 5 by investigating the discrete-time analog of quantum state stabilization with quasi-local dynamics. Whilecontinuous-time Markovian dynamics cannot exactly stabilize a target state infinite time, discrete-time dynamics can, in principle. We develop necessary andsufficient conditions for establishing if a given target state can be stabilized by afinite sequence of quasi-local dynamical maps. Then we investigate the efficientscheme of robust finite-time stabilization, whereby the target state is stabilizedregardless of the implementation order of the dynamical maps. A main theme inthis chapter is the role that certain “commuting structures” play in facilitatingrobust stabilization.
• Chapter 7 turns to the task of preparing quantum information in a quantumerror correcting code. As in the previous chapter, we explore several exampleswhereby this task can be achieved exactly, in principle, by a finite sequenceof quasi-local dynamical maps. We develop a number of principles which aidthe construction of such finite-time dissipative encoders utilizing the quantum
7
Introduction
stabilizer formalism.
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Chapter 2
Quantum marginals: sharability andjoinability
9
Quantum marginals: sharability and joinability
This chapter presents material that appeared in Physical Review A, 88:032323(2013), in an article titled “Compatible quantum correlations: Extension problems forWerner and isotropic states”, which is joint work with Lorenza Viola.
2.1 IntroductionUnderstanding the nature of quantum correlations in multiparty systems and thedistinguishing features they exhibit relative to classical correlations is a central goalacross quantum information processing (QIP) science [17], with implications rang-ing from condensed-matter and statistical physics to quantum chemistry, and thequantum-to-classical transition. From a foundational perspective, exploring whatdifferent kinds of correlations are, in principle, allowed by probabilistic theories moregeneral than quantum mechanics further helps to identify under which set of physicalconstraints the standard quantum framework may be uniquely recovered [36, 37].
In this context, entanglement provides a distinctively quantum type of correla-tion, that has no analogue in classical statistical mechanics. A striking feature ofentanglement is that it cannot be freely distributed among different parties: if a bi-partite system, say, A(lice) and B(ob), is in a maximally entangled pure state, thenno other system, C(harlie), may be correlated with it. In other words, the entan-glement between A and B is monogamous and cannot be shared [38, 39, 40, 41, 42].This simple tripartite setting motivates two simple questions about bipartite quan-tum states: given a bipartite state, we ask whether it can arise as the reduced stateof A-B and of A-C simultaneously; or, more generally, given two bipartite states, weask if one can arise as the reduced state of A-B while the other arises as the reducedstate of A-C. It should be emphasized that both of these are questions about theexistence of tripartite states with given reduction properties. While formal (and moregeneral) definitions will be provided later in the chapter, these examples serve to in-troduce the notions of sharing (1-2 sharing) and joining (1-2 joining), respectively.In its most general formulation, the joinability problem is also known as the quantummarginal problem (or local consistency problem), which has been heavily investigatedboth from a mathematical-physics [27, 43, 44] and a quantum-chemistry perspective[45, 46] and is known to be QMA-hard [47]. Our choice of terminology, however,facilitates a uniform language for describing the joinability/sharability scenarios. Forinstance, we say that the joinable correlations of A-B and A-C are joined by a joiningstate on A-B-C.
The limited sharability/joinability of entanglement was first quantified in the sem-inal work by Coffman, Kundu, and Wootters, in terms of an exact (CKW) inequalityobeyed by the entanglement across the A-B, A-C and A-(BC) bipartitions, as mea-sured by concurrence [38]. In a similar venue, several subsequent investigations at-tempted to determine how different entanglement measures can be used to diagnose
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2.1 Introduction
failures of joinability, see e.g. [48, 49, 41]. More recently, significant progress hasbeen made in characterizing quantum correlations more general than entanglement[50, 51], in particular as captured by quantum discord [52]. While it is now establishedthat quantum discord does not obey a monogamy inequality [53], different kinds oflimitations exist on the extent to which it can be freely shared and/or communicated[54, 55]. Despite these important advances, a complete picture is far from beingreached. What kind of limitations do strictly mark the quantum-classical correlationboundary? What different quantum features are responsible for enforcing differentaspects of such limitations, and how does this relate to the degree of resourcefulnessthat these correlations can have for QIP?
While the above are some of the broad questions motivating this work, our spe-cific focus here is to make progress on joinability and sharability properties in low-dimensional multipartite settings. In this context, reference [56] has obtained a nec-essary condition for three-party joining in finite dimension in terms of the subsystementropies, and additionally established a sufficient condition in terms of the trace-norm distances between the states in question and known joinable states. For thespecific case of qubit Werner states [57], Werner himself established necessary andsufficient conditions for the 1-2 joining scenario [58]. With regards to sharability,necessary and sufficient conditions have been found for 1-2 sharing of generic bipar-tite qubit states [59], as well as for specific classes of qudit states [60]. To the best ofour knowledge, no conditions that are both necessary and sufficient for the joinabilityof generic states are available as yet. In this chapter, we obtain necessary and suffi-cient conditions for both the three-party joinability and the 1-n sharability problems,in the case that the reduced bipartite states are either Werner or isotropic states ond-dimensional subsystems (qudits).
Though our results are restricted in scope of applicability, they provide key in-sights as to the sources of joinability limitations. Most importantly, we find thatstandard measures of quantum correlations, such as concurrence and quantum dis-cord, do not suffice to determine the limitations in joining quantum correlations.Specifically, we find that the joined states need not be entangled or even discordantin order not to be joinable. Further to that, although separable states may have join-ability limitations, they are, nonetheless, freely (arbitrarily) sharable. By introducinga one-parameter class of probability distributions, we provide a natural classical ana-logue to qudit Werner and isotropic quantum states. This allows us to illustratehow classical joinability restrictions carry over to the quantum case and, more inter-estingly, to demonstrate that the quantum case demands limitations which are notpresent classically. Ultimately, this feature may be traced back to complementarityof observables, which clearly plays no role in the classical case. It is suggestive tonote that the uncertainty principle was also shown to be instrumental in constrainingthe sharability of quantum discord [54]. It is our hope that further pursuits of more
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Quantum marginals: sharability and joinability
general necessary and sufficient conditions may be aided by the methods and findingsherein.
The content is organized as follows. In Sec. 2.2 we present the relevant mathemat-ical framework for defining the joinability and sharability notions and the extensionproblems of interest, along with some preliminary results contrasting the classicaland quantum cases. Sec. 2.3 contains the core results of our analysis. In particular,after reviewing the defining properties of Werner and isotropic states on qudits, inSec. 2.3.1 we motivate the appropriate choice of probability distributions that serveas a classical analogue, and determine the resulting classical joinability limitations inSec. 2.3.2. Necessary and sufficient conditions for three-party joinability of quantumWerner and isotropic states are established in Sec. 2.3.3, and contrasted to the clas-sical scenario. Sec. 2.3.4 shows how the results on isotropic state joinability are infact related to known results on quantum cloning, whereas in Sec. 2.3.5 we establishsimple analytic expressions for the 1-n sharability of both Werner and isotropic states,along with discussing constructive procedures to determinem-n sharability propertiesfor m > 1. In Sec. 2.4, we present additional remarks on joinability and sharabilityscenarios beyond those of Sec. 2.3. In particular, we outline generalizations of ouranalysis to N -party joinability, and show how bounds on the sharability of arbitrarybipartite states follow from the Werner and isotropic results. For ease and clarity ofpresentation, we have omitted the technical proofs of the results in Sec. 2.3 from thisthesis. These proofs can be found in the appendix of [61].
2.2 Joining and sharing classical vs. quantum states
Although our main focus will be to quantitatively characterize simple low-dimensionalsettings, we introduce the relevant concepts with a higher degree of generality, in orderto better highlight the underlying mathematical structure and to ease connectionswith existing related notions in the literature. We are interested in the correlationsamong the subsystems of a N -partite composite system S. In the quantum case, wethus require a Hilbert space with a tensor product structure:
H(N) 'N⊗i=1
H(1)i , dim(H(1)
i ) ≡ di,
where H(1)i represents the individual “single-particle” state spaces and, for our pur-
poses, each di is finite. In the classical scenario, to each subsystem we associate asample space Ωi consisting of di possible outcomes, with the joint sample space being
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2.2 Joining and sharing classical vs. quantum states
given by the Cartesian product:
Ω(N) ' Ω1 × . . .× ΩN .
Probability distributions on Ω(N) are the classical counterpart of quantum densityoperators on H(N).
2.2.1 Joinability
The input to a joinability problem is a set of subsystem states which, in full generality,may be specified relative to a “neighborhood structure” on H(N) (or Ω(N)) [31, 34].That is, let neighborhoods Nj be given as subsets of the set of indexes labelingindividual subsystems, Nk ( ZN . We can then give the following:
Definition 2.2.1. [Quantum Joinability] Given a neighborhood structure N1,N2,. . . ,N` on H(N), a list of density operators
(ρ1, . . . , ρ`) ∈ (D(HN1), . . . ,D(HN`))
is joinable if there exists an N-partite density operator w ∈ D(H(N)), called a joiningstate, that reduces according to the neighborhood structure, that is,
TrNk (w) = ρk, ∀k = 1, . . . , `, (2.1)
where Nk ≡ ZN \ Nk is the tensor complement of Nk.
The analagous definition for classical joinability is obtained by substituting corre-sponding terms, in particular, by replacing the partial trace over Nk with the corre-sponding marginal probability distribution. As remarked, the question of joinabilityhas been extensively investigated in the context of the classical [62] and quantum[27, 63, 64, 56] marginal problem. A joining state is equivalenty referred to as anextension or an element of the pre-image of the list under the reduction map, whilethe members of a list of joinable states are also said to be compatible or consistent.
Clearly, a necessary condition for a list of states to be joinable is that they “agree”on any overlapping reduced states. That is, given any two states from the list whoseneighborhoods are intersecting, the reduced states of the subsystems in the intersec-tion must coincide. From this point of view, any failure of joinability due to a dis-agreement of overlapping reduced states is a trivial case of non-compatible N -partycorrelations. We are interested in cases where joinability fails despite the agreementon overlapping marginals. This consistency requirement will be satisfied by construc-tion for the Werner and isotropic quantum states we shall consider in Sec. 2.3.
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Quantum marginals: sharability and joinability
One important feature of joinability, which has recently been investigated in [65],is the convex structure that both joinable states lists and joining states enjoy. Theset of lists of density operators satisfying a given joinability scenario is convex undercomponent-wise combination; this is because the same convex combination of theirjoining states is a valid joining state for the convex combined list of states. Similarly,the set of joining states for a given list of joined states is convex by the linearity ofthe partial trace.
As mentioned, one of our goals is to shed light on limitations of quantum vs.classical joinability and the extent to which entanglement may play a role in thatrespect. That quantum states are subject to stricter joinability limitations thanclassical probability distributions are, can be immediately appreciated by consideringtwo density operators ρAB = |ΨB〉〈ΨB| = ρAC , where |ΨB〉 is any maximally entangledBell pair on two qubits: no three-qubit joining state wABC exists, despite the reducedstate on A being manifestly consistent. In contrast, as shown in [56, 62], as longas two classical distributions have equal marginal distributions over A, p(A,B) andp(A,C) can always be joined. This is evidenced by the construction of the joiningstate: w(A,B,C) = p(A,B) p(A,C)/p(A). As pointed out in [56], although the abovechoice is not unique, it is the joining state with maximal entropy and represents aneven mixture of all valid joining distributions.
Although any two consistently-overlapped classical probability distributions maybe joined, limitations on joining classical probability distributions do typically arisein more general joining scenarios. This follows from the fact that any classicalprobability assignment must be consistent with some convex combination of purestates. Consider, for example, a pairwise neighboorhood structure, with an associatedlist of states p(A,B), p(B,C), and p(A,C), which have consistent single-subsystemmarginals. Clearly, if each subsystem corresponds to a bit, no convex combinationof pure states gives rise to a probability distribution w(A,B,C) in which each pairis completely anticorrelated; in other words, “bits of three can’t all disagree”. In Sec.2.3.3, we explicitly compare this particular classical joining scenario to analogousquantum scenarios.
While all the classical joining limitations may be expressed by linear inequalities,the quantum joining limitations are significantly more complicated. The limitationsarise from demanding that the joining operator be a valid density operator, namely,trace-one and non-negative (which clearly implies Hermiticity). This fact is demon-strated by the following proposition, which may be readily generalized to any joiningscenario:
Proposition 2.2.2. For any two trace-one Hermitian operators QAB and QAC whichobey the consistency condition TrB (QAB) = TrC (QAC), there exists a trace-one Her-mitian joining operator QABC.
Proof. Consider an orthogonal Hermitian product basis which includes the identity
14
2.2 Joining and sharing classical vs. quantum states
for each subsystem, that is, Ai ⊗ Bj ⊗ Ck, where A0 = B0 = C0 = I. Then wecan construct the space of all valid joining operators QABC as follows. Let dABC bethe dimension of the composite system. The component along A0 ⊗B0 ⊗ C0 is fixedas 1/dABC , satisfying the trace-one requirement. The components along the two-body operators of the form Ai ⊗ Bj ⊗ I are fixed by the required reduction to QAB,and similarly the components along the two-body operators of the form Ai ⊗ I⊗ Ckare determined by QAC . The components along the one-body operators of the formAi ⊗ I⊗ I, I⊗Bi ⊗ I, and I⊗ I⊗ Ci are determined from the reductions of QAB andQAC . This leaves the coefficients of all remaining basis operators unconstrained, sincetheir corresponding basis operators are zero after a partial trace over systems B orC.
Thus, requiring the joining operator to be Hermitian and normalized is not alimiting constraint with respect to joinability: any limitations are due to the non-negativity constraint. Understanding how non-negativity manifests itself is extremelydifficult in general and far beyond our scope here. We can nevertheless give an ex-ample in which the role of non-negativity is clear. Part of the job of non-negativity isto enforce constraints that are also obeyed by classical probability distributions. Forexample, in the case of a two-qubit state ρ, if 〈X ⊗ I〉ρ = 1 and 〈I ⊗ X〉ρ = 1, then〈X ⊗ X〉ρ must equal 1. More generally, consider a set of mutually commuting ob-servables Miki=1 and any basis |m〉 in which all Mi are diagonal. Any valid statemust lead to a list of expectation values (Tr (ρM1) , . . . ,Tr (ρMk)), whose values areelement-wise convex combinations of the vertexes (〈m|M1|m〉, . . . , 〈m|Mk|m〉)|∀m.The interpretation of this constraint is that since commuting observables have simul-taneously definable values, just as classical observables do, probability distributionson them must obey the rules of classical probability distributions. We call on thisfact when we compare the quantum joining limitations to the classical analogue onesin Sec. 2.3.3.
Non-negativity constraints that do not arise from classical limitations on compat-ible observables may be labeled as inherently quantum constraints, the most familiarbeing provided by uncertainty relations for conjugate observables [66, 67]. Althoughcomplementarity constraints are most evident for observables acting on the samesystem, complementarity can also give rise to a trade-off in the information abouta subsystem observable vs. a joint observable. This fact is essentially what allowsBell’s inequality to be violated. For our purposes, the complementarity that comesinto play is that between “overlapping” joint observables (e.g., between ~S1 · ~S2 and~S1 · ~S3 for three qubits). We are thus generally interested in understanding the inter-play between purely classical and quantum joining limitations, and in the correlationtrade-offs that may possibly emerge.
Historically, as already mentioned, a pioneering exploration of the extent to whichquantum correlations can be shared among three parties was carried out in [38],
15
Quantum marginals: sharability and joinability
yielding a characterization of the monogamy of entanglement in terms of the well-known CKW inequality:
C2AB + C2
AC ≤ (C2)minA(BC),
where C denotes the concurrence and the right hand-side is minimized over all pure-state decompositions. Thus, with the entanglement across the bipartition A and (BC)held fixed, an increase in the upper bound of the A-B entanglement can only come atthe cost of a decrease in the upper bound of the A-C entanglement. One may wonderwhether the CKW inequality may help in diagnosing joinability of reduced states. Ifa joining state wABC is not a priori determined (in fact, the existence of such a stateis the entire question of joinability), the CKW inequality may be used to obtain anecessary condition for joinability, namely, if ρAB and ρAC are joinable, then
C2AB + C2
AC ≤ 1. (2.2)
However, there exist pairs of bipartite states – both unentangled (as the followingProposition shows) and non-trivially entangled (as we shall determine in Sec. III.B,see in particular Fig. 2.2a) – that obey the “weak” CKW inequality in Eq. (2.2), yetare not joinable. The key point is that while the limitations that the CKW capturesare to be ascribed to entanglement, entanglement is not required to prevent two statesfrom being joinable. In fact, weaker forms of quantum correlations, as quantified byquantum discord [52], are likewise not required for joinability limitations. Consider,specifically, so-called “classical-quantum” bipartite states, of the form
ρ =∑i
pi|i〉〈i|A ⊗ σiB,∑i
pi = 1,
where |i〉A is some local orthogonal basis on A and σiB is, for each i, an arbitrarystate on B. Such states are known to have zero discord [68]. Yet, the following holds:
Proposition 2.2.3. Classical-quantum correlated states need not be joinable.
on the pairs A-B and A-C, respectively. Both have a completely mixed reduced stateover A and thus it is meaningful to consider their joinability. Let wABC be a joiningstate. Then the outcome of Bob’s X measurement would correctly lead him to predictAlice to be in the state | ↑X〉 or | ↓X〉, while at the same time the outcome of Charlie’sZ measurement would correctly lead him to predict Alice to be in the state | ↑Z〉 or
16
2.2 Joining and sharing classical vs. quantum states
| ↓Z〉. Since this violates the uncertainty principle, wABC cannot be a valid joiningstate.
The existence of separable but not joinable states has been independently reportedin [56]. While formally our example is subsumed under the more general one presentedin Thm. 4.2 therein (strictly satisfying the necessary condition for joinability givenby their Eq. (2.2)), it has the advantage of offering both a transparent physicalinterpretation of the underlying correlation properties, and an intuitive proof of thejoinability failure.
2.2.2 Sharability
As mentioned, the second joinability structure we analyze is motivated by the conceptof sharability. In our context, we can think of sharability as a restricted joiningscenario in which a bipartite state is joined with copies of itself. If H(2) ' H(1)
1 ⊗H(1)2 ,
consider a N -partite space that consists of m “left” copies of H(1)1 and n “right”
copies of H(1)2 , with each neighborhood consisting of one right and one left subsystem,
respectively (hence a total of mn neighborhoods). We then have the following:
Definition 2.2.4. [Quantum Sharability] A bipartite density operator ρ ∈ D(HL⊗HR) is m-n sharable if there exists an N-partite density operator w ∈ D(H⊗mL ⊗H
⊗nR ),
called a sharing state, that reduces left-right-pairwise to ρ, that is,
where the partial trace is taken over the tensor complement of neighborhood ij.
Each m-n sharability scenario may be viewed as a specific joining structure with theadditional constraint that each of the joining states be equal to one another, the listbeing (ρ, ρ, . . . , ρ). In what follows, we shall take arbitrarily sharable to mean ∞-∞sharable, whereas finitely sharable means that ρ is not m-n sharable for some m, n.Also, each property “m-n sharable” (sometimes also referred to as a “m-n extendible”)is taken to define a sharability criterion, which a state may or may not satisfy.
It is worth noting the relationship between sharability and N-representability. TheN -representability problem asks if, for a given (symmetric) p-partite density operatorρ on (H(1)
1 )⊗p, there exists an N -partite pre-image state for which ρ is the p-particlereduced state. N -representability has been extensively studied for indistinguishablebosonic and fermionic subsystems [69, 45, 46] and is a very important problem inquantum chemistry [70]. We can view N -representability as a variant on the shara-bility problem, whereby the distinction between the left and right subsystems is lifted,and m+n = N . Given the p-partite state ρ as the shared state, we ask if there existsa sharing N -partite state which shares ρ among all possible p-partite subsystems. In
17
Quantum marginals: sharability and joinability
the setting of indistinguishable particles, the associated symmetry further constrainsthe space of the valid N -partite sharing states.
Just as with 1-2 joinability, any classical probability distribution is arbitrarilysharable [37]. Likewise, similar to the joinability case, convexity properties playan important role towards characterizing sharability. If dim(H(1)
1 ) = d1 ≡ dL anddim(H(1)
2 ) = d2 ≡ dR, then it follows from the convexity of the set of joinable stateslists that m-n sharable states form a convex set, for fixed subsystem dimensions dLand dR. This implies that if ρ satisfies a particular sharability criterion, then anymixture of ρ with the completely mixed state also satisfies that criterion, since thecompletely mixed state is arbitrarily (∞-∞) sharable.
Besides mixing with the identity, the degree of sharability may be unchanged undermore general transformations on the input state. Consider, specifically, completely-positive trace-preserving bipartite maps M(ρ) that can be written as a mixture oflocal unitary operations, that is,
M(ρ) =∑i
λiUi1 ⊗ V i
2ρUi1
† ⊗ V i2
†,∑i
λi = 1, (2.4)
where U i1 and V i
2 are arbitrary unitary transformations on HL and HR, respectively.These (unital) maps form a proper subset of general Local Operations and ClassicalCommunication (LOCC) [17]. We establish the following:
Theorem 2.2.5. If ρ is m-n sharable, then M(ρ) is m-n sharable for any map Mthat is a convex mixture of unitaries.
Proof. LetM(ρ) be expressed as in Eq. (2.4). By virtue of the convexity of the setof m-n sharable states (for fixed subsystem dimensions), it suffices to show that eachterm, UV ρU †V †, inM(ρ) is m-n sharable. Let w be a sharing state for ρ, and define
w′=(U1 . . .UmVm+1 . . .Vm+n
)w(U †1 . . .U
†mV
†m+1 . . .V
†m+n
).
Then, for any left-right pair of subsystems i and j, it follows that
Tri,j (w′) = UiVjTri,j (w)U †i V†j = U ⊗ V ρU † ⊗ V † = ρUV .
Hence, w′ is an m-n-sharing state for ρUV , as desired.
This result suggests a connection between the degree of sharability and the en-tanglement of a given state. In both cases, there exist classes of states for whichthese properties cannot be “further degraded” by locally acting maps (or any map forthat matter). Obviously, LOCC cannot decrease the entanglement of states with noentanglement, and convex unitary mixtures as above cannot increase the sharability
18
2.2 Joining and sharing classical vs. quantum states
of states with ∞-∞ sharability (because they are already as sharable as possible).These two classes of states can in fact be shown to coincide as a consequence of thefact that arbitrary sharability is equivalent to (bipartite) separability. This result hasbeen appreciated in the literature [40, 36, 71, 37] and is credited to both [72] and [73].We reproduce it here in view of its relevance to our work:
Theorem 2.2.6. A bipartite quantum state ρ on HL ⊗HR is unentangled (or sepa-rable) if and only if it is arbitrarily sharable.
Proof. (⇐) Let ρ be separable. Then for some set of density operators ρLi , ρRi , itcan be written as ρ =
∑i λiρ
Li ⊗ ρRi , with
∑i λi = 1. Let n and m be arbitrary, and
let the N -partite state w, be defined as follows:
w =∑i
λi(ρLi )⊗m ⊗ (ρRi )⊗n,
with N = m + n. By construction, the state of each L-R pair is ρ, since it followsstraighforwardly that Eq. (2.3) is obeyed for each i, j. Thus, w is a valid sharingstate.
(⇒) Since ρ is arbitrarily sharable, there exists a sharing state w for arbitraryvalues of m, n. In particular, we need only make use of a sharing state w for m = 1and arbitrarily large n, whence we let n → ∞. Given w, let us construct anothersharing state w, which is invariant under permutations of the right subsystems, thatis, let
w =1
|Sn|∑π∈Sn
V †πwVπ,
where Sn ≡ π is the permutation group of n objects, acting on H⊗nR via the naturaln-fold representation, Vπ(
∏i |ψi〉) = ⊗i|ψπ(i)〉, i = 1, . . . , n. It then follows that w
shares ρ:
TrL,R (w) =1
|Sn|∑π∈Sn
TrL,R(V †πwVπ
)=
1
|Sn|∑π∈Sn
TrL,π(Ri)(w) =
1
|Sn|∑π∈Sn
ρ = ρ.
Having established the existence of a symmetric sharing state w ∈ D(HL ⊗ H⊗∞R ),Fannes’ Theorem (see section 2 of [73]) implies the existence of a unique representationof w as a sum of product states, w =
∑i λiρ
iL ⊗ ρiR ⊗ ρiR ⊗ . . . . Reducing w to any
L-R pair leaves a separable state. Thus, if ρ is 1-n sharable it must be separable.
As we alluded to before, a Corollary of this result is that in fact 1-∞ sharabilityimplies ∞-∞ sharability. In closing this section, we also briefly mention the concept
19
Quantum marginals: sharability and joinability
of exchangeability [74, 75]. A density operator ρ on (H(1)1 )⊗p is said to be exchangeable
if it is symmetric under permutation of its p subsystems and if there exists a symmetricstate w on (H(1)
1 )⊗(p+q) such that the reduced states of any subset of p subsystems isρ for all q ∈ N. Similar to sharability, exchangeability implies separability. However,the converse only holds in general for sharability: clearly, there exist states which areseparable but not exchangeable, because of the extra symmetry requirement. Thus,the notion of sharability is more directly related to entanglement than exchangeabilityis.
2.3 Joining and sharing Werner and isotropic states
Even for the simplest case of two bipartite states with an overlapping marginal, ageneral characterization of joinability is extremely non-trivial. As remarked, no con-ditions yet exist which are both necessary and sufficient for two arbitrary density op-erators to be joinable; although, conditions that are separately necessary or sufficienthave been recently derived [56]. In this Section, we present a complete characteriza-tion of the three-party joining scenario and the 1-n sharability problem for Wernerand isotropic states on arbitrary subsystem dimension d. We begin by introducingthe relevant families of quantum and classical states to be considered.
2.3.1 Werner and isotropic qudit states, and their classicalanalogues
The usefulness of bipartite Werner and isotropic states is derived from their simpleanalytic properties and range of mixed state entanglement. For a given subsystemdimension d, Werner states are defined as the one-parameter family that is invariantunder collective unitary transformations [57] (see also [75]), that is, transformationsof the form U ⊗ U , for arbitrary U ∈ U(d). The parameterization which we employis given by
ρ(Ψ−) =d
d2 − 1
[(d−Ψ−)
Id2
+(
Ψ− − 1
d
)Vd
],
where V is the swap operator, defined by its action on any product ket, V |ψφ〉 ≡ |φψ〉.This parameterization is chosen because Ψ− is a Werner state’s expectation value withrespect to V , Ψ− = Tr[V ρ(Ψ−)]. Non-negativity is ensured by −1 ≤ Ψ− ≤ 1, andthe completely mixed state corresponds to Ψ− = 1/d. Furthermore, the concurrenceof Werner states is simply given by [76]
C(ρ(Ψ−)) = −Tr[V ρ(Ψ−)
]= −Ψ−, Ψ− ≤ 0. (2.5)
20
2.3 Joining and sharing Werner and isotropic states
For Ψ− > 0, the concurrence is defined to be zero, indicating separability. Wernerstates have been experimentally characterized for photonic qubits, see e.g. [77]. In-terestingly, they can be dissipatively prepared as the steady state of coherently drivenatoms subject to collective spontaneous decay [78].
Isotropic states are defined, similarly, as the one-parameter family that is invariantunder transformations of the form U∗ ⊗ U [79]. We parameterize these states as
ρ(Φ+) =d
d2 − 1
[(d− Φ+)
Id2
+(
Φ+ − 1
d
)|Φ+〉〈Φ+|
],
where |Φ+〉 =√
1/d∑
i |ii〉. The value of the parameter is given by the expectationvalue with respect to the partially transposed swap operator, Φ+ = Tr
[V TA
(AB)ρ(Φ+)],
and is related to the so-called “singlet fraction” [80] by Φ+ = dF . Non-negativity isnow ensured by 0 ≤ Φ+ ≤ d, whereas the concurrence is given by [81],
C(ρ(Φ+)) =
√2
d(d− 1)(Φ+ − 1), Φ+ ≥ 1, (2.6)
and is defined to be zero for Φ+ ≤ 1.Before introducing probability distributions that will serve as the analogue classi-
cal states, we present an alternative way to think of Werner states, which will proveuseful later. First, the highest purity, attained for the Ψ− = −1 state, is 2/[d(d− 1)],with the absolute maximum of 1 corresponding to the pure singlet state for d = 2.Second, collective projective measurements on a most-entangled Werner state returnonly disagreeing outcomes (e.g., corresponding to |1〉 ⊗ |3〉, but not |1〉 ⊗ |1〉). Thefollowing construction of bipartite Werner states demonstrates the origin of both ofthese essential features. For generic d, the analogue to the singlet state is the followingd-partite fully anti-symmetric state:
|ψ−d 〉 =1√d!
∑π∈Sd
sign(π)Vπ|1〉|2〉 . . . |d〉, (2.7)
where, as before, Sd ≡ π denotes the permutation group and |`〉 is an orthonor-mal basis on H(1) ' Cd. The above state has the property of being “completelydisagreeing”, in the sense that a collective measurement returns outcomes that dif-fer on each qudit with certainty. The most-entangled bipartite qudit Werner stateis nothing but the two-party reduced state of |ψ−d 〉. Thus, we can think of generalbipartite qudit Werner states as mixtures of the completely mixed state with thetwo-party-reduction of |ψ−d 〉. The inverse of 2/[d(d− 1)] (the purity) is precisely thenumber of ways two “dits” can disagree. Understanding bipartite Werner states to
21
Quantum marginals: sharability and joinability
arise from reduced states of |ψ−d 〉 will inform our construction of the classical analoguestates, and also help us understand some of the results of Sec. 2.3.3 and 2.3.5.
For Werner states, increased entanglement corresponds to increased “disagree-ment” for collective measurement outcomes. For isotropic states, increased entangle-ment corresponds to increased “agreement” of collective measurements, but only withrespect to the computational basis |i〉 relative to which such states are defined. Itis this expression of agreement vs. disagreement of outcomes which carries over to theclassical analogue states, which we are now ready to introduce. The relevant probabil-ity distributions are defined on the outcome space Ωd×Ωd = 1, . . . , d× 1, . . . , d.To resemble Werner and isotropic quantum states, these probability distributionsshould have completely mixed marginal distributions and range from maximal dis-agreement to maximal agreement. This is achieved by an interpolation between aneven mixture of “agreeing pure states”, namely, (1, 1), (2, 2), . . . , (d, d), and an evenmixture of all possible “disagreeing pure states”, namely, (1, 2), . . . , (1, d), (2, 1), . . . ,(d, d− 1). That is:
p(A = i, B = j)α =α
dδi,j +
1− αd(d− 1)
(1− δi,j), (2.8)
where α is the probability that the two outcomes agree.
To make the analogy complete, it is desirable to relate α to both Ψ− and Φ+. Wedefine α in the quantum cases to be the probability of obtaining |k〉 on system A,conditional to outcome |k〉 on system B for the projective measurement |ij〉〈ij|.For Werner states, this probability is related to Ψ− by
p(|k〉A | |k〉B)W =Ψ− + 1
d+ 1≡ αW , (2.9)
and, similarly for isotropic states, we have
p(|k〉A | |k〉B)I =Φ+ + 1
d+ 1≡ αI . (2.10)
We may thus re-parameterize both the Werner and isotropic states in terms of theirrespective above-defined “probabilities of agreement”, namely:
ρ(αW ) =d
d− 1
[(1− αW )
Id2
+(αW −
1
d
)Vd
], (2.11)
ρ(αI) =d
d− 1
[(1− αI)
Id2
+(αI −
1
d
)|Φ+〉〈Φ+|
], (2.12)
22
2.3 Joining and sharing Werner and isotropic states
subject to the conditions
0 ≤ αW ≤2
d+ 1,
1
d+ 1≤ αI ≤ 1.
For Werner states, αW can rightly be considered a probability of agreement be-cause it is independent of the choice of local basis vectors in the projective measure-ment U ⊗ U |ij〉〈ij|U † ⊗ U †. For isotropic states, αI does not have as direct aninterpretation. We may nevertheless interpret α as a probability of basis-independentagreement if we pair local basis vectors on A with their complex conjugates on B. Inother words, αI can be thought of as the probability of agreement for local projectivemeasurements of the form U∗ ⊗ U |ij〉〈ij|U∗† ⊗ U † 1.
2.3.2 Classical joinability limitations
In order to determine the joinability limitations in the classical case, we begin by not-ing that any (finite-dimensional) classical probability distribution is a unique convexcombination of the pure states of the system. In our case, there are five extremalthree-party states, for which the two-party marginals are classical analogue states, asdefined in Eq. (2.8). These are
p(A,B,C agree) =1
d
∑i
(i, i, i),
p(A,B agree) =1
d(d− 1)
∑i 6=j
(i, i, j),
p(A,C agree) =1
d(d− 1)
∑i 6=j
(i, j, i),
p(B,C agree) =1
d(d− 1)
∑i 6=j
(j, i, i),
p(all disagree) =1
d(d− 1)(d− 2)
∑i 6=j 6=k
(i, j, k),
1In principle, one might question the validity of our analogy on account of the quantum statesbeing more pure than the mixed classical states in Eq. (2.8), e.g., the classical analogue with α = 1may seem closer to the separable quantum state (|11〉〈11|+ . . .+ |dd〉〈dd|)/d. We note, however, thatany POVM gives a mapping from a quantum state to a probability distribution; we have chosenhere two families of quantum states for which the resulting probability distribution is minimallydependent on the choice of measurement, as argued. Furthermore, we aim to understand whichfeatures of joinability limitations are of a quantum origin and which are simply due to classicallimitations. In contrast, the family of “classical-like” quantum states p
∑i |ii〉〈ii|/d + (1 − p)I/d2
would be un-interesting, as manifestly only constrained by classical joining limitations.
23
Quantum marginals: sharability and joinability
where (i, j, k) stands for the pure probability distribution p(A,B,C) = δA,iδB,jδC,k.The first four of these states are valid for all d ≥ 2 and each corresponds to a vertexof a tetrahedron, as depicted in Fig. 2.1(left). The fifth state is only valid for d ≥ 3and corresponds to the point (αAB, αAC , αBC) = (0, 0, 0) in Fig. 2.1(right). Anyvalid three-party state for which the two-party marginals are classical analogue statesmust be a convex combination of the above states. Therefore, the joinable-unjoinableboundary is delimited by the boundary of their convex hull. For the d = 2 case, theinequalities describing these boundaries are explicitly given by the following:
p(A,B,C agree) ≥ 0⇒ αAB + αAC + αBC ≥ 1,
p(C disagrees) ≥ 0⇒ −αAB + αAC + αBC ≤ 1,
p(B disagrees) ≥ 0⇒ αAB − αAC + αBC ≤ 1,
p(A disagrees) ≥ 0⇒ αAB + αAC − αBC ≤ 1,
where each inequality arises from requiring that the corresponding extremal state hasa non-negative likelihood. In the d ≥ 3 case, the inequality p(A,B,C agree) ≥ 0 isreplaced by αAB, αAC , αBC ≥ 0.
2.3.3 Joinability of Werner and isotropic qudit states
We now present our results on the three-party joinability of Werner and isotropicstates and then compare them to the classical limitations just found in the previoussection. While, as mentioned, the proofs have been relegated to the published versionof the paper [61], the basic idea is to exploit the high degree of symmetry that theseclasses of states enjoy.
Consider Werner states first. Our starting point is to observe that if a tripartitestate wABC joins two reduced Werner states ρAB and ρAC , then the “twirled state”wABC , given by
wABC =
∫(U ⊗ U ⊗ U)wABC (U ⊗ U ⊗ U)†dµ(U), (2.13)
is also a valid joining state. In Eq. (2.13), µ denotes the invariant Haar measureon U(d), and the twirling super-operator effects a projection into the subspace ofoperators with collective unitary invariance [82]. By invoking the Schur-Weyl duality[83], the guaranteed existence of joining states with these symmetries allows one tonarrow the search for valid joining states to the Hermitian subspace spanned byrepresentations of subsystem permutations, that is, density operators of the form
w =∑π∈S3
µπVπ, w ∈ WW , (2.14)
24
2.3 Joining and sharing Werner and isotropic states
(a) Werner joinability limitations (d=2) (b) Werner joinability limitations (d=5)
Figure 2.1: Three-party quantum and classical joinability limitations for Werner andisotropic states, and their classical analogue, as parameterized by Eqs. (2.11), (2.12),(2.8), respectively. a) Qubit case, d = 2. The Werner state boundary is the surface ofthe darker cone with its vertex at (2/3, 2/3, 2/3), whereas the isotropic state boundaryis the surface of the lighter cone with its vertex at (1/3, 1/3, 2/3). The classicalboundary is the surface of the tetrahedron. b) Higher-dimensional case, d = 5. TheWerner state boundary is the surface of the bi-cone with vertices at (0, 0, 0) and(1/3, 1/3, 1/3), whereas the isotropic state boundary is the flattened cone with itsvertex at (1/6, 1/6, 1/3). The classical boundary is the surface of the two joinedtetrahedra. In both panels the grey line resting on top of the cones indicates thecolinearity of the cone surfaces along this line segment.
25
Quantum marginals: sharability and joinability
where Hermiticity demands that µ∗π = µπ−1 . Given wABC which joins Werner states,each subsystem pair is characterized by the expectation value with the respectiveswap operator, Ψ−ij = Tr[wABC(Vij ⊗ Iij)], where i, j ∈ A,B,C with i 6= j. Hence,the task is to determine for which (Ψ−AB,Ψ
−BC ,Ψ
−AC) there exists a density operator
wABC consistent with the above expectations. Our main result is the following:
Theorem 2.3.1. Three Werner qudit states with parameters Ψ−AB,Ψ−BC ,Ψ
−AC are join-
able if and only if (Ψ−AB,Ψ−BC ,Ψ
−AC) lies within the bi-cone described by
1±Ψ− ≥ 2
3
∣∣Ψ−BC + ωΨ−AC + ω2Ψ−AB∣∣ , (2.15)
for d ≥ 3, or within the cone described by
1−Ψ− ≥ 2
3
∣∣Ψ−BC + ωΨ−AC + ω2Ψ−AB∣∣ , Ψ− ≥ 0, (2.16)
for d = 2, where
Ψ− =1
3(Ψ−AB + Ψ−BC + Ψ−AC), ω = ei
2π3 . (2.17)
Similarly, if a tripartite state wABC joins isotropic states ρAB and ρAC , then the“isotropic-twirled state” wABC , given by
wABC =
∫(U∗ ⊗ U ⊗ U)wABC (U∗ ⊗ U ⊗ U)†dµ(U), (2.18)
is also a valid joining state. A clarification is, however, in order at this point: althoughwe have been referring to the isotropic joinability scenario of interest as three-partyisotropic state joining, this is somewhat of a misnomer because we effectively considerthe pair B-C to be in a Werner state, as evident from Eq. (2.18). Compared to Eq.(2.14), the relevant search space is now partially transposed relative to subsystem A,that is, consisting of density operators of the form
w =∑π∈S3
µπVTAπ , w ∈ Wiso. (2.19)
Our main result for three-party joinability of isotropic states is then contained in thefollowing:
Theorem 2.3.2. Two isotropic qudit states ρAB and ρAC and qudit Werner stateρBC with parameters Φ+
AB,Φ+AC ,Ψ
−BC are joinable if and only if (Φ+
AB,Φ+AC ,Ψ
−BC) lies
within the cone described by
Φ+AB + Φ+
AC −Ψ−BC ≤ d , (2.20)
26
2.3 Joining and sharing Werner and isotropic states
1 + Φ+AB + Φ+
AC −Ψ−BC ≥ (2.21)∣∣∣∣d(Ψ−BC − 1) +
√2d
d− 1(eiθΦ+
AB + e−iθΦ+AC)
∣∣∣∣,e±iθ = ±i
√(d+ 1)/(2d) +
√(d− 1)/(2d),
or, for d ≥ 3, within the convex hull of the above cone and the point (0, 0,−1).
The results of Theorems 2.3.1 and 2.3.2 as well as of Sec. 2.3.2 are pictorially sum-marized in Fig. 2.1.
We now compare these quantum joinability limitations to the joinability limita-tions in place for classical analogue states. As described in Sec. 2.3.2, the non-negativity of p(A,B,C agree) and p(A disagrees) is enforced by the two inequalitiesαAB + αAC + αBC ≥ 1 and −αAB − αAC + αBC ≥ 1, respectively. We expect thesame requirement to be enforced by the analogue quantum-measurement statistics.For d = 2, the bases of the Werner and isotropic joinability-limitation cones aredetermined by Ψ−AB + Ψ−AC + Ψ−BC ≥ 0 and Φ+
AB + Φ+AC − Ψ−BC ≤ 2, respectively.
Writing down each of these parameters in terms of the appropriate probability ofagreement α, as defined in Eqs. (2.9) and (2.10), we obtain αAB + αAC + αBC ≥ 1and −αAB − αAC + αBC ≥ 1. Hence, for qubits, part of the quantum joining limita-tions are indeed derived from the classical joining limitations. This is also illustratedin Fig. 2.1(left). Of course, one would not expect the quantum scenario to exhibitviolations of the classical joinability restrictions; still, it is interesting that stateswhich exhibit manifestly non-classical correlations may nonetheless saturate boundsobtained from purely classical joining limitations.
For d ≥ 3, the only classical boundary which plays a role is the one which boundsthe base of the isotropic joinability-limitation cone: Φ+
AB +Φ+AC−Ψ−BC ≤ d. Again, in
terms of the agreement parameters, this is (just as for qubits) −αAB−αAC+αBC ≥ 1.In the Werner case, the quantum joinability boundary is not clearly delineated by theclassical joining limitations. We can nevertheless make the following observation.By the non-negativity of Werner states, the three-party joinability region in Fig.2.1(right) is required to lie within a cube of side-length 2/(d+ 1) with one corner at(0, 0, 0). Consider the set of cubes obtained by rotating from this initial cube aboutan axis through (0, 0, 0) and (2/(d+ 1), 2/(d+ 1), 2/(d+ 1). It is a curious fact thatthe exact quantum Werner joinability region (the bi-cone) is precisely the intersectionof all such cubes.
Another interesting feature is that there exist trios of unentangled Werner stateswhich are not joinable. For example, the point (Ψ−AB,Ψ
−AC ,Ψ
−BC) = (1, 1, 0) cor-
responds to three separable Werner states that are not joinable. This point is ofparticular interest because its classical analogue is joinable. Translating (1, 1, 0) intothe agreement-probability coordinates, (αAB, αAC , αBC) = (2/3, 2/3, 1/3), we see that
27
Quantum marginals: sharability and joinability
this point is actually on the classical joining limitation border. Thus, these three sepa-rable, correlated states are not joinable for purely quantum mechanical reasons. Notethat the point (αAB, αAC , αBC) = (2/3, 2/3, 1/3) does correspond to a joinable trio ofpairs in the isotropic three-party joining scenario: this point lies at the center of theface of the isotropic joinability cone, as seen in Fig. 2.1(left). The same fact holds for(2/3, 1/3, 2/3) or (1/3, 2/3, 2/3) when the Werner state pair in the isotropic joiningscenario describes A-C or A-B, respectively; in both cases, we would have obtainedyet another cone in Fig. 2.1 that sits on a face of the classical tetrahedron boundary.
Having determined the joinable trios of both Werner and isotropic states, we arenow in a position to also answer the question of what pairs A-B and A-C of states arejoinable with one another. In the Werner state case, this is obtained by projectingthe Werner joinability bicone down to the Ψ−AB-Ψ
−AC plane, resulting in the following:
Corollary 2.3.3. Two pairs of qudit Werner states with parameters Ψ−AB and Ψ−ACare joinable if and only if Ψ−AB,Ψ
−AC ≥ −1
2, or if the parameters satisfy
(Ψ−AB + Ψ−AC)2 +1
3(Ψ−AB −Ψ−AC)2 ≤ 1, (2.22)
or additionally, in the case d ≥ 3, if Ψ−AB,Ψ−AC ≤ 1
2.
For isotropic states, we may similarly project the cone of Eq. (2.21) onto theΦ+AB-Φ
+AC plane to obtain the 1-2 joining boundary. This yields the following:
Corollary 2.3.4. Two pairs of qudit isotropic states with parameters Φ+AB and Φ+
AC
are joinable if and only if they lie within the convex hull of the ellipse
(Φ+AB/d+ Φ+
AC/d− 1)2
(1/d2)+
(Φ+AB/d− Φ+
AC/d)2
(d2 − 1)/d2= 1, (2.23)
and the point (Φ+AB,Φ
+AC) = (0, 0).
Lastly, by a similar projection of the isotropic cone given by Eqs. (2.20)-(2.21),we may explicitly characterize the Werner-isotropic hybrid 1-2 joining boundary:
Corollary 2.3.5. An isotropic state with parameter Φ+AB and a Werner state with
parameter Ψ−BC are joinable if and only if they lie within the convex hull of the ellipse
(Φ+AB/d+ Ψ−BC/d− 1)2
(1/d2)+
(Φ+AB/d−Ψ+
BC/d)2
(d2 − 1)/d2= 1, (2.24)
and the point (Φ+AB,Ψ
−BC) = (0, 1), and, for d ≥ 3, within the additional convex hull
introduced by the point (Φ+AB,Ψ
−BC) = (0, 1).
28
2.3 Joining and sharing Werner and isotropic states
(a) Werner state joinability (b) Isotropic state joinability
Figure 2.2: Two-party joinability limitations for Werner and isotropic qudit states.(a) Werner states. The shaded region corresponds to joinable Werner pairs, withthe lighter region being valid only for d ≥ 3. The rounded boundary is the ellipsedetermined by Eq. (2.22). This explicitly shows the existence of pairs of entangledWerner states that are within the circular boundary determined by the weak CKWinequality, Eq. (2.2), yet are not joinable. (b) Isotropic states. The three regionscorrespond to the joinable pairs of isotropic states for d = 2, d = 3 and d = 1000.This shows how, in the limit of large d, the trade-off in isotropic state parametersbecomes linear, consistent with known results on d-dimensional quantum cloning [84].
29
Quantum marginals: sharability and joinability
The above results give the exact quantum-mechanical rules for the two-pair join-ability of Werner and isotropic states, as pictorially summarized in see Figs. 2.2a and2.2b. A number of interesting features are worth noticing. First, by restricting to theline where Ψ−AB = Ψ−AC , we can conclude that qubit Werner states are 1-2 sharable ifand only if Ψ− ≥ −1/2, whereas for d ≥ 3, all qudit Werner states are 1-2 sharable.As we shall see, this agrees with the more general analysis of Sec. 2.3.5.
Second, some insight into the role of entanglement in limiting joinability may begained. In the first quadrant of Fig. 2.2a, where neither pair is entangled, it is nosurprise that no joinability restrictions apply. Likewise, it is not surprising to see that,in the third quadrant where both pairs are entangled, there is a trade-off between theamount of entanglement allowed between one pair and that of the other. But, inthe second and fourth quadrants we observe a more interesting behavior. Namely,these quadrants show that there is also a trade-off between the amount of classicalcorrelation in one pair and the amount of entanglement in the other pair. In fact, thesmoothness of the boundary curve as it crosses from one of the pairs being entangledto unentangled suggests that, at least in this case, entanglement is not the correctfigure of merit in diagnosing joinability limitations.
2.3.4 Isotropic joinability results from quantum cloning
Interestingly, the above results for 1-2 joinability of isotropic states can also be ob-tained by drawing upon existing results for asymmetric quantum cloning, see e.g.[84, 85] for 1-2 and 1-3 asymmetric cloning and [86, 87, 88] for 1-n asymmetric cloning.One approach to obtaining the optimal asymmetric cloning machine is to exploit theChoi isomorphism [89] to translate the construction of the optimal cloning map tothe construction of an optimal operator (or a “telemapping state”). This connection ismade fairly clear in [86, 90]; in particular, “singlet monogamy” refers to the trade-offin fidelities of the optimal 1-n asymmetric cloning machine or, equivalently, to thetrade-off in singlet fractions for a (1 + n) qudit state. We describe how the approachto solving the optimal 1-n asymmetric cloning problem may be rephrased to solve the1-n joinability problem for isotropic states.
The state |Ψ〉 described in Eq. (4) of [86] is a 1-n joining state for n isotropicstates characterized by singlet fractions F0,j (related to the isotropic state parameterby F0,j = Φ+
0,j/d, as noted). The bounds on the singlet fractions are determinedby the normalization condition of |Ψ〉, together with the requirement that |Ψ〉 bean eigenstate of a certain operator R defined in Eq. (3) of [86]. That |Ψ〉 is anisotropic joining state is readily seen from its construction, and that it may optimizethe singlet fractions (hence delineate the boundary in the F0,j space) is proven in[90]. Our contribution here is the observation that this result provides the solutionto the 1-n joinability of isotropic states. The equivalence is established by the fact
30
2.3 Joining and sharing Werner and isotropic states
that optimality is preserved in either direction by the Choi isomorphism.Quantitatively, the boundary for 1-n optimal asymmetric cloning, is given by Eq.
(6) in [86] in terms of singlet fractions. Specializing to the 1-2 joining case andrewriting in terms of Φ+, we have
Φ+AB + Φ+
AC ≤ (d− 1) +1
n+ d− 1
(√Φ+AB +
√Φ+AC
)2
.
As one may verify, this is equivalent to the result of Corollary 2.3.4. In light ofthis connection, the fact that, as d increases, the isotropic-joinability cone of Fig.2.1(right) becomes flattened down to the αAB-αAC plane is directly related to thelinear trade-off in the isotropic state parameters for the semi-classical limit d→∞,as discussed in [84]. Within our three-party joining picture, we can give a partialexplanation of this fact: namely, it is a consequence of the classical joining boundaryin tandem with the upper limit on the agreement parameter αBC for the Werner stateon B-C: αBC ≤ 2/(d + 1). In the limit of d → ∞, these two boundaries conspireto limit the (A-B)-(A-C) isotropic state joining boundary to a triangle, as explicitlyseen in Fig. 2.2b(right).
For the general 1-n isotropic joining scenario, the quantum-cloning results addi-tionally imply the following:
Theorem 2.3.6. A list of n isotropic states characterized by parameters Φ+0,1, . . . ,Φ
+0,j
is 1-n joinable if and only if the (positive-valued) parameters satisfy
n∑j=1
Φ+0,j ≤ (d− 1) +
1
n+ d− 1
( n∑j=1
√Φ+
0,j
)2
. (2.25)
Interestingly, similar to our discussion surrounding Eq. (2.2), the authors of [86]argue how the “singlet monogamy” bound can lead to stricter predictions (e.g., onground-state energies in many-body spin systems) than the standard monogamy ofentanglement bounds based on CKW inequalities [38, 41].
2.3.5 Sharability of Werner and isotropic qudit states
We next turn to sharability of Werner and isotropic states in d dimension, beginningfrom the important case of 1-n sharing. For Werner states, a proof based on arepresentation-theoretic approach is given in [61]. Although we expect a similar proofto exist for isotropic states, we obtain the desired 1-n sharability result by building onthe relationship with quantum cloning problems highlighted above. We then outlinea constructive procedure for determining the more general m-n sharability of Wernerstates.
31
Quantum marginals: sharability and joinability
Our main results are contained in the following:Theorem 2.3.7. A qudit Werner state with parameter Ψ− is 1-n sharable if and onlyif
Ψ− ≥ −d− 1
n. (2.26)
Theorem 2.3.8. A qudit isotropic state with parameter Φ+ is 1-n sharable if andonly if
Φ+ ≤ 1 +d− 1
n. (2.27)
Proof. Specializing Eq. (2.25) to the case of equal parameters for all n isotropicstates, the above result immediately follows. As stated in [86], this is consistent withthe well known result for optimal 1-n symmetric cloning.
We depict the qubit case (d = 2) of the above result in Fig. 2.3. In the case ofWerner state sharing, Eq. (2.26) implies that a finite parameter range exists wherethe corresponding Werner states are not sharable. In contrast, for d ≥ 3, everyWerner state is at least 1-2 sharable. This simply reflects the fact that |ψ−d 〉 (recallEq. (2.7)) provides a 1-(d−1) sharing state for a most-entangled qudit Werner state.With isotropic state sharing, for all d there is, again, a finite range of isotropic stateswhich are not sharable.
The simplicity of the results in Eqs. (2.26)-(2.27) is intriguing and begs for intu-itive interpretations. Consider a central qudit surrounded by n outer qudits. If thecentral qudit is in the same Werner or isotropic state with each outer qudit, thenTheorems 2.3.7 and 2.3.8 can be reinterpreted as providing a bound on the sums ofconcurrences. For Werner states, we have that the sum of all the central-to-outerconcurrences cannot exceed the number of modes by which the systems may disagree(i.e., d−1). In the isotropic state case, the sum of the n pairwise concurrences cannotexceed the maximal concurrence value given by Cmax,d =
√2(d− 1)/d. These rules
do not hold in more general joining scenarios, as we already know from Sec. 2.3.3.There, we found that the trade-off between A-B concurrence and A-C concurrence isnot a linear one, as such a simple “sum rule” would predict; instead, it traces out anellipse (recall Fig. 2.2a).
Starting from the proof of Thm. 2.3.7 found in [61] in conjunction with similarrepresentation-theoretic tools, it is possible to devise a constructive algorithm fordetermining the m-n sharability of Werner states. The basic observation is to re-alize that the most-entangled m-n sharable Werner state corresponds to the largesteigenvalue of a certain Hamiltonian operator Hm,n, which is in turn expressible interms of Casimir operators. Calculation of these eigenvalues may be obtained usingYoung diagrams. Although we lack a general closed-form expression for max(Hm,n),the required calculation can nevertheless be performed numerically. Representativeresults for n-m sharability of low-dimensional Werner states are shown in Table 2.1.
32
2.4 Further remarks
Figure 2.3: Pictorial summary of sharability properties of qubit Werner and isotropicstates, according to Eqs. (2.26) and (2.27). The arrow-headed lines depict the pa-rameter range for which states satisfy each of the sharability properties displayedto the right and left, respectively. The vertical ticks between end points of theseranges indicate the points at which subsequent 1-n sharability properties begin to besatisfied.
2.4 Further remarks
2.4.1 Joinability beyond the three-party scenario
In Sec. 2.3, we focused on considering joinability of three bipartite (Werner orisotropic) states in a “triangular fashion”, namely, relatively to the simplest over-lapping neighborhood structure N1 = A,B, N2 = A,C on H(3). In a moregeneral N -partite scenario, other neighborhood structures and associated joinabilityproblems may naturally emerge. For instance, we may want to answer the follow-ing question: Which sets of N(N − 1)/2 Werner-state (or isotropic-state) pairs arejoinable? The approach to solving this more general problem parallels the specificthree-party case we discussed.
If each pair is in a Werner state, then if a joining state exists, there must exista joining state with collective invariant symmetry (that is, invariant under arbitrarycollective unitaries U⊗N). Thus, we need only look in the set of states respecting thissymmetry. Any such operator may be decomposed into a sum of operators, whicheach have support on just a single irreducible subspace. This is useful because posi-
33
Quantum marginals: sharability and joinability
Table 2.1: Exact results for n-m sharability of Werner states for different subsystemdimension, with m and n increasing from left to right and from top to bottom ineach table, respectively. For each sharability setting, the value −Φ is given. Asteriskscorreponds to entries whose values have not been explicitly computed.
tivity of the joining operator when restricted to each irreducible subspace is sufficientfor positivity of the overall operator. The joining operators may then be decomposedinto the projectors on each irreducible subspace and corresponding bases of tracelessoperators on the projectors. The basis elements will be combinations of permuta-tion operators and the dimension of each such operator subspaces is given by thesquare of the hook length of the corresponding Young diagram [91]. The remainingtask is to obtain a characterization of the positivity of the operators on each irre-ducible subspace. In [92], for example, a method for characterizing the positivity oflow-dimensional operator spaces is presented. As long as the number of subsystemsremains small, this approach grants us a computationally friendly characterization ofpositivity of the joining states. The bounds on the joinable Werner pairs may thenbe obtained by projecting the positivity characterization boundary onto the space ofWerner pairs, analogous to the space of Fig. 2.2a.
While a complete analysis is beyond our scope, a similar method may in principle
34
2.4 Further remarks
be followed to determine more general joinability bounds for isotropic states. How-ever, a twirling operation that preserves the joining property only exists for certainisotropic joining scenarios. For instance, we took this issue into consideration whenwe required the B-C system to be in a Werner state while A-B and A-C were isotropicstates; it would not have been possible to take the same approach if all three pairswere isotropic states.
2.4.2 Sharability of general bipartite qubit states
For qubit Werner states, one can use the methods of the proof of Thm. III.6 to showthat 1-n sharability does imply n-n sharability [cf. Table I.(a)]. This property neitherholds for Werner qudit states nor bipartite qubit states in general. The simplestexample of a Werner state which disobeys this property is the most-entangled qutritWerner state ρ(Ψ− = −1)d=3. This state is 1-2 sharable, as evidenced by the point(−1,−1,−1) lying within the bi-cone described by Eq. (2.15). The correspondingsharing state is the totally antisymmetric state on three qutrits as given by Eq.(2.7). This is the unique sharing state because the collective disagreement betweenthe subsystems of each joined bipartite Werner state forces collective disagreementamong the subsystems of the tripartite joining state; the totally antisymmetric stateis the only quantum state satisfying this property. Since the only 1-2 sharing statefor ρ(Ψ− = −1)d=3 is pure and entangled, clearly there can exist no 2-2 sharing.
Additionally, we present below a counter-example that involves qubit states offthe Werner line:
Proposition 2.4.1. For a generic bipartite qubit state ρ, 1-n sharability does notimply n-n sharability.
Proof. We claim that the following bipartite state on two qubits,
ρ =1
3
[(|00〉+ |11〉
)(〈00|+ 〈11|
)+ |10〉〈10|
]≡ ρL1R1 ,
is 1-2 sharable but not 2-2 sharable. To show that ρ is 1-2 sharable, direct calculationshows that the two relevant partial-trace constraints uniquely identify w3 ≡ |ψ〉〈ψ|as the only valid sharing state, with
|ψ〉 ≡ 1√3
(|000〉+ |101〉+ |110〉).
The above state may in turn be equivalently written as
|ψ〉 =1√3|0〉 ⊗ |00〉+
√2
3|1〉 ⊗ 1√
2(|01〉+ |10〉) .
35
Quantum marginals: sharability and joinability
In order for ρ to be 2-2 sharable, a four-partite state w4 must exist, such thatTrLiLj(w4) = ρ, for i, j = 1, 2. Any state which 2-2 shares ρ must then 1-2 sharethe pure entangled state w3. That is, in constructing the 2-2 sharing state for ρ, webring in a fourth system L2 which must reduce (by tracing over L1 or L2) to w3. But,since w3 is a pure entangled state, it is not sharable. Thus, there cannot exist a 2-2sharing state for ρ.
We conclude by stressing that our Werner and isotropic state sharability resultsallow in fact to put bounds (though not necessarily tight ones) on the sharability ofan arbitrary bipartite qudit state. It suffices to observe that any bipartite state can betransformed into a Werner or isotropic state by the action of the respective twirlingmap (either Eq. (2.13) or (2.18)). Theorem 2.2.5 proves that the sharability of a statecannot be decreased by a unitary mixture map, and hence twirling cannot decreasesharability. This thus establishes the following:
Corollary 2.4.2. A bipartite qudit state ρ is no more sharable than the Werner state
ρ ≡∫U ⊗ UρVU † ⊗ U †dµ(U),
and the isotropic state
ρ ≡∫U∗ ⊗ UρVUT ⊗ U †dµ(U),
for any ρV = I⊗ V ρ I⊗ V †, with V ∈ U(d).
In the qubit case, for instance, any maximally entangled pure state can be trans-formed into |Ψ−〉 or |Φ+〉 by the action of some local unitary I ⊗ V . Thus, allmaximally entangled pure qubit states and their “pseudo-pure” versions, obtainedas mixtures with the fully mixed states, have the same sharability properties as theWerner/isotropic states.
36
Chapter 3
Joinability of causal and acausalrelationships
37
Joinability of causal and acausal relationships
This chapter presents material that appeared in Journal of Physics A: Mathemat-ical and Theoretical, 48:035307 (2015), in an article titled “On state versus channelquantum extension problems: exact results for U ⊗ U ⊗ U symmetry”, which isjoint work with Lorenza Viola.
3.1 IntroductionIt has long been appreciated that many of the intuitive features of classical probabilitytheory do not translate to quantum theory. For instance, every classical probabilitydistribution has a unique decomposition into extremal distributions, whereas a gen-eral density operator does not admit a unique decomposition in terms of extremaloperators (pure states). Entanglement is responsible for another distinctive trait ofquantum theory: as vividly expressed by Schrödinger back in 1935 [93], “the bestpossible knowledge of a total system does not necessarily include total knowledge ofall its parts,” in striking contrast to the classical case. Certain features of classi-cal probability theory do, nonetheless, carry over to the quantum domain. While itis natural to view these distinguishing features as a consequence of quantum theorybeing a non-commutative generalization of classical probability theory in an appropri-ate sense, thoroughly understanding how and the extent to which the purely quantumfeatures of the theory arise from its mathematical structure remains a longstandingcentral question across quantum foundations, mathematical physics, and quantuminformation processing (QIP), see e.g. Refs. [94, 95, 96, 97].
In this chapter, we investigate a QIP-motivated setting which allows us to directlycompare and contrast features of quantum theory with classical probability theory,namely, the relationship between the parts (subsystems) of a composite quantum sys-tem and the system as a whole. Specifically, building on our earlier work [98], wedevelop and investigate a general framework for what we refer to as quantum join-ability, which addresses the compatibility of different statistical correlations amongquantum measurements on different systems. Arguably, the most familiar case ofjoinability is provided by the “quantum marginal” (aka “local consistency”) problem[27, 99]. In this case, we ask whether there exists a joint quantum state compatiblewith a given set of reduced states on (typically non-disjoint) groupings of subsys-tems. The quintessential example of a failure of joinability is the fact that two pairsof two-level systems (qubits), say, Alice-Bob (A-B) and Alice-Charlie (A-C), cannotsimultaneously be described by the singlet state, |ψ−〉 =
√1/2(|↑↓〉 − |↓↑〉). A semi-
nal exploration of this observation was carried out by Coffman, Kundu, and Wootters[38] and later dubbed the “monogamy of entanglement” [40]. In classical probabilitytheory, a necessary and sufficient condition for marginal probability distributions onA-B and A-C to admit a joint probability distribution (or “extension") on A-B-C isthat the marginals over A be equal [27, 62]. The analogous compatibility condition
38
3.1 Introduction
remains necessary in quantum theory, but, as demonstrated by the above example, isclearly no longer sufficient. The identification of necessary and sufficient conditionsin general settings with overlapping marginals remains an actively investigated openproblem as yet [61, 56, 100].
Physically, standard state-joinability problems as formulated above for density op-erators, may be regarded as characterizing the compatibility of statistical correlationsof two (or more) different subsystems at a given time. However, correlations betweena single system before and after the action of a quantum channel – a completely posi-tive trace-preserving (CPTP) dynamical map – may also be considered, for example,in order to characterize the “location” of quantum information that one subsystemmay carry about another [101] and/or the causal structure of the events on whichprobabilities are defined [96, 102]. The work in [102] thoroughly explores, in par-ticular, the idea of placing kinematic and dynamic correlations on equal footing, byintroducing a formalism of “quantum conditional states” to represent the correlationsof either bipartite quantum states or quantum channels as bipartite operators. Withthese ideas in mind, one may want to formulate a quantum marginal problem forquantum channels (see also Ref. [103]). For example, given two quantum channelsMAB : B(HA) → B(HB) and MAC : B(HA) → B(HC) (with B(H) denoting thespace of bounded linear operators on H), one may ask whether there exists a quan-tum channelMABC : B(HA)→ B(HB ⊗HC), whose reduced channels areMAB andMAC , respectively.
A motivation for considering such channel-joinability problems is that questionsregarding the optimality of paradigmatic QIP tasks such as quantum cloning [84, 85]or broadcasting [104] may be naturally recast as such. A fundamental tool here is theChoi-Jamiolkowski isomorphism [105, 89], which may been used to translate optimalcloning problems into quantum marginal problems [86, 106], and vice-versa [61]. Bothmonogamy of entanglement and the no-cloning theorem [14] have significant impli-cations for the behavior of quantum systems: the former effectively constrains thekinematics of a multipartite quantum system, while the latter constrains the dynam-ics of a quantum system (composite or not). As both of these fundamental conceptsare closely related to respective quantum joinability problems, we are prompted toexplore in more depth their similarities and differences. Identifying a general join-ability framework, able to encompass all such quantum marginal problems, is one ofour main aims here.
The content is organized as follows. In Section 3.2, we introduce and motivatethe use of what we term the homocorrelation map as our main tool for representingquantum channels as bipartite operators. Despite the different motivation, this repre-sentation will share suggestive points of contact with the conditional-state formalismof [102]. Formally, we show how it enables a notion of quantum joinability that in-corporates all joinability problems of interest, and discuss ways in which different
39
Joinability of causal and acausal relationships
joinability problems may be (homomorphically) mapped into one another. In Section3.3, we obtain a complete analytical characterization of some archetypal examples oflow-dimensional quantum joinability problems. Namely, we address three-party join-ability of quantum states, quantum channels, and block-positive (or “local-positive”)operators, in the case that the relevant operators are invariant under the group ofcollective unitary transformations, that is, under the action of arbitrary transforma-tions of the form U ⊗ U ⊗ U . These examples allow us to distinguish the joinabilitylimitations stemming from classical probability theory from those due to quantumtheory and, furthermore, to contrast the joinability properties of quantum channelsvs. states. In Section 3.4, we investigate a possible source for the stricter joinabilitybounds in quantum theory, as compared to classical probability theory. We introducethe notion of degree of agreement (disagreement), that is, the probability that a ran-dom local collective measurement yields same (different) outcomes, as given by anappropriate two-value POVM. We find that quantum theory places different bounds onthe degree of agreement arising from quantum states than it does on that of quantumchannels: while quantum states are limited in their degree of agreement, quantumchannels are limited in their degree of disagreement. The differences in these boundspoint to a crucial distinction between quantum channels and states. At least in theexamples of Section 3.3 and a few others, these limitations suffice in fact to determinethe bounds of joinability exactly. Possible implications of such bounds with regardsto joinability properties of general quantum states and channels are also discussed.
3.2 General quantum joinability framework
We begin by reviewing the standard state-joinability (quantum marginal) problem,framing it in a language suitable for generalization. Given a composite Hilbert spaceH(N) =
⊗Ni=1Hi, a joinability scenario is defined by a list of partial traces Tr`k, with
each `k ⊆ [1, . . . , N ], along with a set of allowed “joining operators,” W , which in thiscase is the set of positive trace-one operators acting on H(N); accordingly, we mayassociate a joinability scenario with a 2-tuple (W, Tr`k). For a given joinabilityscenario, the images of W under the Tr`k define a set of reduced states Rk =Trk(W ). For any list of states ρk ∈ Rk, the following definition then applies:
Definition 3.2.1. [State-Joinability] Given a joinability scenario described by thepair (W ≡ w| w ≥ 0, Tr`k), the reduced states ρk ∈ Rk are joinable if thereexists a joining state w ∈ W such that Tr`k(w) = ρk for all k.
The first step toward achieving the intended generalization of the above definitionto quantum channels is to represent the latter as bipartite operators. In the followingsubsection, we establish a tool to achieve this and highlight its broader utility.
40
3.2 General quantum joinability framework
3.2.1 Homocorrelation map and positive cones
One way to identify channels with bipartite operators is by use of the Choi-Jamiolkowski(CJ) isomorphism [89, 107]. This isomorphism, denoted J , identifies each mapM ∈ L(HA,HB) with the state resulting from the map’s action on one memberof a maximally entangled state:
J (M) ≡ [IA ⊗M](|Φ+〉〈Φ+|) =1
dA
∑ij
|i〉〈j| ⊗M(|i〉〈j|), (3.1)
where IA is the identity map on B(HA), |Φ+〉 =∑
i |ii〉/√dA and dA = dim(HA).
We note that dA|Φ+〉〈Φ+| = V TA , where V is the swap operator on HA⊗HA and TAdenotes partial transposition on subsystem A. The transformation is an isomorphismin that it preserves the positivity of the objects it maps to and from; namely, quantumchannels (CPTP maps) are mapped to quantum states (positive trace-one operators).Consequently, J is a useful diagnostic tool for determining whether a map is CP 1.
Here, we employ an alternative means of identifying quantum channels with bi-partite operators, building on an identification that was introduced for the specialcase of qubits in [108]. In this approach, basis-dependence is avoided by replacing thereference state with the normalized swap operator V/d. Since the latter is not a den-sity operator, this correspondence lacks an interpretation as a physical process. But,for our purposes, this comes at the greater benefit of yielding a resulting bipartiteoperator that bears the statistical properties of the corresponding channel. Formally,we define a homocorrelation map, H, which takes any map M ∈ L(HA,HB) (withL(HA,HB) being the set of linear maps, or “superoperators”, from B(HA) to B(HB)),to a “channel operator” MH ∈ B(HA ⊗HB) according to
H(M) ≡ [IA ⊗M](V/dA) =1
dA
∑ij
|i〉〈j| ⊗M(|j〉〈i|), (3.2)
where V =∑
i,j |ij〉〈ji| with respect to any orthonormal basis |i〉2. While the CJisomorphism is a handy diagnostic tool, the homocorrelation map serves a differentpurpose. It does not take CP maps to positive operators. Instead, it takes each mapto an operator which exhibits the same statistical correlations as that map:
1Note that J depends on a choice of local basis, needed to define |Φ+〉 and TA. In order for theisomorphism to hold, the reference state (|Φ+〉〈Φ+| above) must be maximally entangled; again ford > 2, any such state reflects a choice of local bases.
2The homocorrelation map H is closely related to the causal conditional states defined in [102].Namely, a channel operator resulting from the homocorrelation map is precisely the causal condi-tional state conditioned on ρA being the completely mixed state. Note that Eq. (3.2) above differsfrom Eq. (29) in [102] by a factor of dA.
41
Joinability of causal and acausal relationships
(a) (b)
Figure 3.1: (a) Commutativity diagram summarizing the relationship between theChoi-Jamiolkowski isomorphism and the homocorrelation map defined in Eqs. (3.2)-(3.1). In (b), the corresponding actions are given in terms of tensor network diagramnotation [109]. Proposition 3.2.2 may be straightforwardly proved using this notation.
Proposition 3.2.2. A bipartite state ρ ∈ B(HA ⊗ HB) and a quantum channelM : B(HA)→ B(HB) exhibit the same correlations, that is,
Tr[ρA⊗B] =1
dATr[M(A)B], ∀A ∈ B(HA), B ∈ B(HB). (3.3)
if and only if the equality H(M) = ρ holds.
Proof. The two operators ρ and H(M) are equal if and only if their expectationsTr[ρA⊗B] = Tr[H(M)A⊗B] for all A,B. Thus, it suffices to show that Tr[H(M)A⊗B] = 1
dATr[M(A)B] for all A,B. This equality may be established as follows:
Tr[H(M)A⊗B] = 1dA
∑i,j Tr[|i〉〈j| ⊗M(|j〉〈i|)A⊗B]
= 1dA
∑i,j Tr[(|i〉〈j|A)⊗ (M(|j〉〈i|)B)]
= 1dA
∑i,j〈j|A|i〉Tr[M(|j〉〈i|)B] = 1
dATr[M(A)B]. 2
Equation (3.3) may be taken as the defining property of the homocorrelation map.An example may explicitly demonstrate the utility of this representation. Considerthe one-parameter family of qudit depolarizing channels [17], defined as
Dη(ρ) = (1− η)Tr(ρ)Id
+ ηρ. (3.4)
The action of this channel commutes with all unitary channels in that Dη(UρU †) =UDη(ρ)U †. Under the homocorrelation map, the depolarizing channels are taken to
42
3.2 General quantum joinability framework
operators with U ⊗ U symmetry, namely,
H(Dη) = (1− η)I⊗ Id2
+ ηV
d, (3.5)
where V is, again, the swap operator. Trace-one, positive operators of this form arethe well-known Werner states [57] (see also Sec. 3.3.1). Imagine that an observer doesnot know a priori whether her two measurements are made on distinct systems in aWerner state or if they are made on the same system before and after a depolarizingchannel has been applied. If presented with a Werner state or depolarizing channelhaving η = − 1
d2−1to 1
d+1, the observer will not be able to distinguish between the
two cases. The homocorrelation map makes this operational identification explicit.To contrast, the CJ map takes the depolarizing channels to so-called isotropic states[80],
J (Dη) = (1− η)I⊗ Id2
+ η|Φ+〉〈Φ+|, (3.6)
where as before |Φ+〉 is the maximally entangled state. The isotropic states aredefined by their symmetry with respect to U ⊗ UT transformations. An observer inthe scenario above would certainly be able to distinguish between the correlations ofthe depolarizing channel and the isotropic states, as long as η 6= 0.
The distinction between the CJ isomorphism and the homocorrelation map can befurther appreciated by contrasting the sets of operators they produce. The set of CPmaps forms a cone in the set of superoperators L(HA,HB). Both the CJ isomorphismand the homocorrelation map are cone-preserving maps (by linearity) from L(HA,HB)to B(HA⊗HB). While in the case of the CJ isomorphism, the resulting cone is exactlythe cone of bipartite states, in the case of the homocorrelation map, the cone is distinctfrom the cone of states. One of the main findings of this chapter is that the correlationsexhibited by bipartite states and the ones exhibited by quantum channels need notbe equivalent. Furthermore, we find that this difference plays a role in their distinctjoinability properties. The homocorrelation representation of channels provides uswith a natural framework for exploring this difference: a channel and a state withdiffering correlations will be represented as distinct operators in the same operatorspace; as an example, the classes of bipartite Werner states and depolarizing channelsare depicted in Fig. 3.2. These notions and their use in joinability are fleshed out inwhat follows.
The cone of positive operators plays a central role in defining joinability of quan-tum states. Analogously, the cone of homocorrelation-mapped channels (or “channel-positive operators”) will play a central role in defining joinability of quantum channels.
Definition 3.2.3. [State-positivity] An operator M ∈ B(H) is state-positive ifTr(MP ) ≥ 0 for all Hermitian projectors P = P † = P 2 ∈ B(H). We notate this
43
Joinability of causal and acausal relationships
condition as M ≥st 0 and emphasize that the resulting set is a self-dual cone.
Recall that a mapM is a valid quantum channel if Tr[J (M)P ] ≥ 0 for all P = P 2 ∈B(HA ⊗HB) [89]. Using the relationships of Fig. 3.1, we translate this condition toone on the homocorrelation-mapped operator M = H(M). Specifically, we define:
Definition 3.2.4. [Channel-positivity] An operator M ∈ B(HA⊗HB) is channel-positive with respect to the A-B bipartition if Tr(MP TA) ≥ 0 for all Hermitian projec-tors P = P † = P 2 ∈ B(HA⊗HB). We notate this condition M ≥ch 0, and emphasizethat the resulting set is, again, a self-dual cone.
I
Tr M = 1
0
V
θ
I – V/d
θ
Figure 3.2: State- and channel-positive cones for two qudit Werner operators. The re-gion of the solid arc (blue) corresponds to state-positive operators, while the region ofthe dashed arc (pink) corresponds to channel-positive operators. The overlapping re-gion, seen as purple, corresponds to PPT operators; of these, the normalized operatorsare also unentangled state-positive operators. The self-dual nature of the state- andchannel-positive cones is consistent with the right angles of each cone’s vertex. TheYoung diagrams represent the corresponding projectors into the symmetric 1
2(I + V )
and antisymmetric 12(I−V ) subspaces, respectively. For qudit dimension d, the angle
θ is calculated to be cos θ = Tr[V (I + V )]/√
Tr[V 2]Tr[(I + V )2] =√
(d+ 1)/2d.
In the general case, we can give a characterization of the intersection of the twocones and their complements. This is aided by the fact that the CJ isomorphismand the homocorrelation map are related to one another by partial transpose. Acommutivity diagram of these relationships is given in Fig. 3.1, where the tensornetwork diagram calculus [109] may be used to concisely demonstrate that, up tonormalization,
J −1 H = H−1 J = TA.
44
3.2 General quantum joinability framework
Proposition 3.2.5. A bipartite state ρ ∈ B(HA ⊗ HB) and a quantum channelM : B(HA)→ B(HB) exhibit the same correlations if and only if the density operator(or equivalently, channel operator) has a positive partial transpose (PPT).
Proof. By Prop. 3.2.2, if a bipartite state and a quantum channel exhibit the samecorrelations, then ρ = H(M). Since J is related to H by a partial trace, we alsohave H(M)TA = J (M)/dA. By the positivity preservation of J , M being CPTPimplies that J (M) is a positive operator. Thus, we have that ρTA = J (M)/dA ispositive.
This result may be used to directly connect quantum channels to entanglement:
Corollary 3.2.6. If the correlations of a bipartite state ρ ∈ B(HA ⊗HB) cannot beexhibited by a quantum channel, then the state is entangled.
Proof. Since the correlations cannot be exhibited by a quantum channel, the operatoris not PPT, by Prop. 3.2.5. Then, by the Peres-Horodecki criterion [110], the stateis necessarily entangled.
3.2.2 Generalization of joinability
We are now poised to use the homocorrelation representation to define the joinabilityof channels. The channel-positive operators provide an alternative set with whichto define the allowed joining operators W . As a warm-up, we rephrase the channel-joinability problem that was posed in the Introduction. Consider quantum channelsfrom HA to HB⊗HC . Under the homocorrelation map, these correspond to tripartiteoperators lying in the channel-positive cone, notated WA|BC . The partial traces TrCand TrB take channel-positive operators in WA|BC to channel-positive operators inWA|B and WA|C , respectively; that is, operators in WA|B and WA|C correspond tovalid quantum channels via the homocorrelation map. The corresponding channel-joining scenario is then defined as (WA|BC , TrC ,TrB). A channel-joinability problempresents two channel operatorsMAB ∈ WA|B andMAC ∈ WA|C and seeks to determinethe existence of a channel operator MABC ∈ WA|BC which reduces to the two channeloperators in question. In general, we thus have the following:
Definition 3.2.7. [Channel-Joinability] Given a joinability scenario described bythe pair (W ≥ch 0, Tr`k), the reduced operators Mk ∈ Rk are joinable if thereexists a joint operator M ∈ W such that Tr`k(M) = Mk for all k.
We note that a channel joinability (or extension) problem can be stated usingthe CJ isomorphism instead of the homocorrelation map, as done in [103]. However,as we argued, the homocorrelation map provides a platform to directly compare thejoinability of states and channels of equivalent correlations. For instance, it will allow
45
Joinability of causal and acausal relationships
us to simultaneously compare the joinability of local-unitary-invariant quantum statesand channels, and consequently to compare these both to the joinability of analogousclassical probability distributions (c.f. Fig. 3.5).
Before proceeding to the general notion of joinability, we also remark that allowedjoining operators in W have thus far been considered to be either state-positive orchannel-positive. However, from a mathematical standpoint, a sensible joinabilityproblem only needs W to be a convex cone. To investigate this generalization and(as motivated later) to meld state and channel joining, we consider a third typeof positivity that we call local-positivity. This notion is equivalent to both block-positivity [111] and to map-positivity (not necessarily CP) [112, 113], in that byrepresenting linear maps using the homocorrelation map, the cone of (transformed)positive maps is equal to the cone of bipartite block-positive operators. Formally:
Definition 3.2.8. [Local-positivity] An operatorM ∈ B(HA⊗HB) is local-positivewith respect to the A-B factorization if Tr(MPA ⊗ PB) ≥ 0 for all pure states PA =P 2A ∈ B(HA) and PB = P 2
B ∈ B(HB). We notate this condition M ≥loc 0.
The set of channel-positive operators and state-positive operators are each sub-conesof the local-positive operators, as local-positivity clearly is a weaker condition. Local-positive operators are directly relevant to QIP, in particular because they may serve asan entanglement witnesses [114]. Moreover, in comparing quantum joinability limita-tions to analogous limitations stemming from classical probability theory, joinabilityscenarios defined with respect to W ≥loc 0 may allow the identification of quantumlimitations in a “minimally constrained" setting, closer to the (less strict) classicalboundaries. In Sec. 3.3.2, we find that local-positivity does nevertheless providestricter-than-classical limitations on joinability.
Another way of viewing the various definitions of positivity is to understand thesubscript on the inequality to indicate the dual cone from which inner products withM must be positive. For M ≥st 0, M ≥ch 0, and M ≥loc 0, the respective dualcones are the positive span of rank-one projectors, the positive span of partially-transposed projectors, and the positive span of product projectors (from which thetrace-one condition confines to the set of separable states). We note that the firsttwo cones are self-dual (and, furthermore, symmetric [115]), while the local-positivecone is not. With several important examples of positivity established, each being adifferent convex set with which to defineW , we are in a position to give the following:
Definition 3.2.9. [General Quantum Joinability] Let W be a convex cone inB(H(N)), and Tr`k be partial traces with `k ⊂ ZN . Given the joinability scenario(W, Tr`k), the operators Mk ∈ Rk are joinable if there exists a joining operatorw ∈ W such that Tr`k(w) = Mk for all k.
This general definition naturally encompasses the various joinability problems ref-erenced in the Introduction. Specifically, in the case where W is the set of quan-
46
3.2 General quantum joinability framework
tum states on a multipartite system, the joinability problem reduces to the quan-tum marginal problem, while if W consists of channel-positive operators describingquantum channels from one multipartite system to another, one recovers the channel-joining problem instead. Specific instances of this problem are the optimal asymmet-ric cloning problem [84, 85, 88], the symmetric cloning problem [116, 117], and thek-extendibility problem for quantum maps [118]. In addition to providing a unifiedperspective, our approach has the important advantage that different classes of join-ability problems may be mapped into one another, in such a way that a solution toone provides a solution to another. This is made formal in the following:
Proposition 3.2.10. Let W and W ′ be two positive cones of operators acting onthe space H(N), let Tr`k be a set of partial traces that apply to both cones, andlet φ : W → W ′ be a positivity-preserving (homo)morphism, which permits reducedactions φk satisfying φk Tr`k = Tr`k φ. If Mk ∈ Tr`k(W ) is joinable withrespect to W , then φk(Mk) ∈ Tr`k(W
′) is joinable with respect to W ′.
Proof. Assume that w is a valid joining operator for the operators Mk ∈ Tr`k(W ).Then, the set of operators φk(Mk) ∈ Tr`k(W
′) is joined by the operator φ(w),since Tr`k [φ(w)] = φk(Tr`k [w]) = φk(Mk) and φ(w) ∈ W ′.
This is shown in the commutative diagram of Fig. 3.3. We shall use a strongercorollary of this result in the remaining sections:
Corollary 3.2.11. Let φ be a one-to-one positivity-preserving map from W to W ′,with invertible reduced actions φk satisfying φk Tr`k = Tr`k φ (and similarly fortheir inverses). Then a set of operators Mk ∈ Tr`k(W ) is joinable if and only ifthe set of operators φk(Mk) is joinable.
Proof. The forward implication follows from Proposition 3.2.10, while the backwardsimplication follows from the fact that φ and the φk are invertible, along with thecontrapositive of Proposition 3.2.10.
The joinability-problem isomorphism we make use of is the partial transpose map.The latter permits a natural reduced action, namely, partial transpose on the remain-ing of the previously transposed subsystems. As explained, the partial transpose is apositivity-preserving bijection between states and channel operators (via H). Thus,if we determine the joinable-unjoinable demarcation for a class of states, we willdetermine the joinable-unjoinable demarcation for a corresponding class of channel-operators.
47
Joinability of causal and acausal relationships
Figure 3.3: Commutivity diagram showing a homomorphism of joinability problems.
3.3 Three-party joinability settings with collectiveinvariance
In this Section, we obtain an exact analytical characterization of the state-joining,channel-joining, and local-positive joining problems in the three-party scenario, bytaking advantage of collective unitary invariance. That is, we determine what trio ofbipartite operators MAB, MAC , and MBC may be joined by a valid joining operatorwABC , subject to the appropriate symmetry constraints. As noted, the most familiarcase is state joinability, whereby the bipartite operators along with the joining tri-partite operator are state-positive. The next case considered is referred to as “1-2channel joinability”: here, we specify a bipartition of the systems (say, A|BC) andconsider the bipartite operators which cross the bipartition (MAB and MAC), alongwith the joining operator, to be channel-positive with respect to the bipartition,while the remaining bipartite operator (MBC) is state-positive. Since each of the threepossible bipartition choices (A|BC, B|AC, and C|AB) constitutes a different channeljoinability scenario, a total of four possibilities arise for three-party state/channeljoinability. Lastly, motivated by the suggestive symmetry arising from these resultsand their relation to classical joining, we consider the weaker notion of local-positivejoining, in which all operators involved are only required to be local-positive.
3.3.1 Joinability limitations from state-positivity and channel-positivity
We begin by describing the operators which are to be joined. The bipartite reducedoperators inherit the collective unitary invariance from the tripartite operators fromwhich they are obtained. Therefore, by a standard result of representation theory[119], the operators which are to be joined are of the following form:
ρ(η) = (1− η)Id2
+ ηV
d, (3.7)
48
3.3 Three-party joinability settings with collective invariance
where V is the swap operator defined earlier. The above operators are known tobe state-positive for the range − 1
d−1≤ η ≤ 1
d+1, corresponding to the d-dimensional
(qudit) Werner states we already mentioned. The parameterization is chosen so thatη is a “correlation” measure: if d = 2, η = −1 corresponds to the singlet state, η = 0to the maximally mixed state, while η = 1 is not a valid quantum state, but expressesperfect correlation for all possible collective measurements. Note that a value η = 1,for instance, does correspond to a valid quantum channel. Intuitively, channel-positiveoperators with U ⊗ U -invariance correspond to depolarizing channels. It is knownthat complete positivity (or channel-positivity) of the depolarizing channel is ensuredprovided that − 1
d2−1≤ η ≤ 1 [120]. However, we find it instructive to independently
establish state- and channel- positivity bounds using the CJ isomorphism.To this end, we enlarge the above class of U ⊗ U -invariant operators to the class
of operators with collective orthogonal invariance, namely, invariance under transfor-mations of the more general form O ⊗ O, belonging to the so-called Brauer algebra[121, 122]3. In addition to U ⊗ U -invariant operators, the Brauer algebra also con-tains U∗ ⊗ U -invariant operators. The latter class of operators, which includes thewell-known isotropic states, are spanned by the operators I and V TA . Thus, the setof O ⊗O-invariant operators are of the form
ρ(η, β) = (1− η − β)Id2
+ ηV
d+ β
V TA
d. (3.8)
In particular, the operator ρ(0, 1) is a generic Bell state on two qudits, ρ(0,−1/(d−1))is the maximally entangled Werner state (namely, the singlet state for d = 2), ρ(1, 0)is the identity channel, and ρ(0, 0) is the completely mixed state (or the completelydepolarizing channel). We can then establish the following:
Proposition 3.3.1. A bipartite operator ρ(η) with collective unitary invariance ischannel-positive if and only if − 1
d2−1≤ η ≤ 1.
Proof. The Brauer algebra includes all the state-positive operators which are mapped,via the CJ isomorphism J , to the U ⊗U -invariant channel-positive operators; underJ , η and β in Eq. (3.8) are swapped with one another. Since J takes state-positiveoperators to channel-positive operators, we need only obtain the set of state-positiveoperators. State-positivity of these operators is enforced by the inner products with
3Operators in this algebra have been extensively analyzed in [123, 124], and recent work char-acterizing their irreducible representations may be found in [125, 126]. The Brauer algebra act-ing on N d-dimensional Hilbert spaces is spanned by representations of subsystem permutationsVπ|π ∈ SN, along with their partial transpositions with respect to groupings of the subsystemsV Tl
π |π ∈ SN , l ⊆ 1, . . . , N. In terms of tensor network diagrams, each element of this basis isrepresented by a set of disjoint pairings of 2N vertices, with the vertices arranged in two rows, bothcontaining N of them.
49
Joinability of causal and acausal relationships
respect to their (operator) eigenspaces being non-negative. Such eigenspaces are PA,P+, and PY , independent of η and β: the first is the anti-symmetric subspace, thesecond is the one-dimensional space spanned by |Φ+〉, and the third is the spacespanned by vectors |y〉 satisfying 〈y|(|y〉)∗ = 04. The eigenvalues are as follows:
ρ(η, β)PA = [(1− η − β)/d2 − η/d]PA,
ρ(η, β)P+ = [(1− η − β)/d2 + η/d+ β]P+,
ρ(η, β)PY = [(1− η − β)/d2 + η/d]PY .
Hence, state-positivity of the bipartite Brauer operators is ensured by
The inequalities bounding channel-positivity are obtained by swapping the ηs and βs.In particular, we recover that the state-positive range for U ⊗ U -invariant operatorsis − 1
d−1≤ η ≤ 1
d+1, whereas the channel-positive range is − 1
d2−1≤ η ≤ 1.
In a similar manner, we can also obtain the ranges of local-positivity:
Proposition 3.3.2. A bipartite operator ρ(η) with collective unitary invariance islocal-positive if and only if − 1
d−1≤ η ≤ 1.
Proof. Local positivity is ensured by the non-negativity of expectation values withrespect to the product vectors |xx〉, |xx〉, |yy〉, |yy〉, satisfying |x〉∗ = |x〉 and |y〉∗ =|y〉, where the bar indicates a vector orthogonal to the original vector. In terms of ηand β, these constraints read
0 ≤ 〈ρ(η, β)〉xx = (1− η − β)/d2 + η/d+ β/d,
0 ≤ 〈ρ(η, β)〉xx = (1− η − β)/d2,
0 ≤ 〈ρ(η, β)〉yy = (1− η − β)/d2 + η/d,
0 ≤ 〈ρ(η, β)〉yy = (1− η − β)/d2 + β/d.
More compactly, these boundaries are given by
− 1
d− 1≤ η + β ≤ 1, −(d− 1)η + β ≤ 1, η − (d− 1)β ≤ 1.
Thus, for bipartite Brauer operators, local-positive operators are equivalent to convexcombinations of state- and channel-positive operators5. In particular, for the local-
4As noted, both the definition of |Φ+〉 and the use of complex conjugation are basis-dependentnotions. It is understood that all usages of either refer to the same (arbitrary) choice of basis.
5This property is known as decomposability [112]. Interestingly, such an equivalence also holdsfor arbitrary bipartite qubit states [127].
50
3.3 Three-party joinability settings with collective invariance
(a) (b)
Figure 3.4: Positivity regions for bipartite Brauer operators: (a) d = 2. (b) d = 5.The solid triangle (blue) encloses the state-positive region, the dashed triangle (pink)encloses the the channel-positive region, and the outer boundary encloses the local-positive region.
positive range of U ⊗ U -invariant operators, it follows that − 1d−1≤ η ≤ 1, as stated.
A pictorial summary of the three positivity bounds is presented in Fig. 3.4.Having characterized all types of positivity for the (bipartite) operators to be
joined, we now turn to characterize the positivity for the (tripartite) joining opera-tors. For each positive tripartite set (W ≥st 0, W ≥ch 0, and W ≥loc 0), we obtainthe trios of joinable bipartite operators by simply applying the three partial traces(TrA,TrB,TrC) to each positive operator. In more detail, our approach is to obtain anexpression for the positivity boundary of the tripartite operators in terms of operatorspace coordinates, and then re-express this boundary in terms of reduced-state param-eters (the three Werner parameters in this case). For state- and channel-positivity,the desired characterization follows directly from the analysis reported in our previouswork [61].
Corollary 3.3.3. With reference to the parameterization of Eq. (3.7), we have that:(i) Three Werner states with parameters (ηAB, ηAC , ηBC) are joinable with respect tothe (WABC ≥st 0, TrA,TrB,TrC) scenario if and only if
(ii) Three U ⊗ U-invariant operators with parameters (ηAB, ηAC , ηBC) are channel-joinable with respect to the (WA|BC ≥ch 0, TrA,TrB,TrC) scenario if and only if
1
d− 1+ ηAB + ηAC − ηBC ≥
∣∣∣∣ 2
d− 1+ dηBC +
√2d
d− 1(eiθηAB + e−iθηAC)
∣∣∣∣,eiθ ≡
√(d− 1)/2d± i
√(d+ 1)/2d.
The channel-joinability limitations in the other two scenarios B|AC and C|AB maybe obtained by permuting the ηs accordingly.
Proof. Result (i) corresponds to Theorem 3 in [61], re-expressed in terms of theparametrization of Eq. (3.7) (with reference to the notation of Eqs. (15)–(17) in [61],one has η` = (d/(d2 − 1))(Ψ−` − 1/2), ` = AB,AC,BC).
In order to establish (ii), note that J may be used to translate any U∗⊗U -invariantstate-positive joinability problem into a U ⊗ U -invariant channel-positive joinabilityproblem, drawing on Corollary 3.2.11. Explicitly, under J (partial transpose in thecase of operators), the U∗ ⊗ U ⊗ U -invariant state-positive operators WA∗BC are inone-to-one correspondence with the U ⊗ U ⊗ U -invariant channel-positive operatorsWA|BC . Hence, by the joinability isomorphism induced by the partial transpose, thesolution to a joinability problem of the scenario (WA∗BC , TrA,TrB,TrC) gives asolution to a corresponding joinability problem of (WA|BC , TrA,TrB,TrC). Thus,to obtain the depolarizing channel-joinability boundaries, we simply translate theisotropic state parameters of Eqs. (20)-(21) in [61] into η parameters.
The joinability limitations of all four scenarios are depicted in Fig. 3.5. Asstressed in [61], the quantum joinability limitations must adhere to the analogousclassical joinability limitations (seen as the tetrahedra in Fig. 3.5). In the qubit case,we find it intriguing that the inclusion of the quantum channel-joinability limitationsallows us to regain the tetrahedral symmetry imposed by the classical limitations;whereas each scenario on its own expresses a continuous rotational symmetry that isnot reflected classically. In other words, if we consider the joinability scenario definedby
(spanWABC ,WA|BC ,WB|AC ,WC|AB, TrA,TrB,TrC),
the joinable bipartite operators respect the tetrahedral symmetry suggested by theclassical joinability bounds. This amounts to asking the question: what trios ofbipartite correlations – as derivable from either quantum states or channels, or fromprobabilistic combinations of the two – may be obtained from the measurements onthree systems? Though the result expresses the tetrahedral symmetry of the classicaljoinability limitations, these classical joinability limitations do not suffice to enforcethe stricter quantum joinability limitations, as manifest in the fact that the corners
52
3.3 Three-party joinability settings with collective invariance
(a) (b)
Figure 3.5: Joinability of operators on the β = 0 line for (a) d = 2 and (b) d = 5(each from a different perspective). State-positivity, along with channel-positivitywith respect to each of the three bipartitions, obtains the four cones depicted here.The joinability limitations for classical probability distributions are given by (a) thetetrahedron with black edges and (b) the union of the two tetrahedra with blackedges.
of the classical joinability tetrahedron are not reached by the quantum boundaries.We diagnose such limitations as strictly quantum features that do not have classicalanalogues – as we will discuss later in this work.
3.3.2 Joinability limitations from local-positivity
We now explore how local-positive joinability (a strictly weaker restriction, as noted)relates to the state/channel-joinability limitations above, as well as to the underlyingclassical limitations. As of yet, we only know that the local-positive limitations willlie between the classical and the quantum boundaries. Since obtaining a simpleanalytical characterization for arbitrary subsystem dimension d appears challengingin the local-positive setting, and useful insight may already be gained in the lowest-dimensional (qubit) setting, we focus on d = 2 in this section. Our main result iscontained in the following:
Theorem 3.3.4. With reference to Eq. (3.7), three qubit Werner operators (con-strained by local-positivity) with parameters (ηAB, ηAC , ηBC) are joinable by a local-
53
Joinability of causal and acausal relationships
Figure 3.6: Boundary of local-positive Werner operators which are joinable via local-positive operators, as described by the Roman surface, see Theorem 3.3.4.
positive tripartite Werner operator w if an only if the following conditions hold:
1 + ηAB + ηAC + ηBC ≥ 0, 1 + ηAB − ηAC − ηBC ≥ 0,
1− ηAB + ηAC − ηBC ≥ 0, 1− ηAB − ηAC + ηBC ≥ 0,
and
2ηABηACηBC − η2ABη
2AC − η2
ABη2BC − η2
ACη2BC ≥ 0.
The proof is rather lengthy and can be found in the appendix of [98]. The resultingboundary is depicted in Fig. 3.6; the shape and its determining equation is recognizedas the convex hull of the Roman surface (aka Steiner surface) [112, 128]. Comparingwith Fig. 3.5(a), we see that, still, the quantum joinability limitation arising fromfrom local-positivity is stricter than the corresponding classical one. However, it iscloser to the classical limitations than the state/channel-positive limitations obtainedin the previous section for d = 2. To shed light on the cause of the quantum bound-ary here, we can explicitly construct a product-state projector, whose probabilitywould be negative if joinability outside of this shape were allowed. The family ofjoining states w that we need to consider (see appendix of [98] for details) may beparameterized in terms of the bipartite reduced state Werner parameters as
w(ηAB, ηAC , ηBC) =1
8I +
ηAB4
(VAB − I/2) +ηAC
4(VAC − I/2) +
ηBC4
(VBC − I/2).
54
3.3 Three-party joinability settings with collective invariance
Consider, in particular, the following state on A-B-C:
|ψ〉 =
[10
]⊗[
cos 2π/3sin 2π/3
]⊗[
cos 4π/3sin 4π/3
], (3.9)
which corresponds to the pure product state with the local Bloch vectors as anti-parallel with one another as possible. Computing its expectation with respect tow(ηAB = ηAC = ηBC ≡ η), the largest value of η that admits a non-negative valueis η = 2/3. Hence, local-positivity limits the simultaneous joining of these Werneroperators to a maximum of η = 2/3. The operational interpretation of this resultdeserves attention. Consider a local projective measurement made on each of threequbit systems. Furthermore, consider the three systems to have a collective unitarysymmetry, in the sense that there are no preferred local bases. In our general picture,where local positivity is considered, the systems need not be three distinct systems– they may also be the same system at two different points in time. Local positivityenforces the rule that “all probabilities arising from such measurements must be non-negative”. In the example above (i.e. ηAB = ηAC = ηBC), this rule implies thatthe three equal correlations (as measured by the ηs) can never exceed 2/3. As thisexample and Fig. 3.6 show, local-positivity enforces joinability limitations more strictthan those of classical probability theory. Notwithstanding, these limitations reflectthe same symmetry as the classical limitations do, namely, symmetry with respectto individually inverting two axes. The state-joining and channel-joining scenariosreflected a preference towards the negative axis (anticorrelation) and the positive axis(correlation) of the ηs, respectively.
Before concluding this section, we connect the above discussion to the relationshipbetween local-positivity and separability. As mentioned earlier, the cone of localpositive operators and the cone of separable operators are dual to one another. Theoperator subspace we are dealing with is spanned by the orthonormal operators 1√
8I,
1√6(VAB − I/2), 1√
6(VAC − I/2), and 1√
6(VBC − I/2) with coordinates 1√
8,√
38ηAB,√
38ηAC , and
√38ηBC , respectively. In Theorem 3.3.4, we determined the algebraic
surface bounding the local positive operators; hence, the dual to this surface willbound the separable operators within this space. The dual to the Roman surface isknown as the Cayley’s cubic surface [129], which, for a given scale parameter w ischaracterized by ∣∣∣∣∣∣
w x yx w zy z w
∣∣∣∣∣∣ = 0.
We first set x =√
38ηAB, y =
√38ηAC , and z =
√38ηBC . Then we must set w so that
55
Joinability of causal and acausal relationships
the Cayley surface delimits the separable states. For each extremal separable state inour space, there is a corresponding local-positive operator acting as an entanglementwitness; a state is separable if the inner product with its entanglement witness isnonnegative.
Consider the extremal local-positive operator ηAB = ηAC = ηBC = 2/3 that wemade use of previously. This operator will act as an entanglement witness for anotheroperator with ηAB = ηAC = ηBC = σ. We obtain σ by solving
1√8√38
23√
38
23√
38
23
·
1√8√38σ√
38σ√
38σ
= 0,
to arrive at σ = −16. With this, the only value of w allowing the Cayley surface
to be solved by σ = −16is w = 1√
24. Setting the scaling value and evaluating the
determinant, we find that the separable states are bound by the surface
1 + 54ηABηACηBC − 9(ηAB + ηAC + ηBC)2
+18(ηABηAC + ηABηBC + ηACηBC) ≥ 0.
This inequality may also be obtained using Theorem 1 in [82]. The shape of theseparable states is depicted in Figure 3.7. Several remarks may be made. First,the set of separable states exhibits the tetrahedral symmetry shared by the classicaljoinability boundary and the local-positive joinability boundary. Thus, among thevarious boundaries we have considered in this three dimensional Euclidean space,the state- and channel-positive boundaries are the only ones not obeying tetrahedralsymmetry. However, both the convex hull and the intersection of the state- andchannel-positive cones bound regions which recover this symmetry. It is a curiousobservation that the convex hull of these cones is “nearly” the local-positive region,while the intersection is “nearly” the set of separable states. Earlier we found, inthe two-qudit case, that local-positivity coincides with the union of the state- andchannel-positive regions, as well as that the separable region was their intersection.Here we consider the analog for three qubits. The result is that i) the convex hull ofstate- and channel- positive operators is strictly contained in the set of local-positiveoperators; and ii) the intersection of the state- and channel-positive operators isstrictly contained in the set of separable states.
We may further interpret the latter result in terms of PPT considerations. Theoperators which result from a homocorrelation-mapped channel necessarily have PPT.Corollary 1 in Ref. [82] states that the PPT and bi-separable Werner operators co-
56
3.4 Agreement bounds for quantum states and channels
(a) (b)
Figure 3.7: (a) Set of separable operators within the set of local-positive operators.(b) Intersection of the state- and channel-positive cones within the set of local-positiveoperators. While the separable operators are a subset of the intersection set, the twoobjects coincide (only) at their vertices. In panel (a) the closest point in the separableset to the (−1,−1,−1) corner of the figure is (−1/6,−1/6,−1/6), whereas in panel(b), the closest point in the intersection set is (−1/5,−1/5,−1/5).
incide. Thus, any state-positive operator which is also a homocorrelation mappedchannel is necessarily bi-separable. Hence, the intersection of the four cones will bethe set of states which are bi-separable with respect to any of the three partitions.This set is clearly contained in the set of tri-separable states. These observations il-luminate the relationships among entanglement, quantum states, and quantum chan-nels. Specifically, the homocorrelation map allows us to place quantum channels inthe same arena as quantum states, and hence to directly compare and contrast them.Finding that the tri-separable operators are a proper subset of the bi-separable ones,we wonder what features these strictly bi-separable operators possess, and what doesbi-separability imply for the states or channels supporting such correlations.
3.4 Agreement bounds for quantum states and chan-nels
In what remains, we illustrate some crucial differences between channel- and state-positive operators. These differences inform the nature of their respective joinabilitylimitations. In order to directly compare states to channels we restrict our consid-erations here to operators in B(Hd ⊗ Hd). Qualitatively, state-positive operatorsare restricted in the degree to which they can support agreeing outcomes, whereas
57
Joinability of causal and acausal relationships
channel-positive operators are restricted in the degree to which they can supportdisagreeing outcomes. We define the degree of agreement to be the likelihood ofa certain POVM element. Specifically, consider a local projective measurementM = |ij〉〈ij|. We can coarse-grain this into a two-element projective measure-ment with the bipartition into “agreeing” outcomes, EA =
∑i |ii〉〈ii|, and “disagree-
ing” outcomes, ED =∑
i 6=j |ij〉〈ij|, respectively. Lastly, so that these outcomes arebasis-independent, we can “twirl” EA and ED as follows:
EA =
∫dµ(U)U ⊗ U
(∑i
|ii〉〈ii|)U † ⊗ U †
ED =
∫dµ(U)U ⊗ U
(∑i 6=j
|ij〉〈ij|)U † ⊗ U †,
where dµ(U) denotes integration with respect to the invariant (Haar) measure. It issimple to see that these two operators yield a resolution of identity and hence form aPOVM. We can compute these two operators explicitly as follows. By the invarianceof the Haar measure, we can rewrite EA as
EA = d
∫dµ(ψ)|ψ〉〈ψ|⊗2,
for which the above integral is proportional to the projector onto (or identity operatorI+2 in) the totally symmetric subspace H⊗2
+ ⊂ H⊗2 [130]. Explicitly, we can write
EA =d
d+2
I+2 =
d
d+2
I + V
2, d+
2 ≡ dim(H+2 ) =
(2 + d− 1
2
), (3.10)
ED = I− EA. (3.11)
We define the degree of agreement to be the likelihood of EA and, similarly, the degreeof disagreement to be the likelihood of ED. Operationally, these values are the prob-ability that, for a randomly chosen local projective measurement made collectively,the local outcomes will agree or, respectively, disagree.
We now proceed to show how quantum channels differ from quantum states in theirallowed range of agreement likelihood. In the case of a bipartite operator ρ ∈ B(H⊗H), we are familiar with computing this agreement probability as Tr(EAρ). To carryout the same computation for a channel operator, the homocorrelation map becomesexpedient. Given a quantum channelM : B(H) → B(H), we wish to determine theprobability that the outcome of a randomly chosen projective measurement (made onthe completely mixed state) will agree with the outcome of the same measurementafter the application of M. Assume the outcome was |i〉 from an orthogonal basis
58
3.4 Agreement bounds for quantum states and channels
|i〉. Then the post-channel state is M(|i〉〈i|), and the likelihood that the post-channel measurement will also be |i〉 is 〈i|M(|i〉〈i|)|i〉. Lastly, if we want to averagethis likelihood of agreement over all choices of basis we integrate,
p(agree) =∫dµ(U)Tr
(M(U |i〉〈i|U †)U |i〉〈i|U †
)=∫dµ(ψ)Tr
(M(|ψ〉〈ψ|)|ψ〉〈ψ|
).
If we wish to find the bounds on this value, the above form does not make transparentthe fact that we are performing an optimization problem in a convex cone. But,recalling the namesake property of the homocorrelation map, Eq. (3.3), the aboveexpression may be rewritten as
p(agree) = Tr
[H(M)d
∫dµ(ψ)|ψ〉〈ψ| ⊗ |ψ〉〈ψ|
]= Tr[H(M)EA].
Accordingly, the likelihood of agreement is calculated for channel operators in thehomocorrelation representation in just the same way as it is for bipartite densityoperators. With the stage set, the desired bounds are described in the followingtheorem:
Theorem 3.4.1. Let w be an operator in B(Hd ⊗Hd), and consider a POVM withoperation elements EA, ED as in Eqs. (3.10)-(3.11). Then the degree of agreementfor w ≥st 0 as calculated by the likelihood of EA is bounded by
0 ≤ Tr(wEA) ≤ 2
2 + d− 1, (3.12)
while the degree of agreement for w ≥ch 0 is bounded by
1
2 + d− 1≤ Tr(wEA) ≤ 1. (3.13)
Proof. In the case of state-positive operators, the maximal value of Tr(wEA) isachieved by setting w = EA/Tr(EA), which results in Tr(wEA) = 2/(d+ 1). Forthe lower bound, it is simple to see that choosing w to lie in the complement of theprojector yields a value of zero. Hence, we have obtained the bound of Eq. (3.12).
In the case of channel-positive operators, the value of Tr(wEA), where w ≥ch 0, isunchanged by a partial transposition of both operators. Thus, we may seek boundson the value of Tr(wTAE
TAA ), where wTA is a density operator. By using Eq. (3.10),
59
Joinability of causal and acausal relationships
the partial transposition of EA is
ETAA =
d
d+2
I + V TA
2.
Thus, the upper and lower bounds on Tr(wEA) are achieved by setting wTA = V TA/dand wTA = (I − V TA)/(d2 − d), respectively. Accordingly, the resulting bounds aredd+2
12≤ Tr(wEA) ≤ d
d+2
1+d2, which simplify to those of Eq. (3.13).
By virtue of the homocorrelation map, the above result may be understood geo-metrically. The objects involved are the agreement/disagreement POVM operatorsEA and ED, and the state- and channel- positive cones Wst and Wch, respectively.Theorem 3.4.1 places an upper bound on the inner product between vectors in Wst
and EA, and, similarly, on the inner product between vectors in Wch and ED. Thisgeometric understanding is aided by the example of Werner operators shown in Fig.3.2.
Lastly, we proceed to show that general joinability limitations (though not strictones) can be derived based solely on i) the above agreement bounds of channelsand states; ii) joinability bounds of classical probabilities; and iii) the fact that theagreement likelihoods must obey rules of classical joinability. Ultimately, the reducedstates must satisfy certain limitations arising from joining limitations of classicalprobability distributions. In the three-party joining scenario, the bipartite marginaldistributions of three classical d-nary random variables must have probabilities ofagreement αAB, αAC , and αBC satisfying the following inequalities [61]:
Since Tr(wEA) is a probability of agreement, it too is subject to the above con-straints. Hence, we identify Tr(ρiEA) ≡ α`, where ` = AB, AC, or BC. Considerthe case where systems B-C are state-positive. Theorem 3.4.1 then sets the boundTr(ρBCEA) ≤ 2
d+1. Setting the parameter αBC = Tr(ρBCEA) to this upper limit of
2d+1
, Eq. (3.16) becomes
αAB + αAC ≤d+ 3
d+ 1.
60
3.4 Agreement bounds for quantum states and channels
In the case of αAB = αAC ≡ α, this yields
α ≤ d+ 3
2(d+ 1),
which corresponds precisely to the optimal bound for qudit cloning [131] (cf. Eq.(21) therein, where their F coincides with our α). We can similarly recover the exactbound for the 1-2 sharability of qubit Werner states determined in [61]. Again, weset the B-C agreement to its extremal value Tr(ρBCEA) = 2
d+1, as given by Theorem
3.4.1. For d = 2, Eq. (3.17) applies, and substituting in the extremal value of αBCwe obtain αAB + αAC ≥ 1
3. Again, in the case of αAB = αAC ≡ α, this yields α ≤ 1
6,
which is the exact condition for 1-2 sharability of Werner qubits.While obtaining a full generalization of Theorem 3.4.1 to multiparty systems would
entail a detailed understanding of representation theory for Brauer algebras which isbeyond our current purpose, we can nevertheless establish the following:
Theorem 3.4.2. Let w ∈ B(H⊗Nd ), and consider a POVM with operation elementsEA = d
d+NI+N , ED = I − EA (analogous to Eqs.(3.10)-(3.11)). Then the degree of
agreement for w ≥st 0 as calculated by the likelihood of EA is bounded by
0 ≤ Tr(wEA) ≤ d(d−1+NN
) . (3.18)
Proof. The maximal and minimal values of Tr(wEA) are achieved by setting w =EA/Tr(EA) and w = (dI/d+
N − EA)/Tr(dI/d+N − EA), respectively, which yields the
desired bounds of Eq. (3.18).
From the above multiparty bound, one may attempt to recover, for instance, theknown bounds on 1-n sharability of Werner states [61]. However, thus far we havenot been successful in this endeavor. In the tripartite qudit setting, such boundswere found to be sufficient, but this might be a special feature of this particularcase. Therefore, it remains an open question to determine whether there exists asimple principle (or simple principles) which govern joinability limitations beyondthe tripartite setting.
61
Chapter 4
Towards an alternative approach tojoinability: enforcing positivitythrough purification
63
Towards an alternative approach to joinability: enforcing positivitythrough purification
4.1 Introduction
The aim of this chapter is to follow up on some of the ideas brought up in theprevious two chapters and also to draw further connections using the concept ofjoinability. We develop a new approach to the joinability problem which allows usto formalize the “composition law” that was touched upon in the previous chapter,to easily incorporate the concept of measurement incompatibility into the joinabilityframework, and to find a simple principle that governs joinability failures in boththe classical and quantum cases. The techniques used in this chapter are ratherelementary. Yet, to our knowledge, they have not been applied to the problem ofjoinability. Surprisingly, with basic linear algebra, we unveil important connectionsamong principles that govern the structure of quantum relationships. We emphasizethat the ideas presented in this chapter are just sprouting and that the purpose ofpresenting them is to provide an environment in which they can grow.
In the previous chapters, failures of joinability were due to a failure of positive-semidefiniteness or complete positivity. In this chapter we make positivity manifestby purifying the quantum state with an ancillary system or by “lifting” the quantumchannel to an isometric description in a larger Hilbert space. In this way, failures ofjoinability are only due to failures of trace-normalization, such as the conclusion thatthe likelihood of every possible event is zero. We emphasize that the motivation forpursuing the ideas in this chapter are mostly theoretical, urged by the desire to betterunderstand the fundamental concepts of monogamy of entanglement, no-cloning, andmeasurement incompatibility.
Previously, we presented the following explanation for the fact that qubit pairsA-B and A-C could not both be in the singlet state. The singlet state exhibits per-fectly disagreeing measurement outcomes for the same spin measurement made onboth qubits. Assuming a singlet state were shared between A-B and A-C, we can“compose” the disagreements to conclude that the same spin measurement made onB-C must produce perfectly agreeing outcomes. However, no valid quantum state(i.e. no positive-semidefinite trace-one operator) can produce such perfectly agreeingoutcomes for all collective spin measurements. Thus, we have that the A-B disagree-ment and the A-C disagreement force the contradictory B-C agreement. A similarargument can explain the no-cloning result.
The initial motivation for the ideas in this chapter had been to further explorethe idea of “composing” two conditions, such as the A-B disagreement and the A-Cdisagreement, to obtain a third condition. Such an argument was made to work inthe singlet state sharing example and to obtain the no-cloning principle. But, it is notclear how it could be properly generalized. Furthermore, could such an approach beextended beyond the three-body joinability scenario? In the spirit of generalized en-tanglement [132], could such an approach be applied to general consistency problems
64
4.1 Introduction
which do not admit a natural tensor product structure as in fermionic systems?Before discussing our new approach to joinability we establish the nature of the
constraints that are considered. We consider constraints to be on the manifestlypositive objects such as the purification of the quantum state or the isometry of thequantum channel. We briefly review the construction of these objects.
Given a density operator ρ ∈ B(HA), there always exists a quantum state |ψ〉 ∈H1A ⊗ H2
A such that Tr2 (|ψ〉〈ψ|) = ρ. This is easily verified. ρ ≥ 0 implies √ρ =√ρ† is well-defined. Then |ψ〉 = (
√ρ ⊗ I)|Ω〉 satisfies the partial trace condition,
where |Ω〉 =∑
i |ii〉 is the unnormalized maximally entangled state. The state |ψ〉is unique up to a norm-preserving linear map, or isometry, acting on the ancillarysystem. A similar argument can be used to show that, given a quantum channelE : B(HA) → B(HB), there exists an isometry V : H1
A → H1B ⊗ H2
B ⊗ H3A such
that Tr23
(V · V †
)= E(·). Here, trace-preservation of E corresponds to the isometry
condition on V . The isometry is defined up to the action of another isometry actingon the traced-out ancillary systems.
We consider two types of constraints that we refer to as hard constraints and softconstraints. Hard constraints express knowledge of statistical impossibilities, whilesoft constraints bound the likelihoods of various outcomes. We represent a hardconstraint on a quantum state as a set of linear equations satisfied by its purification:
(〈φ|A ⊗ IB)|ψ〉AB = 0, ∀ |φ〉 ∈ SA ⊆ HA, (4.1)
where SA is the set of states involved in the hard constraint. A hard constraint on aquantum channel is, similarly, represented by a set of linear equations satisfied by itsisometry:
(〈φ|AB ⊗ IC)(IA ⊗ VA→BC)|Ω〉AA = 0, ∀ |φ〉AB ∈ SAB ⊆ HA ⊗HB, (4.2)
where |Ω〉AA =∑
i |ii〉 is the unnormalized maximally entangled state and SAB is,again, the set of vectors used to subject |ψ〉 to the hard constraint.
As a simple example, consider a system of three qubitsHA⊗HB⊗HC and constrainthe density matrix of this system to have reduced state AB described by the (anti-symmetric) singlet state |ψ−〉 = 1√
2(|01〉 − |10〉). Denote the space of symmetric
states on AB as S2(H). This is the space orthogonal to the singlet state. Hence, thecondition that AB is described by the singlet state is expressed by the hard constraint
(〈φ|AB ⊗ ICD)|ψ〉ABCD = 0, ∀ |φ〉 ∈ S2(H) (4.3)
Standing alone, this means of expressing the AB-singlet condition is not very illu-minating. The value of this rephrasing is revealed when we consider multiple hardconstraints in tandem. As we show in the following section, we can recover a number
65
Towards an alternative approach to joinability: enforcing positivitythrough purification
of basic quantum “no-go” principles as stemming from an over-constraining of hardconstraints.
Soft constraints bound the likelihoods of certain projective measurement out-comes. We represent a soft constraint on a quantum state as a set of inequalitiessatisfied by its purification:
‖(〈φ|A ⊗ IB)|ψ〉AB‖2 ≤ ε|| |φ〉||2|| |ψ〉||2, ∀ |φ〉 ∈ SA ⊆ HA, (4.4)
where ε represents the likelihood bound associated to the set of states SA. A softconstraint on a quantum channel is, similarly, represented by a set of inequalitiessatisfied by its isometry:
‖(〈φ|AB ⊗ IC)(IA ⊗ VA→BC)|Ω〉‖ ≤ ε|| |φ〉||2|| (IA ⊗ VA→BC)|Ω〉||2,∀ |φ〉AB ∈ SAB ⊆ HA ⊗HB. (4.5)
4.2 Joinability limitations from hard constraints
In this section we explore joinability where the partial descriptions of the system aregiven by hard constraints. We begin with a trivial example from classical probabilitytheory. Consider the simplest classical system, the probabilistic bit. The probabilitydistribution is given by the likelihoods p(0) and p(1). An example of inconsistenthard constraints is p(0) = 0 and p(1) = 0. No distribution on outcomes 0 and 1 cansatisfy these constraints while also satisfying a non-trivial normalization condition.
This example seems too trivial to bear insight. However, we will argue that allinconsistent sets of hard constraints fail to be consistent for the same reason as theabove example fails: the likelihood of any outcome is zero. Surprisingly, this samereasoning explains the non-sharability of maximally entangled states, the no-cloningprinciple, the incompatibility of certain observables, as well as the simple classicalprinciple that three bits cannot disagree.
Consider three classical probabilistic bits A, B, and C with the constraints, A andB disagree, B and C disagree, and A and C disagree. These conditions are equivalentto the three hard constraints, A and B cannot agree, B and C cannot agree, and A
66
4.2 Joinability limitations from hard constraints
and C cannot agree. These three conditions exclude all possible configurations:
(4.6)The constraints imply that the likelihood of any outcome is zero. Therefore, nodistribution can satisfy these three conditions.
Consider a single qubit S with the constraints “|0〉 is certain upon making a Pauli-Z measurement” and “|+〉 is certain upon making a Pauli-X measurement”. We cantranslate each into a hard constraint on the purification of the density operator to anancillary system A: 〈1|S ⊗ IA|ψSA〉 = 0 and 〈−|S ⊗ IA|ψSA〉 = 0. Any state vector|φ〉 ∈ HS can be written as some linear combination |φ〉 = α|1〉 + β|−〉. Hence, forany |φ〉 ∈ HS , there exists α and β such that
Only |ψSA〉 = 0 satisfies 〈φ| ⊗ IA|ψSA〉 = 0 for all |φ〉. Thus, no non-trivial state cansatisfy both constraints.
It seems that we have taken a simple fact of quantummechanics (i.e. uncertainty ofZ and X measurements cannot both be zero) and given an unfamiliar or cumbersomeargument for reaching it. However, a failure of the trace-one condition seems to bemore direct or basic than one involving algebraic properties of observables, especiallywhen the example is more complicated than the one above.
Next, we use the concept of hard constraints to easily prove a fact used in theprevious chapter: there is no bipartite quantum state that exhibits perfectly agreeingoutcomes for all collective projective measurements. Considering the system to bedescribed by the Hilbert space H = Cd ⊗ Cd we can rephrase the condition in termsof hard constraints as,
Since the set of vectors |φ〉 ⊗ |φ⊥〉 span Cd ⊗ Cd (this can be taken as an exercise forthe enthusiastic reader), there is no non-trivial solution to the set of hard constraints,verifying the statement. Note that we did not have to make any reference to apositive-semidefinite constraint as we did while making this argument in the previous
67
Towards an alternative approach to joinability: enforcing positivitythrough purification
chapter. The key insight was to phrase the condition in terms of a set of negativestatements, or impossibilities, as opposed to a number of positive expectation values.
With a parallel argument we can show that no quantum channel can map everyinput pure state to an orthogonal pure state. Considering the quantum channel tobe described by an isometry V : Cd → Cd ⊗ HA we rephrase the condition in termsof hard constraints as follows. The condition states that the output of the channel,TrA
(V |φ〉〈φ|V †
), should be orthogonal to |φ〉, or, equivalently, 〈φ|V |φ〉 = 0 for all
|φ〉 ∈ Cd. Defining 〈φT | ≡ T (|φ〉), and noting that |φ〉 = (〈φT | ⊗ I)|Ω〉, we write thehard constraints as
Then, because span〈φT | ⊗ 〈φ| = (Cd⊗Cd)† (this, too, can be taken as an exercise),there is no non-trivial solution for V .
These two previous observations address the quantum distinction between causaland acausal relationships. There is certainly a one-to-one correspondence betweenthe vectors |φ〉 ⊗ |φ〉 and matrices |φ〉 ⊗ 〈φ|. The former linear objects inhabit thespace H ⊗ H, which corresponds to acausal quantum relationships, while the latterinhabit the space B(H) ∼= H⊗H∗, which corresponds to causal quantum relationships.Despite the one-to-one correspondence, the span of the |φ〉 ⊗ |φ〉 is a strict subspaceof H ⊗ H, while the span of the |φ〉 ⊗ 〈φ| is equal to H ⊗ H. In particular thevectors |φ〉 ⊗ |φ〉 ∈ Cd ⊗ Cd only span a d(d + 1)/2-dimensional space, while the setof |φ〉〈φ| ∈ Cd ⊗ Cd† spans Cd ⊗ Cd†.
Next we consider the impossibility of having entanglement between both A-B andB-C. We return to the three-qubit system with the constraint that A-B and B-Care each described by the singlet state |ψ−〉 = 1√
2(|01〉 − |10〉). We can rephrase this
in terms of a set of hard constraints on the purified state |ψ〉ABCD,
and similarly for the BC system. Since span|φφτ〉, |τφφ〉 = (C2)⊗3, again, there isno non-trivial solution. Thus, we can view the impossibility of singlet sharing to bedue to the constraints having ensured zero likelihood for any measurement outcome.In this sense, the singlet sharing is impossible for the same reason that three bitsdisagreeing is impossible.
For completeness, we prove the no-cloning theorem using the hard constraintapproach. Consider a quantum channel from system A to BC, where HA = HB =HC ≡ H, represented by the isometry V : HA → HB ⊗ HC ⊗ HD. We can phrasethe cloning condition as a negative statement by requiring that the output of thechannel applied to |φ〉 be orthogonal to span(|φ⊥〉)⊗HC and HB⊗ span(|φ⊥〉), where
along with the analogous statement from switching the roles of B and C. Sincespan〈φTφτ |, 〈φT τφ| = H† ⊗H† ⊗H†, no non-trivial solution exists for V . As withthe no singlet sharing example, we can explain the impossibility of cloning as due tothe set of constraints ruling out all possibilities.
As a final example, we consider the incompatibility of measuring devices. Aquantum measurement can be described by a quantum channel from the measuredsystem A to a classical measuring device D. The measuring device is made “classical”by requiring that operators appearing on B(HD) are restricted to being diagonalin a particular basis. Equivalently, operators on D must be in the commutativealgebra C⊕d, where d is the number of outcomes. As with any quantum channel, suchmeasurement channels admit an isometric representation V : HA → HD ⊗ HE forsome ancillary system E. For a complete projective measurement on HA = Cd withoutcomes corresponding to the basis of vectors |j〉 ∈ HA, an isometric representationof the channel is V =
∑j |jj〉〈j|, where we have let the |j〉 also correspond to an
orthonormal basis of vectors in HD and in HE.
With this formalism in place, we consider the case of having a single system bemeasured by two different measurement devices simultaneously. Following [133], wesay that the two measurements are compatible if there exists a single measurementdevice from which the two measurements can be obtained via partial trace of theoutputs. Note that this is equivalent to the definition of channel joinability. Re-stricting to the qubit case, consider an X and a Z measurement device, from thequantum system A to the devices DX and DZ . We label the basis vectors of theoutput systems as |+〉, |−〉 and |0〉, |1〉, respectively, such that the labels corre-spond to the states of the input system. We can ensure the X and Z measurementconditions by requiring that, with input |±〉, the output on DX must be orthogonalto |∓〉 and that, with input |0/1〉, the output on DZ must be orthogonal to |1/0〉(the “/” indicates “or”). Letting V : HA → HDX ⊗ HDZ ⊗ HE be the isometryof the quantum measurement, the hard constraints are (〈∓| ⊗ 〈φ| ⊗ IE)V |±〉 = 0for all |φ〉 ∈ HDZ and (〈φ| ⊗ 〈1/0| ⊗ IE)V |0/1〉 = 0 for all |φ〉 ∈ HDX . Sincespan〈∓|⊗〈φ|⊗ |±〉, 〈φ|⊗〈1/0|⊗ |0/1〉 = H†DX ⊗H
†DZ⊗HA, no non-trivial solution
exists.
Thus, we have shown that many of the important quantum no-go principles, in-cluding the impossibility of singlet sharing, no-cloning, and measurement incompati-bility can be explained with the same argument used to show p(0) = 0 and p(1) = 0are incompatible. The approach used is not restricted to needing a tensor productstructure. Therefore, it would be interesting to explore these ideas for fermionic or
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Towards an alternative approach to joinability: enforcing positivitythrough purification
bosonic systems, making better connection, for instance, with the N -representabilityproblem [134, 28, 27].
4.3 Joinability limitations from soft constraints
We now turn to the case where, instead of ruling out any possibilities, we simplyplace bounds on their likelihoods. As mentioned, the motivation for the approachto joinability, developed in this chapter, was to elucidate the composition rule thatwe previously used to show the impossibility of singlet sharing. The first examplereturns to the singlet sharing scenario.
Consider a tripartite system of qubits ABC. We aim to show that the correla-tions of AB in tandem with the correlations of BC ensure certain correlations on AC.For instance, we used the “perfect disagreement” of the singlet state to ensure thatthe singlet describing AB and BC ensures the correlations on AC to be perfectlyagreeing. After much effort towards trying to develop and generalize this “compo-sition observation”, we have arrived at a simple idea which makes this possible. Akey insight, which was conveyed in the previous section, is the utility of expressingconstraints in terms of negative statements. There, instead of formalizing the com-position (disagree)AB + (disagree)BC ⇒ (agree)AC , we found it simpler to formalizethe composition (not agree)AB + (not agree)BC ⇒ (not disagree)AC as represented byhard constraints. Then, we expect the corresponding composition rule in the soft con-straint case to be (low agreement)AB + (low agreement)BC ⇒ (low disagreement)AC .
We enforce some degree of singlet correlation on a pair of subsystems by upperbounding the inner product of the state with the set of |φφ〉, which span the symmetricsubspace. Assume that, for the purified state |ψ〉 on ABCD, the soft constraint onAB is given by
‖(〈φφτ | ⊗ ID)|ψ〉‖2 ≤ αAB, ∀|φ〉, |τ〉 ∈ C2 (4.12)
while the soft constraint on BC is given by
‖(〈τφφ| ⊗ ID)|ψ〉‖2 ≤ αBC , ∀|φ〉, |τ〉 ∈ C2. (4.13)
We expect to be able to bound the likelihood of “disagreeing” outcomes on AC, suchas |φ〉⊗ |φ⊥〉, where the two states are orthogonal. Consider, then, the expression forthe likelihoods on AC, ∥∥(〈φτφ⊥| ⊗ ID)|ψ〉
∥∥2. (4.14)
Since the Hilbert space of system B is two-dimensional, an arbitrary vector |τ〉B canbe written as a linear combination |τ〉 = λ|φ〉 + µ|φ⊥〉. Making this replacement in
70
4.3 Joinability limitations from soft constraints
the likelihood expression, we obtain∥∥(λ〈φφφ⊥| ⊗ ID + µ〈φφ⊥φ⊥| ⊗ ID)|ψ〉∥∥2. (4.15)
We can bound the value of this expression using the triangle inequality,∥∥(〈φτφ⊥| ⊗ ID)|ψ〉∥∥2 ≤
∥∥(λ〈φφφ⊥| ⊗ ID)|ψ〉∥∥2
+∥∥(µ〈φφ⊥φ⊥| ⊗ ID)|ψ〉
∥∥2. (4.16)
Each term of the right-hand side of the inequality is the form of the constraints weplace on AB and BC. Letting the maximum value of
∥∥(〈φτφ⊥| ⊗ ID)|ψ〉∥∥2 be δAC
and noting that |λ|, |µ| ≤ 1, we then obtain the monogamy-like inequality
δAC ≤ αAB + αBC . (4.17)
This inequality captures how an “agreement-bound” on AB and on BC (i.e. αAB andαBC) enforce a “disagreement-bound” on AC (i.e. δAC). Note that the hard-constraintcase is recovered by setting αAB = αBC = 0, which forces δAC = 0 (an impossibility).Also, using the fact that span|φφ⊥〉 = C2 ⊗ C2, there is no vector which can beorthogonal to all |φφ⊥〉. Thus, there is a lower bound to the value of δAC , which canbe computed to be 1
2(show as an exercise). This expresses a “trade-off” in the values
that αAB and αBC may take,1
2≤ αAB + αBC . (4.18)
This inequality gives a necessary condition for determining if the soft constraintsare consistent. It does not diagnose all inconsistent sets of constraints. This is,in part, due to the fact that the inequality is weakened by upper-bounding λ andµ. In future work, we hope to properly compare these observations to our previousWerner joinability findings. We only briefly make a few observations. First, applyingthe inequality in Eq. (4.18) to Werner states gives a linear trade-off for the Wernerparameters (see Chapters 2 and 3 for various parameterizations). We found, however,that the exact trade-off between the Werner parameters for AB and BC is quadratic(see Eq. (2.22)).
We can now understand another opportunity afforded by phrasing the constraintsas negative statements (or upper bounds on likelihoods): the upper-bounds of theconstraints allow us to use the triangle inequality to upper bound the derived con-straint. If instead, we had used lower-bounds, the direction of these inequalities wouldhave opposed the direction of the triangle inequality.
The singlet state is not particular with respect to monogamy constraints. Weexpect that other Bell states should lend themselves to similar composition laws.Sticking to the qubit case, each maximally entangled state is obtained by applyinga particular unitary transformation to, without loss of generality, system B: |ψU〉 =
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Towards an alternative approach to joinability: enforcing positivitythrough purification
(I ⊗ U)|ψ−〉. With this, the hard constraint that ensures two qubits are describedby |ψU〉 is (〈φφ| ⊗ IC)(IA ⊗ U †B ⊗ IC)|ψ〉ABC = 0 for all |φ〉 ∈ C2. This is a simplemodification of the singlet state hard constraints.
Accordingly, the soft constraints which ensure a state is “close to” the state |ψU〉are stated as ∥∥∥(〈φφ| ⊗ IC)(IA ⊗ U †B ⊗ IC)|ψ〉ABC
∥∥∥2
≤ αUAB, ∀ |φ〉 ∈ C2. (4.19)
Consider the three-party joining scenario, let the above soft constraint apply to AB(where we purify the quantum state using an ancilla system D). Let the followingsoft constraint apply to BC,∥∥∥(〈τφφ| ⊗ ID)(IAB ⊗ V †C ⊗ ID)|ψ〉ABCD
∥∥∥2
≤ αVBC , ∀ |φ〉 ∈ C2. (4.20)
We follow the reasoning of the singlet state example. Consider the following expressionfor certain likelihoods on AC∥∥(〈φτφ⊥| ⊗ ID)(IAB ⊗ (U †V †)C ⊗ ID)|ψ〉
∥∥2. (4.21)
Now, we can write |τ〉 = λU |φ〉+ µU |φ⊥〉 for any |τ〉, giving∥∥∥(λ〈φφφ⊥| ⊗ ID + µ〈φφ⊥φ⊥| ⊗ ID)(IA ⊗ U †B ⊗ (U †V †)C ⊗ ID)|ψ〉∥∥∥2
. (4.22)
We can bound the value of this expression using the triangle inequality,∥∥(〈φτφ⊥| ⊗ ID)(IAB ⊗ (U †V †)C ⊗ ID)|ψ〉∥∥2 (4.23)
≤∥∥∥(λ〈φφφ⊥| ⊗ ID)(IA ⊗ U †B ⊗ (U †V †)C ⊗ ID)|ψ〉
∥∥∥2
+∥∥∥(µ〈φφ⊥φ⊥| ⊗ ID)(IA ⊗ U †B ⊗ (U †V †)C ⊗ ID)|ψ〉
∥∥∥2
. (4.24)
By setting |τ ′〉 = V U |φ⊥〉 and |φ′〉 = U |φ⊥〉, we simplify the inequality∥∥(〈φτφ⊥| ⊗ ID)(IAB ⊗ (U †V †)C ⊗ ID)|ψ〉∥∥2 (4.25)
≤∥∥∥(λ〈φφτ | ⊗ ID)(IA ⊗ U †B ⊗ ICD)|ψ〉
∥∥∥2
+∥∥∥(µ〈φφ′φ′| ⊗ ID)(IAB ⊗ V †C ⊗ ID)|ψ〉
∥∥∥2
. (4.26)
Each term on the right-hand side of the inequality can be recognized as the expressions
72
4.3 Joinability limitations from soft constraints
in the soft constraints on AB and BC, respectively. Define
δV UAC ≡ max|φ〉,|τ〉
∥∥(〈φτφ⊥| ⊗ ID)(IAB ⊗ (U †V †)C ⊗ ID)|ψ〉∥∥2. (4.27)
Using |λ|, |µ| ≤ 1, we obtain the inequality
δV UAC ≤ αUAB + αVBC . (4.28)
This generalized composition rule simplifies to the singlet state case by setting U =V = I. This inequality demonstrates how, by ensuring certain quantum correlationsfor AB and for BC, certain correlations are forced upon AC. We parameterized themaximally entangled qubit states with a unitary transformation. This shows that, ifAB is close to the maximally entangled state |ψU〉 and BC is close to the maximallyentangled state |ψV 〉, then the likelihood of any measurement outcome |φ〉 ⊗ V U |φ⊥〉is upper bounded. It would be interesting to explore further generalizations of theseobservations to qudits and to more than three systems.
The above “composition rule” inequality does not, on its own, tell us when theinitial soft constraints, themselves, are inconsistent with one another. As with thesinglet example, we must further evaluate or place a lower bound on the disagreementparameter δAC , in order to obtain necessary conditions on the consistency of the softconstraints. We have yet to provide a general intuition behind joinability failures inthe soft-constraint case. For instance, it would be useful to directly obtain an inequal-ity involving αAB and αBC , or involving a general set of soft constraint parameters εi.We outline an approach to obtaining such inequalities and show that joinability fail-ures, in this case too, are on account of failures of the trace-normalization condition.Thus, we can view this cause of joinability failures as being a more general version ofthe “zero-total-probability” explanation in the hard-constraint case.
As with the quantum case, for a classical probability distribution the sum ofall likelihoods must be 1. Considering the classical soft constraints of p(0) ≤ ε andp(1) ≤ δ, the constraints are inconsistent unless 1 ≤ ε+δ. A simple quantum exampleobtains analogous inconsistency bounds. Consider a qubit system A (with ancilla B)subject to the soft constraints ‖(〈+| ⊗ I)|ψ〉AB‖2 ≤ δ and ‖(〈0| ⊗ I)|ψ〉AB‖2 ≤ ε.Writing the trace as 〈0| · |0〉+ 〈1| · |1〉, we replace |1〉 =
where ρ = TrB (|ψ〉〈ψ|AB). Thus, the trace-normalization condition is impossible
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Towards an alternative approach to joinability: enforcing positivitythrough purification
unless 12≤ δ + ε +
√2δε. The inequality expresses the fact if δ and ε are both too
small, the total probabilistic weight cannot amount to the proper normalization value.Intuitively, the constraints have “ruled out too much”.
We can apply this same technique to the soft constraints in the singlet sharingexample. Consider the following soft constraint that ensures proximity to the singletstate on AB
where S2(C2) denotes the symmetric subspace. Defining SAB ≡ S2(C2)⊗ C2, we canequivalently express this soft constraint as
‖(〈ν|ABC ⊗ ID)|ψ〉‖2 ≤ αAB, ∀ |ν〉 ∈ SAB. (4.32)
Defining SBC ≡ C2 ⊗ S2(C2), we can give the analogous soft constraint on BC as
‖(〈ω|ABC ⊗ ID)|ψ〉‖2 ≤ αBC , ∀ |ω〉 ∈ SBC . (4.33)
Since spanSAB,SBC = (C2)⊗3, there are no states |ψ〉 which admit likelihood zerofor all states in SAB and SBC . Furthermore, this ensures that, just like the previousqubit example, we can express the trace operation in terms of vectors chosen from SABand SBC . Let |j〉 be an arbitrary basis for (C2)⊗3. Each basis vector can be writtenas a linear combination |j〉 = xj|νj〉 + yj|ωj〉, where |νj〉 ∈ SAB and |ωj〉 ∈ SBC .Letting ρ = TrD (|ψ〉〈ψ|ABCD), we can write the trace of ρ as
Thus, the agreement parameters are bound by 1 ≤ λαAB + η√αABαBC + µαBC ,
where, the expressions for λ, η, and µ are given in the last line above. Crucially,the values of λ, η, and µ depend on the choice of basis |j〉 and the choice ofdecomposition |j〉 = xj|νj〉 + yj|ωj〉. Some choices will lead to less-strict joinabilitylimitations than other choices. Unfortunately, there is not a single choice of basisand decomposition which diagnoses all joinability failures. Rather, we expect thereto be family of “optimal” decompositions. Nevertheless, any choice of decomposition
74
4.3 Joinability limitations from soft constraints
gives finite values for λ, η, and µ and, therefore, leads to non-trivial constraints onthe agreement parameters 1 ≤ λαAB + η
√αABαBC + µαBC .
This approach can certainly be extended to more general settings. Furthermore,it would be valuable to investigate the structure of choosing the decompositions |j〉 =xj|νj〉+yj|ωj〉, and to understand the features of the optimal family of decompositions.We anticipate that convex geometry might play a role in determining such optimalfamilies. If so, we will have returned to an undesirable vantage point, in that, one ofthe motivations of the work in this chapter was to avoid the use of convex geometryfor understanding joinability. We concede that, it is possible (and probably likely),that any complete exploration of the concept of joinability must resolve to using thetools of convex geometry. Regardless, we have shown how the failure of the trace-normalization condition is responsible for certain failures of joinability.
As we emphasized in the introduction to this chapter, the ideas presented hereare very preliminary and represent a starting point for further investigation. Whilein the previous two chapters we have emphasized the role that positivity plays inlimiting joinability, here, we have argued that an alternative explanation exists. Inparticular, we showed that failures of the consistency of hard or soft constraints canbe explained by over-constraints on the total probabilistic weight of the quantumstate or channel. We achieved this by enforcing the quantum state or channel to bemanifestly positive-semidefinite or completely positive by means of purification or anisometric extension, respectively. When positive semi-definiteness is made manifest,joinability failures can be diagnosed with failures of the trace-normalization condition.
75
Chapter 5
Asymptotic stabilization of quantumstates with continuous-timequasi-local dynamics
77
Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
This chapter presents material that appeared in Quantum Information and Compu-tation, 7:0657, (2016), in an article titled “General fixed points of quasi-local frustration-free quantum semigroups: from invariance to stabilization”, which is joint work withFrancesco Ticozzi and Lorenza Viola.
5.1 IntroductionConvergence of a dynamical system to a stable equilibrium point is a hallmark ofdissipative, irreversible behavior. In particular, rigorously characterizing the na-ture and stability of equilibrium states of irreversible quantum evolutions is a long-standing problem central to both the mathematical theory of open quantum sys-tems and the foundations of quantum statistical mechanics [135, 136]. In recentyears, renewed interest in these issues has been fueled by the growing theoreti-cal and experimental significance of techniques for quantum reservoir engineering[137] and dissipative quantum control [21] within Quantum Information Process-ing (QIP). Representative applications that benefit from engineered dissipation in-clude robust quantum state preparation, with implications for steady-state entan-glement [32, 138, 139, 140, 141], non-equilibrium topological phases of matter [142],and ground-state cooling [143, 144, 145]; as well as open-system quantum simulation[146, 147], steady-state dissipation-driven quantum computation [148, 149], dissipa-tive quantum gadgets and autonomous quantum error correction [150, 151], alongwith quantum-limited sensing and amplification [152, 153].
While applications are often developed by making reference to a specific physi-cal setting, a common theme is the key role played by constraints, that may restrictthe allowed dynamical models and the extent of the available manipulations. Thismotivates seeking a rigorous system-theoretic framework for characterizing controlledopen-quantum system dynamics subject to given resource constraints. In this work,we focus on dissipative multipartite quantum systems described by time-independentquasi-local (QL) semigroup dynamics, capturing the fact that, in many physicallyrelevant scenarios, both the coherent (Hamiltonian) and irreversible (Lindblad) con-tributions to the semigroup generator must act non-trivially only on finite subsets ofsubsystems, determined for instance by spatial lattice geometry. The main questionwe address is to determine what properties an arbitrary target state of interest mustsatisfy in order to be the unique stationary (“fixed”) point for a given QL constraint,thereby making the state globally QL-stabilizable in principle, in an asymptotic sense.
In previous work [31, 34], this question has been addressed under the assumptionthat the target state is pure, providing in particular a necessary and sufficient linear-algebraic condition for the latter to be stabilizable without requiring Hamiltoniandynamics. Such pure states are called purely Dissipatively Quasi-Locally Stabilizable(DQLS). While restricting to a pure target state is both a natural and adequate
78
5.2 Preliminaries
first step in the context of dissipatively preparing paradigmatic entangled states ofrelevance to QIP (such as W or GHZ states), allowing for a general mixed fixed-point is crucial for a number of reasons. On the one hand, since mixed quantumstates represent the most general possibility, this is a prerequisite for mathematicalcompleteness. On the other hand, from a practical standpoint, QL stabilization ofa mixed state which is sufficiently close to an “unreachable” pure target may still bevaluable for QIP purposes, a notable example being provided by thermal graph statesat sufficiently low temperature [154]. Furthermore, as physical systems in thermalequilibrium are typically far from pure, characterizing mixed-state QL stabilizationmight offer insight into thermalization dynamics as occurring in Nature and on aquantum computer [155]. From this point of view, a stability analysis of thermalstates of QL Hamiltonians is directly relevant to developing efficient simulation andsampling algorithms for the quantum canonical ensemble, so-called “quantum Gibbssamplers,” as analyzed in [156] for commuting Hamiltonians.
In the mixed-state scenario, the problem of QL stabilization involves qualitativelydifferent features and is substantially more complex. This is largely due to the factthat the analysis tools used in the pure-state setting do not lend themselves to a for-mulation where the invariance property of the globally defined target state translatesdirectly at the level of QL generator components. We bypass this difficulty by restrict-ing to the important class of frustration-free (FF) semigroup dynamics [157, 156], forwhich global invariance of a state also implies its invariance under each QL compo-nent. Physically, the FF property is known to hold within standard derivations ofMarkovian semigroup dynamics, for instance based on Davies’ weak coupling limit or“heat-bath” approaches generalizing classical Glauber dynamics [136, 156].
5.2 Preliminaries
5.2.1 Notation and background
Consider a finite-dimensional Hilbert space H, dim(H) = d, and let B(H) be theset of linear operators on H. X† shall denote the adjoint of X ∈ B(H), with self-adjoint operators X = X† representing physical observables. The adjoint operationcorresponds to the transpose conjugate when applied to a matrix representation ofX, with the simple transpose being denoted by XT and the entry-wise conjugationby X∗. To avoid confusion, we shall use I to indicate the identity operator on B(H),whereas I will indicate the identity map (or super-operator) from B(H) to itself. Weshall use X ≡ Y to say that X is defined as Y .
The convex subset D(H) ⊂ B(H) of trace-one, positive-semidefinite operators,called density operators, is associated to physical states. We are concerned with statechanges in the Schrödinger picture between two arbitrary points in time, say 0 and
79
Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
t > 0, which are described by a completely-positive trace-preserving (CPTP) linearmap (or quantum channel) on B(H) [136]. A map Tt is CP if and only if it admitsan operator-sum representation,
ρ(t) = Tt(ρ0) =∑k
Mkρ0M†k , ρ0 ∈ D(H), (5.1)
for some Mk ⊂ B(H), and is also TP if in addition∑
kM†kMk = I. The oper-
ators Mk are referred to as (Hellwig-)Kraus operators or operation elements [158].The operator-sum representation of a CPTP map is not unique, and new decompo-sitions may be obtained from unitary changes of the operators Mk. Dual dynamics1with respect to the Hilbert-Schmidt inner product on B(H) (Heisenberg picture) areassociated to unital CP maps T †, that is, obeying the condition T †(I) = I.
A continuous one-parameter semigroup of CPTP maps Ttt≥0, with T0 = I,characterized by the Markov composition property Tt Ts = Tt+s, for all t, s ≥ 0, willbe referred to as a Quantum Dynamical Semigroup (QDS) [136]. We shall denote byL the corresponding semigroup generator, Tt = eLt, with the corresponding dual QDST †t t≥0 being described by the generator L†. It is well known that L (also referred toas the “Liouvillian”) can be always expressed in Lindblad canonical form [159, 160],that is, in units where ~ = 1:
ρ(t) = L (ρ(t)) ≡ −i[H, ρ(t)] +∑k
(Lkρ(t)L†k −
1
2L†kLk, ρ(t)
), t ≥ 0, (5.2)
where H = H† is a self-adjoint operator associated with the effective Hamiltonian(generally resulting from the bare system Hamiltonian plus a “Lamb shift” term), andthe Lindblad (or noise) operators Lk specify the non-Hamiltonian component of thegenerator, resulting in non-unitary irreversible dynamics. Equivalently, L defines avalid QDS generator if and only if it may be expressed in the form (see e.g. Theorem7.1 in [161])
L(ρ) ≡ E(ρ)− (κρ+ ρκ†), κ ≡ iH +1
2E†(I), (5.3)
where E is a CP map and the anti-Hermitian part of κ identifies the Hamiltonianoperator.
We shall denote by L(H, Lk) the QDS generator associated to HamiltonianH and noise operators Lk. Throughout this chapter, both H and all the Lk willbe assumed to be time-independent, with (5.2) thus defining a linear time-invariant
1While from a probabilistic and operator-algebra viewpoint it would be more natural to considerthe dynamics acting on the states as (pre-)dual, we follow here the standard quantum physicsnotation as it allows for a more direct connection with existing work as well as a more compactnotation in our context.
80
5.2 Preliminaries
dynamical system. It is important to recall that, as for CPTP maps, the Lind-blad representation is also not unique, namely, the same generator can be associ-ated to different Hamiltonian and noise operators (see e.g. Proposition 7.4 in [161]),and, further to that, the separation between the Hamiltonian and the noise oper-ators is not univocally defined [31, 161]. Specifically, the Liouvillian is unchanged,L(H, Lk) = L(H ′, L′k), if the new operators may be obtained as (i) linear combi-nations of H, Lk and the identity, L′k = Lk + ckI, H ′ = H − (i/2)
∑k(c∗kLk − ckL
†k),
with ck ∈ C; or (ii) unitary linear combinations, L′k =∑
l uklLl, H′ = H, with
U ≡ ukl a unitary matrix (and the smaller set “padded” with zeros if needed),corresponding to a change of operator-sum representation for E in Eq. (5.3).
We will denote a †-closed associative subalgebra A ⊆ B(H) generated by a set ofoperators X1, . . . , Xk as A ≡ algX1, . . . , Xk. If T ≡ T (Mk) and L ≡ L(H, Lk)are a CP map and a QDS generator, then we shall let algT ≡ algMk andalgL ≡ algH,Lk, respectively. These algebras are invariant with respect tothe change of representation in the Kraus or, respectively, Hamiltonian and Lindbladoperators since, as remarked, equivalent representations are linearly related to oneanother. Let ⊕ denote the orthogonal direct sum. It is well known that any †-closedassociative subalgebra A of B(H) admits a block-diagonal Wedderburn decomposi-tion [162], namely, H may be decomposed in an orthogonal sum of tensor-productbipartite subspaces, possibly up to a summand:
H ≡(⊕
`
H`
)⊕HR =
(⊕`
H(A)` ⊗H
(B)`
)⊕HR, (5.4)
in such a way that
A =
(⊕`
B(H(A)` )⊗ I(B)
`
)⊕ OR, (5.5)
where I(B)` represents the identity operator on the factorH(B)
` and OR the zero operatoron HR, respectively. Relative to the same decomposition, the commutant A′ of A inB(H), given by A′ ≡ Y | [Y,X] = 0, ∀X ∈ A, has the dual structure
A′ =(⊕
`
I(A)` ⊗ B(H(B)
` )
)⊕ B(HR). (5.6)
Consider now a density operator ρ ∈ D(H) such that supp(A) ⊆ supp(ρ), wherefor a generic operator space W the support is henceforth defined as supp(W ) ≡∪O∈W supp(O). It then follows that
Aρ ≡ ρ12 A ρ
12 = Y |Y = ρ
12Xρ
12 , X ∈ A ⊆ B(H) (5.7)
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
is a †-closed subalgebra with respect to the modified operator product X1 ρ X2 ≡X1ρ
−1X2, where the inverse is in the sense of Moore-Penrose [163] if ρ is not invertible.In the QIP literature, an associative algebra like in Eq. (5.7), which may be thoughtof as arising from a standard associative algebra A upon replacing each identity factorin Eq. (5.5) with a fixed matrix τ (B)
` in each factor has been termed a distorted algebra[164, 165]. In particular, we shall call Aρ a ρ-distorted algebra, and refer to the mapΦρ(X) ≡ ρ
12Xρ
12 as a “distortion map”. The ρ-distorted algebra generated by a set of
operators X1, . . . , Xk will be correspondingly denoted by Aρ ≡ algρ(X1, . . . , Xk).
5.2.2 Fixed points of quantum dynamical semigroups
States that are invariant (aka stationary or “fixed”) under the dissipative dynamicsof interest will play a central role in our analysis. Let fix(T ) indicate the set of fixedpoints of a CP map T ; when Tt = eLt for t ≥ 0, then clearly fix(Tt) = ker(L). In thissection, we summarize relevant results on the structure of fixed-point sets for CPTPmaps, and slightly extend them to continuous-time QDS evolutions.
Recall that fixed points of unital CPTP maps form a †-closed algebra: this stemsfrom the fact that algT = algT †, along with the following result (see e.g. Theo-rem 6.12 in [161]):
Lemma 5.2.1. Given a CPTP map T , the commutant algT ′ is contained infix(T †). In particular, if there exists a positive-definite state ρ > 0 in fix(T ), then
algT ′ = fix(T †). (5.8)
If T is CPTP and unital, its dual map always admits the identity as a fixed point offull rank. It then follows that fix(T ) = algT ′ [166, 165]. A similar result can beestablished for QDS generators (Theorem 7.2, [161]):
Lemma 5.2.2. Given a QDS generator L, the commutant algL′ is contained inthe kernel of L†. In particular, if L(ρ) = 0 for some ρ > 0, then
algL′ = ker(L†). (5.9)
A key result to our aim is that, in general, the set of fixed points of a QDS has thestructure of a distorted algebra. The following characterization is known for arbitrary(non-unital) CPTP maps (see e.g. Corollary 6.7 in [161]):
Theorem 5.2.3. Given a CPTP map T and a full-rank fixed point ρ,
fix(T ) = ρ12 fix(T †) ρ
12 , (5.10)
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5.2 Preliminaries
Moreover, with respect to the decomposition fix(T †) =⊕
` B(H(A)` )⊗ I(B)
` , we have
ρ =⊕`
ρ(A)` ⊗ τ
(B)` , (5.11)
where ρ(A)` and τ (B)
` are full-rank density operators of appropriate dimension.
Building on the previous results, an analogous statement can be proved for QDSdynamics:
Theorem 5.2.4. (QDS fixed-point sets, full-rank case) Given a QDS generatorL and a full-rank fixed point ρ,
ker(L) = ρ12 ker(L†) ρ
12 . (5.12)
Moreover, with respect to the decomposition ker(L†) =⊕
` B(H(A)` )⊗ I(B)
` , we have
ρ =⊕`
ρ(A)` ⊗ τ
(B)` , (5.13)
where ρ(A)` and τ (B)
` are full-rank density operators of appropriate dimension.
Proof. In order for eLtt≥0 to be a QDS, and thus a semigroup of trace-norm con-tractions [136], L must have spectrum in the closed left-half of the complex planeand no purely imaginary eigenvalues with multiplicity. It is then easy to show, byresorting to its Jordan decomposition [161], that the following limit exists:
T∞ ≡ limt→∞
1
t
∫ t
0
eLτdτ.
Being the limit of convex combination of CPTP maps, which form a closed convex set,T∞ it also CPTP. Furthermore, T∞ projects onto ker(L), namely, fix(T∞) = ker(L),and T∞ has only eigenvalues 0, 1 with simple Jordan blocks. Similarly, it follows thatthe unital CP map T †∞ ≡ (T∞)† projects onto ker(L†). Using these facts along withTheorem 5.2.3, we then have:
ker(L) = fix(T∞) = ρ12 fix(T †∞) ρ
12 = ρ
12 ker(L†) ρ
12 .
The structure of the fixed point, Eq. (5.13), follows from Theorem 5.2.3 applied toT∞.
The above two theorems make it clear that, given discrete- or continuous-timeCPTP dynamics admitting a full-rank invariant state ρ, the fixed-point sets fix(T )
83
Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
and ker(L) are a ρ-distorted algebra with structure
Aρ =⊕`
B(H(A)` )⊗ τ (B)
` , (5.14)
where the states τ (B)` are the same for every element in fix(T ) or ker(L). In addition,
since ρ has a compatible block structure [Eq. (5.13)], it is immediate to see thatfix(T ) and ker(L) are invariant with respect to the action of the linear mapMλ(X) ≡ρλXρ−λ for any λ ∈ C, and in particular for the modular group Miφ [167]. Thesame holds for the fixed points of the dual dynamics.
In fact, we can show that modular invariance is also a sufficient condition for adistorted algebra to be a fixed-point set for a CPTP map that fixes ρ, as relevant tothe problem of designing stabilizing dynamics for ρ. In order to do this, we need aresult by Takesaki [168], which we give in its finite-dimensional formulation (adaptedfrom [167], Theorem 9.2):
Theorem 5.2.5. Let A be a †-closed subalgebra of B(H), and ρ a full-rank densityoperator. Then the following are equivalent:(i) There exists a unital CP map E† such that fix(E†) = A, (E†)2 = E† and E(ρ) = ρ.
(ii) A is invariant with respect toM 12, that is, for every X ∈ A, ρ 1
2Xρ−12 ∈ A.
These conditions are equivalent to saying that the map E† is a conditional expec-tation on A that preserves ρ. We can then prove the following:
Theorem 5.2.6. (Existence of ρ-preserving dynamics) Let ρ be a full-rank den-sity operator. A distorted algebra Aρ admits a CPTP map T such that fix(T ) = Aρif and only if it is invariant forM 1
2.
Proof. First, notice that, if Aρ is a distorted algebra, then it is invariant for M 12if
and only if the “undistorted” algebra A ≡ ρ−12Aρρ−
12 is invariant forM 1
2. This follows
from the fact that M 12commutes with both the distortion map and its inverse. In
particular, if Aρ is invariant forM 12, we have:
M 12(A) =M 1
2(ρ−
12Aρρ−
12 ) = ρ−
12M 1
2(Aρ)ρ−
12 ⊆ ρ−
12Aρρ−
12 = A.
Thus, by Theorem 5.2.5, a unital CP projection E† onto A exists whose adjointpreserves ρ. By Theorem 5.2.3, the CPTP dual E ≡ T is such that fix(T ) = Aρ, asdesired.
To prove the other implication, it is sufficient to notice that Eq. (5.11) impliesthatM 1
2leaves A = fix(T †) invariant, and thus
M 12(Aρ) =M 1
2(ρ
12Aρρ
12 ) = ρ
12M 1
2(A)ρ
12 ⊆ ρ
12Aρ
12 = Aρ.
84
5.2 Preliminaries
If the dynamics admit no full-rank fixed state, we may restrict to the supportof a given fixed point, which is an invariant subspace for the Schrödinger’s-pictureevolution:
Theorem 5.2.7. (QDS fixed-point sets, general case) Given a finite-dimensionalQDS generator L, and a maximal-rank fixed point ρ with H ≡ supp(ρ), let L denotethe reduction of L to B(H). We then have
ker(L) = ρ12 (ker(L†)⊕ O) ρ
12 . (5.15)
Proof. For any ρ ∈ ker(L), the subspace H ≡ supp(ρ) is invariant for the dynam-ics [138]. Assume that L = L(H, Lk) and let Π : H → H denote the partialisometry onto H. Define the reduced (projected) operators ρ ≡ ΠρΠ†, H ≡ ΠHΠ†,and Lk ≡ ΠLkΠ
†. The dynamics inside H is then determined by the correspond-ing projected Liouvillian L(H, Lk) [169], and ρ is, by construction, a full-rankstate for this dynamics. Hence, the fixed-point set ker(L) is the distorted algebraker(L) = ρ
12 ker(L†)ρ 1
2 .
Consider now a maximal-rank fixed point, satisfying supp(ρ) = H = supp(ker(L)).It then follows from Theorem 9 in [138] that H is not only invariant but also attractivefor the dynamics. This means that
limt→∞
Tr(
Π⊥eLt(ρ0))
= 0, ∀ρ0 ∈ D(H).
With H being attractive, we have that ker(L) can have support only in H, and canthus be constructed by appending the zero operator on H⊥, so that, using Theorem5.2.4:
ker(L) = ker(L)⊕O = ρ12 (ker(L†)⊕O) ρ
12 .
In the above proof, we made the construction explicit in terms of a representationL = L(H, Lk) in order to make it clear that the result does not hold if we considerρ
12 ker(L†) ρ 1
2 , since H need not be invariant for L†. Again, it follows that ρ admits ablock decomposition as in Eq. (5.13), compatible with that of ker(L†) on its support.
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
5.2.3 Quasi-local semigroup dynamics
Throughout this chapter, the open quantum system of interest will consist of a finitenumber n of distinguishable subsystems, defined on a tensor-product Hilbert space
H =n⊗a=1
Ha, dim(Ha) = da, dim(H) = D.
As in [31, 34], we shall introduce quasi-locality constraints on the system’s evolutionby specifying a list of neighborhoods, namely, groups of subsystems on which operatorsare allowed to “act simultaneously”. Mathematically, neighborhoods Nj may bespecified as subsets of the set of indexes labeling the subsystems, that is,
Nj ⊆ 1, . . . , n, j = 1, . . . ,M.
Each neighborhood induces a bipartite tensor-product structure of H as
H = HNj ⊗HN j , HNj ≡⊗a∈Nj
Ha, HN j ≡⊗a/∈Nj
Ha. (5.16)
Likewise, with a neighborhood structure N ≡ Nj in place, any state ρ ∈ D(H)uniquely determines a list of reduced neighborhood states ρNj:
where TrN j indicates the partial trace over HN j . Quasi-local dynamical constraintsmay be specified by requiring compatibility with the bipartitions in (5.16), in thefollowing sense:
Definition 5.2.8. (Neighborhood operator) An operator X ∈ B(H) is a neigh-borhood operator relative to a given neighborhood structure N if there exists j suchthat the action of X is non-trivial only on HNj , that is:
X = XNj ⊗ IN j ,
where IN j is the identity operator on HN j .
A similar definition may be given for neighborhood CPTP maps and generators. Therelevant quasi-locality notion for QDS dynamics is then the following:
Definition 5.2.9. (QL semigroup) A QDS generator L is Quasi-Local (QL) rela-tive to a given neighborhood structure N if it may be expressed as a sum of neighbor-hood generators:
86
5.2 Preliminaries
L =∑j
Lj, Lj ≡ LNj ⊗ IN j . (5.18)
Quasi-locality of a Liouvillian is well-defined, as the structural property in Eq. (5.18)is defined independently of a particular representation of the generator. In terms ofan explicit representation, the above definition is equivalent to requiring that thereexists some choice L ≡ L(H, Lk), such that each Lindblad operator Lk is a neigh-borhood operator and the Hamiltonian may be expressed as a sum of neighborhoodHamiltonians, namely:
Lk = Lk,Nj ⊗ IN j , H =∑j
Hj, Hj ≡ HNj ⊗ IN j .
A Hamiltonian H of the above form is called a QL Hamiltonian (often “few-body,”in the physics literature)2. Mathematically, this denomination is natural given that,for the limiting case of closed-system dynamics, a QL Hamiltonian so defined au-tomatically induces a QL (Lie-)group action consistent with Eq. (5.18), with Lj ≡i adHj(·) = i [Hj, ·].
Remark 1. The above QL notion is appropriate to describe any locality constraintthat may be associated with a spatial lattice geometry and finite interactions range(e.g., spins living on the vertices of a graph, subject to nearest-neighbor couplings).QL semigroup dynamics have also been considered under less restrictive assumptionson the spatial decay of interactions [171], and yet different QL notions may be po-tentially envisioned (e.g. based on locality in “momentum space” or relative to “errorweight”). The present choice provides the simplest physically relevant setting thatallows for a direct linear-algebraic analysis. We stress that, due to the freedom in therepresentation of the QDS generator, QL semigroup dynamics may still be inducedby Lindblad operators that are not manifestly of neighborhood form. In principle, itis always possible to check the QL property by verifying whether a QDS generatorL has components only in the (super-)operator subspace spanned by QL generators.While it may be interesting to determine more operational and efficient QL criteriain specific cases, in most practical scenarios (e.g. open-system simulators [147]) avail-able Lindblad operators are typically specified in a preferred neighborhood form fromthe outset.
2In particular, the notions of neighborhood Hamiltonian and QL Hamiltonian reduce to thestandard uni-local and local ones for non-overlapping neighborhoods, see e.g. [170].
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
Our main focus will be on determining conditions under which a certain state ofinterest, ρ, is guaranteed to be invariant and the unique asymptotically stable statefor some QL dynamics. Formally, an invariant state ρ ∈ D(H) for a QDS withgenerator L is said to be Globally Asymptotically Stable (GAS) if
limt→+∞
eLt(ρ0) = ρ, ∀ρ0 ∈ D(H). (5.19)
For QDS dynamics not subject to QL constraints it is known that a state is GAS ifand only if it is the unique fixed point [161, 30]. A definition of stabilizable statesrelevant to our constrained setting may be given as follows:
Definition 5.2.10. (QLS state) A state ρ ∈ D(H) is Quasi-Locally Stabilizable(QLS) relative to a neighborhood structure N if there exists a QL generator L forwhich ρ is GAS.
Existing work has so far focused on stabilizability of a pure state, with specialemphasis on steady-state entanglement [31, 32, 34]. While even in this case, in general,a careful balancing of Hamiltonian and dissipative action is essential, a simple yet non-trivial stabilization setting arises by further requiring that the target can be madeQLS by a generator without a Hamiltonian component, namely, by using dissipationalone. Given the freedom in the representation of a QDS generator, in order toformalize this additional constraint we introduced a standard representation for agenerator L(H, Lk) that fixes a pure state ρ ≡ |Ψ〉〈Ψ| ∈ D(H), as in the followingresult (Corollary 1 in [34]):
Proposition 5.2.11. If a generator L(H, Lk) makes ρ = |Ψ〉〈Ψ| GAS, then thesame generator can be represented in a standard form L(H, Lk), in such a way thatH|Ψ〉 = h|Ψ〉, h ∈ R and Lk|Ψ〉 = 0, for all k.
In the standard representation, the target |Ψ〉 ∈ ker(Lk) may thus be seen as acommon “dark state” for all the noise operators, borrowing from quantum-optics ter-minology. With this in mind, a pure state ρ = |Ψ〉〈Ψ| may be defined as DissipativelyQuasi-Locally Stabilizable (DQLS) if it is QLS with H ≡ 0 and QL noise operatorsLk in standard form. Notice that such a definition implies that ρ is invariant for thedynamics relative to each neighborhood, namely, ρ ∈ ker(L(Lk)), for each k. Build-ing on this QL-invariance condition allows for proving the following characterizationof DQLS states [31]:
88
5.2 Preliminaries
Theorem 5.2.12. A pure state ρ = |Ψ〉〈Ψ| ∈ D(H) is DQLS relative to N if andonly if
supp(ρ) =⋂k
supp(ρNk ⊗ IN k). (5.20)
Remark 2. The proof of the above result includes the construction of a set ofstabilizing Lindblad operators Lk that make ρ DQLS, also implying that one suchoperator per neighborhood always suffices. It is easy to show that any rescaledversion of the same operators, L′k ≡ rkLk, also yield a stabilizing QL generatorL′ =
∑k |rk|2Lk ≡
∑k γkLk – incorporating “model (γ-)robustness,” in the terminol-
ogy of [169].However, the reasoning followed for QL stabilization of a pure state does not ex-
tend naturally to a general, mixed target state. The main reason is that the standardform, hence the DQLS definition itself, do not have a consistent analogue for mixedstates. A major simplification if ρ is pure stems from the fact that it is straightforwardto check for invariance, directly in terms of the generator components (see Proposi-tion 1 in [34]); for general ρ, we seek a definition that extends the DQLS notion, andthat similarly allows for explicitly studying what the invariance of ρ means at a QLlevel. A natural choice is to restrict to the class of frustration-free dynamics. Thatis, in addition to the QL constraint, we demand that each QL term in the generatorleave the state of interest invariant. Formally, we define [156]:
Definition 5.2.13. (FF generator) A QL generator L =∑
j Lj is Frustration Free(FF) relative to a neighborhood structure N = Nj if any invariant state ρ ∈ ker(L)also satisfies neighborhood-wise invariance, namely, ρ ∈ ker(Lj) for all j.
Beside allowing for considerable simplification, FF dynamics are of practical interestbecause they are, similar to the DQLS setting, robust to certain perturbations. As inRemark 5.2.4, given a QL generator L =
∑j Lj, define a “neighborhood-perturbed”
QL generator L′ =∑
j λjLj, with λj ∈ R+. If L is FF, then L(ρ) = 0 impliesLj(ρ) = 0 for each j; therefore, λjLj(ρ) = 0 for each j, and thus L′(ρ) = 0 as well.Were L not FF, then the kernel of L would not be robust against such neighborhood-perturbations in general. With these motivations, we introduce the notion of QLstability that we analyze for the remainder of this chapter :
Definition 5.2.14. (FFQLS state) A state ρ ∈ D(H) is Frustration-Free Quasi-Locally Stabilizable (FFQLS) relative to a neighborhood structure N if it is QLS witha stabilizing generator L that is FF.
Remarkably, studying FFQLS states will allow us to recover the results for DQLS purestates as a special case. In fact, a pure state is FFQLS if and only if it is DQLS. Thisclaim is proved in Appendix A of [172]. To summarize, the above definition consists of
89
Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
four distinct mathematical conditions that the generator L of the dissipative dynamicsmust obey for a given target invariant state ρ:
• (QDS): L is a generator of a CPTP continuous semigroup;
• (QL): L is QL, that is, L =∑
j Lj, with Lj = LNj ⊗ IN j ;
• (GAS): ρ is GAS, or equivalently ker(L) = span(ρ);
• (FF): L is FF, namely ker(Lj) ≥ span(ρ) for all Nj.The problem we are interested in is to determine necessary and sufficient conditionsfor a given state to be FFQLS and, if so, to design QL FF dynamics that achievesthe task.
In this section, we derive necessary conditions for a target state to be FFQLS.Frustration-freeness requires such a state to be in the kernel of each neighborhoodgenerator. We show that, if a neighborhood generator is to leave a global state in-variant, the size and structure of its kernel are constrained; in general, the kernel willbe larger than the span of the reduced neighborhood state (as a vector in Hilbert-Schmidt space). However, if the target state is to be the unique fixed point of theQDS dynamics, then the intersection of all the neighborhood-generator kernels mustcoincide with the span of the target state. We shall show in Section 5.4.2 that thiscondition is also sufficient for a generic (full-rank) state to be FFQLS.
5.3.1 Linear-algebraic tools
Recall that, given a tensor product of two inner-product spaces V = VA ⊗ VB and avector v ∈ V , a Schmidt decomposition of v is any decomposition
v =∑i
λiai ⊗ bi,
where ai ∈ VA, bi ∈ VB, λi > 0, and ai, bi are each orthonormal sets of vectors.There are two instances of Schmidt decomposition which are relevant in our context,both well known within QIP [173]. The first is the Schmidt decomposition of abipartite pure state |ψ〉 ∈ HA ⊗HB, namely,
The second is the so-called operator-Schmidt decomposition, whereby a bipartiteoperator M ∈ B(HA ⊗HB) = B(HA) ⊗ B(HB) is factorized in terms of elements inthe vector spaces B(HA) and B(HB), relative to the Hilbert-Schmidt inner product.Specifically,
M =∑i
λiAi ⊗Bi,
where Ai ∈ B(HA), Bi ∈ B(HB), λi > 0, and Tr(A†iAj) = Tr(B†iBj) = δij.Building on the concept of Schmidt decomposition, we introduce the Schmidt
span:
Definition 5.3.1. (Schmidt span) Given a tensor product of two inner productspaces V = VA⊗VB and a vector v ∈ V with Schmidt decomposition v =
∑i λiai⊗ bi,
the Schmidt span of v relative to VA is the subspace
ΣA(v) = spanai ∈ VA | v =
∑i
ai ⊗ bi, bi ∈ VB. (5.21)
Without referring to a particular tensor-product decomposition, it is possible toshow that the Schmidt span is the image of v under partial inner product:
ΣA(v) = a ∈ VA | a = (IA ⊗ b†)v for some b ∈ VB. (5.22)
One example is when VA ⊗ VB is a matrix space, such as CdA ⊗ CdB† (where thelatter factor is meant as a space of row vectors). In this case, the Schmidt span ofa matrix W ∈ CdA ⊗ CdB†, relative to the first factor CdA , is simply the range of thematrix W (namely, the set of all linear combinations of its column vectors), namelyΣA(W ) = range(W ). Similarly, the Schmidt span of W relative to the second factorCdB† is the orthogonal complement of the kernel ofW or, in other words, the support :ΣB(W ) = supp(W ). Another example is when VA ⊗ VB is a bipartite operator space,such as B(HA)⊗B(HB). The Schmidt span ofM ∈ B(HA)⊗B(HB) relative to B(HA),i.e. “on A”, is the operator subspace ΣA(M) = TrB [(IA ⊗B)M ] , B ∈ B(HB).
The Schmidt span is a useful tool because conditions on how a neighborhoodoperator is to affect a global state constrains how such an operator must act onthe entire operator-Schmidt span of that state. This intuition is formalized in thefollowing Lemma:
Lemma 5.3.2. (Invariance of Schmidt span) Given a vector v ∈ VA ⊗ VB andMA ∈ B(VA), if (MA ⊗ IB)v = λv, then (MA ⊗ IB)v′ = λv′ for all v′ ∈ ΣA(v) ⊗ VB.In particular:
Proof. Consider the Schmidt decomposition of v, v =∑
i γiai ⊗ bi, where ai ∈ VA,bi ∈ VB, γi > 0, and ai, bi are each orthonormal sets of vectors. Applying theeigenvalue equation for MA to this yields∑
i
γiMAai ⊗ bi = λ∑i
γiai ⊗ bi.
Multiplying both sides by IA ⊗ b†j, where b†j is the dual vector of bj, selects out the
VA-factor of the ith term, i.e., MAai = λai. This holds for each i and any linearcombination of the ais. By definition, the Schmidt span of v is ΣA(v) = spanai.Denoting βi a basis for VB, we may write any v′ ∈ ΣA(v)⊗VB as v′ =
∑ij µijai⊗βj.
Applying (MA ⊗ IB) to this we obtain
(MA ⊗ IB)v′ =∑ij
µijMAai ⊗ βj = λv′.
Thus, all elements in ΣA(v) ⊗ VB have eigenvalue λ with respect to MA ⊗ IB, asclaimed. Eqs. (5.23) and (5.24) follow by specializing the above result to λ = 0 andλ = 1, respectively.
5.3.2 Invariance conditions for quasi-local generators
As remarked, we require the global dynamics to be FF. This simplifies considerablythe analysis, as global invariance of the target state is possible only if the latter isinvariant for each neighborhood generator. Therefore, we examine the properties ofa neighborhood generator that ensure the target state ρ to be in its kernel. Notethat if ρ is factorizable relative to the neighborhood structure (i.e., a pure or mixedproduct state), ρ is invariant as long as each factor of ρ if fixed. Each such reducedneighborhood state can then be made not only invariant but also attractive by aneighborhood generator, if the reduced states are the only elements in the kernelsof the corresponding LNj . This automatically makes the global factorized state alsoGAS. In other words, if ρ is factorizable, then QL stabilizability is guaranteed. If thetarget state is non-factorizable (in particular, entangled), the above scheme need notwork; a non-factorizable state will have some operator Schmidt spans with dimensiongreater than one. The following Corollary, which follows from Lemma 5.3.2, illustratesthe implication of quasi-locally fixing a state with non-trivial operator-Schmidt spans:
Corollary 5.3.3. Let Lj = LNj ⊗ IN j be a neighborhood Liouvillian. If ρ ∈ ker(Lj),then it must also be that
ΣNj(ρ)⊗ B(HN j) ≤ ker(Lj).
Accordingly, if each neighborhood generator Lj is to fix a non-factorizable ρ (as isnecessary for global invariance with FF dynamics), then each neighborhood generatormust be constructed to leave invariant, in general, a larger space of operators –specifically, the corresponding neighborhood operator-Schmidt span of ρ.
However, leaving only the Schmidt spans invariant is, in general, not possible ifthe dynamics are to be CPTP, since a Schmidt span need not be a distorted algebra(as required by Theorem 5.2.7). We show that, in order for ρ to be in the kernel ofa valid QL generator, it is necessary that the dynamics leave certain “minimal fixed-point sets” generated by the Schmidt spans invariant as well. We give the following:
Definition 5.3.4. (Minimal modular-invariant distorted algebra) Let ρ ∈D(H) be a density operator, and W ⊆ B(H). The minimal modular-invariant dis-torted algebra generated by W is the smallest ρ-distorted algebra generated by Wwhich is invariant with respect to M 1
2(X) = ρ
12Xρ−
12 , where the inverse is in the
sense of Moore-Penrose if ρ is not full-rank.
In the finite-dimensional case that we consider, Fρ(W ) can be constructed by thefollowing iterative procedure: define F0 ≡ algρ(W ), and compute
Fk+1 = algρ(M 12(Fk)),
until Fk+1 = Fk ≡ Fρ(W ). This particular distorted algebra is the smallest structurewhose invariance is required if the dynamics are to be CPTP:
Lemma 5.3.5. (Minimal fixed-point sets) LetW ≤ B(H) be an operator subspacecontaining a positive-semidefinite operator ρ such that supp(ρ) = supp(W ). If W ≤fix(T ) for a CPTP map T : B(H)→ B(H), then
Fρ(W ) ≤ fix(T ). (5.25)
Proof. Given the iterative construction of Fρ(W ), it suffices to show that if someset W ⊆ fix(T ) includes a density operator with supp(ρ) = supp(W ), then bothalgρ(W ) ⊆ fix(T ) andM 1
2(W ) ⊆ fix(T ), and their support is still equal to supp(ρ).
Since T is a CP linear map, fix(T ) is closed with respect to linear combinations and
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
†-adjoint. We are left to show that fix(T ) is closed with respect to the ρ-modifiedproduct. Consider a partial isometry V : supp(ρ)→ H, and define the reduced map
T : B(supp(ρ))→ B(supp(ρ)), T (X) ≡ V †T (V XV †)V.
Since ρ is invariant, the set of operators with support contained in supp(ρ) = supp(W )is an invariant subspace, and thus T (V V †XV V †) = V V †T (V V †XV V †)V V †. Byconstruction, T is CP, TP, and has a full-rank fixed point ρ ≡ V †ρV . It follows fromTheorems 5.2.3 and 5.2.6 that fix(T ) is a ρ-distorted algebra; hence, it is closed withrespect to the modified product, as well as modular-invariant. Now, if X, Y ∈ W arefixed points for T , so are X = V †XV, Y = V †Y V † for T . Since their adjoint, linearcombinations and ρ-distorted products are in fix(T ), we have:
T (Xρ−1Y ) = T (V V †XV V †ρ−1V V †Y V V †) = V V †T (V Xρ−1Y V †)V V †
= V T (Xρ−1Y )V † = V Xρ−1Y V †
= V XV †V ρ−1V †V Y V † = Xρ−1Y.
Hence, it must be algρ(W ) ≤ fix(T ), as desired, and we still have supp(algρ(W )) =supp(ρ).
On the other hand, if X ∈ algρ(W ), then supp(M 12(X)) ∈ supp(W ), and we have:
T (Mρ12(X)) = T (V V †ρ
12Xρ−
12V V †) = V V †T (V ρ
12 Xρ−
12V †)V V †
= V T (ρ12 Xρ−
12 )V † = V ρ
12 Xρ−
12V †
= ρ12Xρ−
12 .
Accordingly,M 12(W ) ∈ fix(T ) and supp(M 1
2(X)) ⊆ supp(ρ) as well, as desired.
5.3.3 From invariance to necessary conditions for stabilizabil-ity
In order to apply the above lemma to our case of interest, namely, finding necessaryconditions for FFQLS, the first step is to show that the reduced neighborhood statesof ρ may be used to generate the minimal ρ-distorted algebra containing the Schmidtspan:
Proposition 5.3.6. Given a neighborhood Nj ∈ N , the support of the correspondingreduced state, ρNj = TrN j (ρ), is equal to the support of the operator-Schmidt spanΣNj(ρ).
Proof. Since ρNj ∈ ΣNj(ρ), supp(ρNj) ≤ supp(ΣNj(ρ)). It remains to show the oppo-site inclusion, that is, by equivalently considering the complements, that ker(ρNj) ≤ker(ΣNj(ρ)). Let |ψ〉 ∈ ker(ρNj). Since ρNj ≥ 0, we then have Tr
(ρNj |ψ〉〈ψ|
)=
Tr (ρ(|ψ〉〈ψ| ⊗ I)) = 0. Let Ei be a positive-operator valued measure (POVM)on HN j which is informationally complete (that is, spanEi = B(HN j)). ThePOVM elements sum to I, giving
∑iTr (ρ(|ψ〉〈ψ| ⊗ Ei)) = 0. Since each term is
non-negative, Tr (ρ(|ψ〉〈ψ| ⊗ Ei)) = 0 for all i. Letting ρi ≡ TrN j (ρ(I⊗ Ei)), we canwrite 0 = Tr (ρ(|ψ〉〈ψ| ⊗ Ei)) = 〈ψ|ρi|ψ〉. Then, ρi ≥ 0 implies ρi|ψ〉 = 0 for all i.Since the Ei span the operator space B(HN j), by using Eq. (5.22), we have that thecorresponding ρi span ΣNj(ρ). Hence, |ψ〉 ∈ ker(ΣNj(ρ)).
The above Proposition, together with Lemma 5.3.2 and Lemma 5.3.5, imply thefollowing:
Corollary 5.3.7. If a state ρ is in the kernel of a neighborhood generator Lj =LNj ⊗ IN j , then the minimal fixed-point set generated by the neighborhood Schmidtspan obeys
FρNj (ΣNj(ρ))⊗ B(HN j) ≤ ker(Lj). (5.26)
Proof. Assume that ρ ∈ ker(Lj). By Lemma 5.3.2, we have ΣNj(ρ) ⊗ B(HN j) ≤ker(Lj). By Proposition 5.3.6, we also know that the support of ρNj is equal to thatof ΣNj(ρ), and hence supp(ρNj ⊗ IN j) = supp(ΣNj(ρ)⊗ B(HN j)). With this and thefact that
ρNj ⊗ IN j ∈ ΣNj(ρ)⊗ B(HN j) ≤ ker(Lj),
Lemma 5.3.5 implies that
FρNj⊗INj
(ΣNj(ρ)⊗ B(HN j)
)≤ ker(Lj),
or, equivalently, FρNj (ΣNj(ρ))⊗ B(HN j) ≤ ker(Lj), as desired.
Summing up the results obtained on invariance so far, and recalling that unique-ness of the equilibrium state is necessary for GAS, we have the following necessarycondition:
Theorem 5.3.8. (Necessary condition for FFQLS) A state ρ is FFQLS relativeto the neighborhood structure N only if
span(ρ) =⋂j
FρNj (ΣNj(ρ))⊗ B(HN j). (5.27)
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
Proof. Let ρ be FFQLS relative to N . Frustration-freeness of L implies that ker(L) =⋂Nj ker(Lj). From QLS, we have ker(L) = span(ρ). Thus, FFQLS implies⋂
j
ker(Lj) = span(ρ).
Corollary 5.3.7 implies that for each neighborhood, FρNj (ΣNj(ρ))⊗B(HN j) ≤ ker(Lj).Hence, ⋂
j
FρNj (ΣNj(ρ))⊗ B(HN j) ≤⋂j
ker(Lj).
By construction, we also have
span(ρ) ≤⋂j
FρNj (ΣNj(ρ))⊗ B(HN j).
Stringing together the three relationships above, we arrive at the desired result, Eq.(5.27).
In this section, we move from necessary conditions for FFQL stabilization to sufficientones, by providing in the process a constructive procedure to design stabilizing semi-group generators. A key step will be to establish a property that arbitrary (convex)sums of Liouvillians enjoy, namely the fact that, as long as the algebras associatedwith individual components of the generator are contained in the algebra associatedto the full generator, the existence of a common full-rank fixed point suffices to provefrustration-freeness. Drawing on this result, we will prove that the necessary condi-tion of the previous section is also sufficient in the generic case where the target stateis full-rank, and then separately address general target states.
5.4.1 A key result on frustration-free Markovian evolutions
Consider a QDS of the form L =∑
k Lk, where individual terms need not, at thisstage, correspond to neighborhood generators. The following general result holds:
Theorem 5.4.1. (Common fixed points of sums of Liouvillians) Let L =∑k Lk be a sum of QDS generators, and assume that the following conditions hold:
(i) algLk ≤ algL for each k;(ii) there exists a positive definite ρ ∈ ker(L) such that ρ ∈ ker(Lk) for all k.
Then ρ′ is invariant under L only if it is invariant under all Lk, that is:
ρ′ ∈ ker(L) =⇒ ρ′ ∈ ker(Lk) ∀ k.
Proof. By linearity of L, we clearly have that ker(L) ≥⋂k ker(Lk). We show that
under the hypotheses, ker(L) ≤⋂k ker(Lk), therefore effectively implying ker(L) =⋂
k ker(Lk). By (ii), ρ is a full-rank state in ker(L) and ρ ∈ ker(Lk) for all k. Theorem5.2.4 implies that
ker(L) = ρ12 ker(L†)ρ
12 and ker(Lk) = ρ
12 ker(L†k)ρ
12 , ∀k.
Then, by Lemma 5.2.2, we also have that
ρ12 ker(L†)ρ
12 = ρ
12 algL′ρ
12 and ρ
12 ker(L†k)ρ
12 = ρ
12 algLk′ρ
12 .
In view of condition (i), the relevant commutants satisfy
algL′ ≤ algLk′, ∀k.
The above inequality may then be used to bridge the previous equalities, yielding:
ker(L) = ρ12 ker(L†)ρ 1
2 = ρ12 algL′ρ 1
2
≤
ker(Lk) = ρ12 ker(L†k)ρ
12 = ρ
12 algLk′ρ
12 ,
for all k. From this we obtain ker(L) ≤⋂k ker(Lk), which completes the proof.
Remark 3. We note that condition (i) above, namely algLk ≤ algL, is onlyever not satisfied due to the presence of Hamiltonian contributions in Lk. In fact,if Lk ≡ Lk(Lj,k) for each k in a given representation, then L =
∑k Lk also has a
purely dissipative representation L(⋃kLj,k), and thus algLk ≤ algL. On the
other hand, suppose that Lk = Lk(Hk, Lj,k), with Hk 6= 0 in some representation.This implies that L = L(H,
⋃kLj,k), with H =
∑kHk. In this case, since alg(H)
need not contain alg(Hk), condition (i) does not hold in general. As a trivial example,consider two generators associated to H1 = M and H2 = −M, with M 6= I. Clearly,O = alg(H) does not contain alg(M). Likewise, if H = (C2)⊗3 and Z is a single-qubit Pauli operator, consider QL Hamiltonians H1 = ZZI and H2 = IZZ. Thenalg(H1) alg(H1+H2). Intuitively, this stems from the fact that since noise operatorsenter “quadratically” (bilinearly) in the QDS, they cannot cancel each other’s action– unlike Hamiltonians, which by linearity may “interfere” with one another.
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
Interestingly, the reasoning leading to Theorem 5.4.1 also applies to CPTP maps,with the simplification that, since no Hamiltonian is present, condition (i) is alwayssatisfied. Formally:
Corollary 5.4.2. (Common fixed points of sums of CPTP maps) Let T =∑k pkTk be a sum of CPTP maps, with pk > 0 and
∑k pk = 1. If there exists a
positive definite ρ ∈ fix(T ) such that ρ ∈ fix(Tk) for all k, then ρ′ is invariant underT only if it is invariant under all Tk, that is:
ρ′ ∈ fix(T ) =⇒ ρ′ ∈ fix(Tk).
The proof is a straightforward adaptation of the one above, and it can actually beextended to a positive semi-definite fixed state ρ, provided that some extra hypotheseson the support of ρ are satisfied. The precise statement and proof of this extendedresult are given in Appendix B of [172].
Another direct corollary of Theorem 5.4.1, which now specializes to locality-constrained dynamics, provides us with a useful tool to ensure that a QL generatorbe FF: the generator itself and all of its QL components must share a full-rank fixedstate.
Corollary 5.4.3. (Frustration-freeness from full-rank fixed point) Let L =∑j Lj be a QL generator, and assume that the following conditions hold:
(i) algLj ≤ algL for each j;(ii) there exists a positive-definite ρ ∈ ker(L) such that ρ ∈ ker(Lj) for all j.Then the QL generator L is FF.
5.4.2 Sufficient conditions for full-rank target states
Theorem 5.4.4. (Sufficient condition for full-rank FFQLS) A full-rank stateρ is FFQLS relative to the neighborhood structure N if
span(ρ) =⋂j
FρNj (ΣNj(ρ))⊗ B(HN j). (5.28)
Proof. To show that this condition suffices for FFQLS, we must show that thereexists some QL FF Liouvillian L for which span(ρ) = ker(L). Our strategy is to firstconstruct a QL generator for which ρ is the unique state in the intersection of theQL-components’ kernels. Then, we use Thm. 5.4.3 to show that this generator is FF,yielding the desired equality.
Fix an arbitrary neighborhood Nj ∈ N , with associated bipartition H = HNj ⊗HN j . We shall construct a neighborhood CPTP map Ej ≡ ENj ⊗ IN j , where ENj
projects onto the minimal fixed-point set containing the neighborhood-Schmidt span(that is, such projection maps are duals to a conditional expectation). Since, byconstruction, FρNj (ΣNj(ρ)) is a modular-invariant distorted subalgebra of B(HNj),Theorem 5.2.6 ensures that there exists a CPTP map ENj such that
fix(ENj) = FρNj (ΣNj(ρ)).
In particular, we take E2Nj = ENj , so that it projects onto its fixed points. Explicitly,
its structure follows from the decomposition in Eq. (5.14):
FρNj (ΣNj(ρ)) =⊕`
B(H(A)`,j )⊗ τ (B)
`,j ,
with a corresponding Hilbert space decomposition HNj ≡⊕
`H`,j =⊕
`H(A)`,j ⊗H
(B)`,j ,
and τ (B)`,j a full-rank state on H(B)
`,j . Introducing partial isometries Π`,j : H`,j → HNj ,the sought-after maps ENj can be constructed as:
ENj(ρ) ≡⊕`
TrH(B)`,j
(Π†`,jρΠ`,j)⊗ τ (B)`,j . (5.29)
It is straightforward to verify that ENj(ρ) is CPTP. Recalling Eq. (5.3), we may thendefine a neighborhood QDS generator by taking κ = E†Nj(I)/2 = I/2 and letting
LNj ≡ ENj − INj , ∀j. (5.30)
Let now L ≡∑
j Lj =∑
j LNj ⊗ IN j define the QL generator of the overalldynamics. We constructed each Lj in such a way that
ker(Lj) = FρNj (ΣNj(ρ))⊗ B(HN j), ∀j.
Hence, by invoking the hypothesis (Eq. (5.28)), it follows that ρ is the unique stateobeying
span(ρ) =⋂j
ker(Lj) ≤ ker(L). (5.31)
A priori, it is still possible that span(ρ) < ker(L). However, since we have chosenκ = κ†, the neighborhood generators Lj defined in Eq. (5.30) do not have anyHamiltonian contribution; recalling Remark 3, it follows that the algebra of the globalgenerator contains the algebra of each neighborhood generator,
algL ≥ algLj, ∀j.
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
Thus, by Corollary 5.4.3, the generator L is FF. From L being FF, it follows in turnthat
ker(L) ≤⋂j
ker(Lj),
which, together with Eq. (5.31), implies span(ρ) = ker(L), as desired.
5.4.3 Sufficient conditions for general target states
If the target state ρ is not full-rank, the necessary condition of Theorem 5.3.8 maystill be shown to be sufficient for FFQLS if an additional condition (referred to as the“support condition” henceforth) is also obeyed:
Theorem 5.4.5. (Sufficient condition for general FFQLS) An arbitrary stateρ is FFQLS relative to the neighborhood structure N if
span(ρ) =⋂j
FρNj (ΣNj(ρ))⊗ B(HN j) (5.32)
andsupp(ρ) =
⋂j
supp(ρNj ⊗ INj). (5.33)
Proof. Our strategy is to use the support condition of Eq. (5.33) to reduce thenon-full-rank case to the full-rank one. As in the proof of the previous theorem, fixan arbitrary neighborhood Nj, and consider the maps ENj , defined in Eq. (5.29).Let PNj ∈ B(HNj) denote the Hermitian projector onto supp(FρNj (ΣNj)(ρ)), andP⊥Nj = INj − PNj the associated orthogonal projector. In this case, we compose eachENj with the corresponding map
E0Nj(·) ≡ PNj(·)PNj +
PNjTr(PNj)
Tr (P⊥Nj ·), (5.34)
where E0Nj is, like ENj , both CP and TP:
Tr (E0Nj(M)) = Tr
(M(PNj + P⊥Nj)
)= Tr(M), ∀M ∈ B(HNj).
With this, consider new CPTP maps given by ENj E0Nj , whereby it follows that new
with the global evolution being driven, as before, by the QL generator L =∑
j Lj.Define now Π to be the projector onto supp(ρ), and consider the positive-semidefinite
function V (τ) = 1 − Tr (Π τ) , τ ∈ B(H). The derivative of V along the trajectoriesof the generator we just constructed is
V (τ) = −∑j
Tr (ΠLj(τ)) .
By LaSalle-Krasowskii theorem [174], the trajectories will converge to the largest in-variant set contained in the set of τ such that the above Lyapunov function V (τ) = 0.We next show that this set must have support only on supp(ρ) =
⋂Nj supp(ρNj⊗IN j).
Since V is defined on global input operators, we first re-express each neighborhoodgenerator Lj in Eq. (5.35) as
Lj = Ej E0j − I, Ej ≡ ENj ⊗ IN j , E0
j ≡ E0Nj ⊗ IN j ,
where we have used the property Ej E0j = (ENj E0
Nj) ⊗ IN j . Additionally, letPj ≡ PNj ⊗ IN j denote the projector onto supp(FρNj (ΣNj(ρ)) ⊗ IN j). Assume nowthat supp(τ) * supp(ρNk ⊗ IN k) for some Nk ∈ N , that is, Tr(τP⊥k ) > 0. By usingthe explicit form of the maps E0
Nj given in Eq. (5.34), we then have:
V (τ) ≤ −Tr (ΠLk(τ))
= −Tr(Π (Ek E0
k )τ − Π τ)
= −Tr (ΠEk(PkτPk))− Tr(τP⊥k
) Tr (ΠEk(Pk))Tr (Pk)
+ Tr (Πτ) . (5.36)
Since the target state ρ is invariant under Ek, its support is also invariant. Thisimplies that
Tr (ΠEk(PkτPk)) ≥ Tr (ΠEk(ΠτΠ)) = Tr (Πτ) .
Hence, the sum of the first and the third term in Eq. (5.36) is less than or equal tozero. The second term, on the other hand, is strictly negative. This is because: (i) weassumed that Tr(τP⊥k ) > 0; (ii) with Π ≤ Pk, and Ek(Pk) having the same supportof Pk by construction, it also follows that Tr (ΠEk(Pk)) > 0. We thus showed thatno state τ with support outside of the support of ρ can be in the attractive set forthe dynamics. Hence, the dynamics asymptotically converges onto the support of ρ,which is invariant for all the Lj. By restricting to this set, the maps E0
j have no effectand the same argument of Theorem 5.4.4 shows that the only invariant set in such asubspace is span(ρ), as desired.
Remark 4. We note that the support condition in Eq. (5.33) is indeed a natural can-
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
didate for a sufficient FFQLS condition, since if ρ is pure, it reduces to the necessaryand sufficient condition of Theorem 5.2.12. However, it is provably not necessary ingeneral, see Sec. 5.5.4. We conjecture that the necessary condition of Theorem 5.3.8is in fact sufficient for general (non-full-rank) states. However, we currently lack acomplete proof.
5.5 Illustrative ApplicationsIn this section, we illustrate the general framework presented thus far through a num-ber of examples, with a twofold goal in mind: to both demonstrate the applicabilityand usefulness of the mathematical tools we have developed, and to gain insight intothe problem of mixed-state QL stabilization, along with appreciating important dif-ferences from the pure-state setting. For simplicity, we shall focus in what followson a multi-partite system consisting of n co-dimensional qudit subsystems, namely,H = ⊗na=1Ha = (Cd)⊗n, with D = dim(H) = dn. In the especially important casecorresponding to qubit (or spin-1/2) subsystems, d = 2, we shall follow standardnotation and denote by |0〉, |1〉 an orthonormal (computational) basis in C2 andby σα|α = 0, 1, 2, 3 ≡ I, X, Y, Z the set of single-qubit Pauli matrices, under thenatural extension to multi-qubit operators, e.g., σ(a)
x ≡ Xa = I⊗ . . . I⊗X⊗ . . . I, withnon-trivial action occurring only on the ath factor.
5.5.1 Some notable failures of quasi-local stabilizability
Before exhibiting explicit classes of states which are provably FFQLS, it may beuseful to appreciate some distinctive features that the mixed nature of the target stateentails and, with that, the failure of some intuitively natural mechanisms to generatecandidate FFQLS states. Recall that an arbitrary pure product (fully factorized)state is always DQLS (or equivalently, as shown, FFQLS) [31] thus, in other words,failure of a pure target state to be FFQLS always implies some entanglement in thestate. In contrast to that, entanglement is not necessary for failures of FFQLS if thetarget is mixed. Consider n qubits arranged on a line, with neighborhood specifiedby nearest-neighbor (NN) pairs, Nj ≡ j, j + 1, j = 1, . . . , n− 1, and the manifestlyseparable target state
ρ ≡ ρsep =1
2(|0〉〈0|⊗n + |1〉〈1|⊗n). (5.37)
Because ρ is already in Schmidt decomposition form for all n, it easily follows thateach Schmidt span has the form
ΣNj(ρ) = span|00〉〈00|j,j+1, |11〉〈11|j,j+1,
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5.5 Illustrative Applications
and is already a ρNj -distorted algebra invariant forM 12. Taking the intersection over
all neighborhoods then leaves the two-dimensional space,⋂j
which was proved to be DQLS relative to the three-body neighborhoodsN1 = 1, 2, 3,N2 = 2, 3, 4 in [31] (see also Sec. 5.5.4 for explicit calculations).
5.5.2 Quasi-local stabilization of graph product states
Multi-qubit pure graph states are an important resource across QIP, with applicationsranging from measurement-based quantum computation [175] to stabilizer quantumerror-correcting codes [176]. More recently, thermal graph states [177, 154] have beenshown to both provide faithful approximations of pure graph states for sufficientlylow temperatures and to support non-trivial multipartite bound entanglement overa temperature range [178]. In this section, we demonstrate that a broader class ofmixed graph states on qudits, which we refer to as graph product states, are FFQLS.
For the special case of qubits, both pure [32, 31] and thermal [154] graph statesare known to be stabilizable with QL FF semigroup dynamics with respect to anatural locality notion induced by the graph. We recover and extend these resultsto d > 2 and a broader class of non-thermal graph states, without making reference,in the mixed-state case, to properties of the Davies QDS generator which is typicallyemployed under weak-coupling-limit assumptions [136, 156, 154]. The key propertyof graph product states is that they can be transformed to a product form relativeto a “logical subsystem factorization,” following a change of basis which is effectedby a sequence of commuting neighborhood unitary transformations (a QL quantumcircuit). Commutativity of the unitaries effectively reduces the problem of FFQLS toone of local stabilization of product states. While for graph states this observationallows to directly obtain QL FF stabilizing dynamics, they nevertheless serve as a
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
first relevant example of FFQLS, in preparation for cases where the tools we proposebecome indispensable.
We formally define general qudit graph states following [179]. Let G = (V,E) be agraph, where vertices j ∈ V are associated to qudits and edges (j, k) ∈ E label theirallowed pairwise interactions. A natural neighborhood structure is derived from G,by letting the jth neighborhood Nj comprise vertex j along with the subset of verticesadjacent to it. Rather than associating the graph to a particular state, the graph isused to construct a set of commuting unitary edge-wise operators, say U(j,k). Theproduct of all such unitaries, UG ≡
∏(j,k)∈E U(j,k), constitutes the quantum circuit
which is used to map an input product state into, in general, an entangled state3.Each U(j,k) is defined as the generalized controlled-Z transformation [179] associatedto a symmetric qudit Hadamard matrix H. Hadamard matrices H are defined bythe conditions H†H = dI, H = HT , and |hij| = 1, where hij ≡ [H]ij. To eachHadamard, there exists a corresponding generalized controlled-Z gate acting on two-qudits, defined by CH |ij〉 = hij|ij〉. With these in place, U(j,k) ≡ CH
(j,k)⊗I(j,k), and theunitary transformation which transforms local operators to neighborhood operatorson Nj is defined by
Uj ≡∏
k∈Nj\j
CH(j,k),
where each operator is defined on the global Hilbert space H, and acting non-triviallyonly on the subsystems by which it is indexed4. Standard pure qudit graph statesmay be defined as [179] |ψG〉 ≡ UG |+〉⊗n. Similarly, we define graph product states as
ρG ≡ UG ρprod UG† = UG
( n⊗j=1
ρj
)UG†, (5.40)
where each ρj is an arbitrary qudit mixed state. We note that graph product states aredistinct from (though overlapping with) so-called graph diagonal states [180], definedas those states obtained by applying the circuit UG to any state ρdiag diagonal in theeigenbasis of the Hadamard matrix.
To construct QL Lindblad operators which stabilize a graph product state ρG, wemay simply construct the local Lindblad operators which prepare each factor ρj of Eq.(5.40) in the un-rotated basis, and then transform these Lindblad operators with UG.Let each factor ρj be diagonalized by ρj = Vj(
∑i γ
ij |i〉〈i|)V
†j , where γij ≥ 0 are the
3In the cluster model of quantum computation, qubit graph states are constructed by applyingthis global unitary (as a sequence of commuting neighborhood-wise actions) on an initial productstate |+〉⊗n.
4 Since H is not uniquely defined, the above U(j,k) depend on the choice of H. For readability,our notation does not make this explicit. For standard qubit graph states, H is the discrete Fouriertransform on C2.
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5.5 Illustrative Applications
ordered (with i) eigenvalues of the qudit density operator ρj and Vj the diagonalizingunitary transformation. Stabilizing Lindblad operators may then be constructed asfollows:
Lji,i+1 =
√γji UjVj|i〉〈i+ 1|V †j U
†j , Lji+1,i =
√γji+1 UjVj|i+ 1〉〈i|V †j U
†j ,
where i, j = 1, . . . , n and each Lindblad operator is defined on the whole H, but byconstruction acts non-trivially only on the neighborhoodNj. That the resulting globaldynamics L =
∑j Lj are FF follows from the commutativity of the neighborhood-
Liouvillians Lj.It is interesting to note that, since any pure state ρdiag in Eq. (5.40) which is
diagonal in the computational basis is necessarily a product, arbitrary pure quditgraph states are FFQLS. In general, however, since ρdiag may be separable but notnecessarily of product form, mixed graph diagonal states need not be FFQLS (in linewith similar conclusions for trivially separable states, as discussed in Sec. 5.5.1).Remark 4: Graph Hamiltonians. Pure graph states may be equivalently definedas a special class of stabilizer states, by assigning to each vertex in G a stabilizergenerator, taken from the generalized Pauli group Gn for n qudits [176]. For qubits,for example, a graph state |ψG〉 may be seen to the the unique ground state of a QLgraph Hamiltonian HG that is a sum of generators of Gn of the form:
HG ≡n∑j=1
HG,j = −∑j∈V
Xj
⊗k∈Nj\j
Zk = −UG†(∑
j
Xj
)UG. (5.41)
By construction, HG is a sum of commuting terms, and may be easily seen to be FF(namely, such that |ψG〉 is also the ground state of each HG,j separately). Furtherto that, the last equality in Eq. (5.41) makes it clear how the graph Hamiltonianis mapped to a (strictly) local one in the “logical basis”, following application of thecircuit UG. A feature that becomes evident from expressing graph-Hamiltonians inthis form, and that is not shared by more general QL commuting Hamiltonians, is thelarge degeneracy of their eigenspaces – precisely 1/d of the global space dimension5.Thermal graph states [177, 154], relative to Hamiltonians as in Eq. (5.41), are aspecial case of graph product states, corresponding to each qudit being in a canonicalGibbs state, namely, ρj ∝ exp(−βjHG,j) in Eq. (5.40) (or, ρG ∝ U †G(⊗jeβjXj)UG),where βj denotes the inverse equilibrium temperature of the jth qubit. Thermal qubitgraph states can thus provide a scalable class of mixed multiparty-entangled states.
The construction leading to graph product states may be generalized to arbi-trary situations where a quantum circuit arising from commuting unitary neighbor-
5For d = 2, this feature is key in enabling graph-state preparation in finite time with discrete-timedynamics designed via splitting-subspace approaches [181].
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
hood operators may be identified, not necessarily stemming from a graph. That is,say that U ≡
∏(j,k)∈E U(j,k), with each U(j,k), as above, being an edge-wise opera-
tor, with [U(j,k), U(j′,k′)] = 0 for all edges (j, k), (j′, k′). Then we may define QL-transformed product states as resulting from the action of U on any product inputstate: ρ ≡ U ρprod U
†. Note that ρprod is, clearly, FFQLS in the strongest sense, rela-tive to strictly local (single-site) neighborhoods, whereas ρ is FFQLS relative to thestructure Nj =
⋃j∩(i,k)6=∅(i, k), which is imposed by the circuit. More generally,
a QL commuting circuit may be used to extend neighborhoods of some input Nin
into larger neighborhoods of some output Nout, associated to weaker QL constraints.FFQLS states maintain their property under this type of transformation, in the fol-lowing sense:
Proposition 5.5.1. (Circuit-transformed FFQLS) Let ρin be FFQLS relative toNin ≡ Nin,i, and let U =
∏j Uj, with Uj ≡ UNj ⊗ IN j and [Uj, Uj′ ] = 0 for all j, j′.
Then the output state ρout = U(ρ) ≡ UρinU† is FFQLS relative to Nout ≡ Nout,k,
whereNout,k ≡ Nin,k ∪
( ⋃Nj∩Nin,i 6=∅
Nj). (5.42)
Proof. This is easily verified by constructing FFQLS dynamics for U(ρ). If Lin =∑i Lin,i is a FFQL stabilizing dynamics for ρin, construct a new Liouvillian by con-
jugation, that is, Lout ≡ U Lin U †.. More explicitly,
Lout(ρ) =∏k
Uk∑i
Lin,i
(∏j
U †j ρ∏j′
Uj′)∏
k
U †k′ ≡∑k
Lout,k(ρ),
where the neighborhood structure of the output Liouvillian relative to the (enlarged,in general) neighborhoods in Eq. (5.42) follows from the commutativity of the circuitunitaries Uj, as conjugation by all but those unitaries constrained by Eq. (5.42) hasno net effect. Both the spectrum and the FF property are preserved as U is unitary,and ρout is stabilized by Lout because its kernel is U(ker(Lin)) = U(span(ρin)).
Physically, the obvious way to construct a QL commuting circuit is via exponen-tiation of commuting QL Hamiltonians, namely, Uj ≡ exp(iHj), with [Hj, Hj′ ] = 0for all j, j′. In fact, any QL commuting circuit arises in this way, in the sense that afamily of QL commuting Hamiltonians may always be associated to U , for instanceby letting Hj = −i logUj in the basis which simultaneously diagonalizes all circuitunitaries.Remark 5: Rapid mixing. As mentioned, for both pure and thermal graph stateson qubits, QL stabilizing dynamics have been thoroughly analyzed in the literature. Inparticular, rigorous upper bounds on the mixing time have been established, showing
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5.5 Illustrative Applications
that such states may be efficiently prepared – that is, the (worst-case) convergencetime scales only (poly-) logarithmically with the system size [182, 154]. Remarkably,rapid mixing has been shown to both lead to stability against QL perturbations ofthe generator [171] and to the emergence of effective area laws [183]. These resultsextend naturally to the broader classes of graph product states and encoded productFFQLS states considered here.
5.5.3 Quasi-local stabilization of commuting Gibbs states
In this subsection, we analyze FFQLS of another class of states derived from com-muting QL Hamiltonians. Consider a Gibbs state:
ρβ ≡e−βH
Tr (e−βH), H ≡
∑j
Hj, β ∈ R+, (5.43)
where each Hj is a neighborhood-operator relative to Nj ∈ N . If the neighborhoodHamiltonians satisfy [Hj, Hj′ ] = 0 for all j, j′, ρβ is also called a commuting Gibbs state[156]. Characterizing QL evolutions that have canonical Gibbs states as their uniquefixed point has both implications for elucidating aspects of thermalization in naturallyoccurring dynamics and for quantum algorithms and simulation – most notably, in thecontext of quantum generalizations of Metropolis sampling [184]. Recent work [156]has shown that Gibbs states of arbitrary QL commuting Hamiltonians are FFQLS,the commutativity property being essential to ensure quasi-locality of either the weak-coupling (Davies) generator or the heat-bath QDS dynamics that dissipatively preparethem. The central result therein establishes an equivalence between the stabilizingdynamics being gapped and the correlations in the Gibbs state satisfying so-called“strong clustering”, implying rapid mixing for arbitrary one-dimensional (1D) latticesystems, or for arbitrary-dimensional lattice systems at high enough temperature.
It is important to appreciate that in the derivation of such results, primitivity ofthe QDS generator is assumed from the outset, and verified, along with the QL andFF properties, by making explicit reference to the structure of the Davies or heat-bath generator (see respectively Lemma 9 and Theorem 10 in [156]). Conversely, theQL notion is not a priori imposed as a design constraint for the dynamics, but againemerges from the structure of the generator itself. In this sense, our framework maybe seen to provide a complementary approach, providing in particular a necessarycondition for thermal dynamics to be primitive relative to a specified neighborhoodstructure. Let us illustrate the potential of our approach by focusing on the simplestsetting of commuting two-body NN Hamiltonians in 1D.
Proposition 5.5.2. A full-rank state ρ > 0 defined on a 1D lattice system is FFQLS
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
relative to neighborhoods Nj = j, j + 1 if and only if
span(ρ) =
(⊗j odd
Fρj,j+1Σj,j+1(ρ)
)⋂(Fρ1Σ1(ρ) ⊗
⊗k even
Fρk,k+1Σk,k+1(ρ)
).
Proof. The proof simply follows from noting that we can group the neighborhoodsj, j + 1 with odd and even j and, since neighborhoods in the same group are notoverlapping, that the intersection of the minimal fixed-point sets corresponding toneighborhoods in the same group corresponds to their product.
One consequence of the above simplification is the following:
Proposition 5.5.3. Let ρ > 0 be FFQLS with respect to the above 1D NN neighbor-hood structure. If, for any neighborhood l, l+1, the minimal fixed-point set is the fullalgebra, that is, Fρl,l+1
Σl,l+1(ρ) = B(Hl,l+1), then ρ must factor as ρ = ρ1...l⊗ρl+1...n.
Proof. Assume that ρ is FFQLS and Fρl,l+1Σl,l+1(ρ) = B(Hl,l+1). Then, ρ satisfies
the intersection condition, which is simplified to a tensor product of two intersections
)]and, similarly, (int right)l+1,...,n is given by[(
B(Hl+1)⊗n⊗
j odd,j=l+2
Fρj,j+1Σj,j+1(ρ)
)⋂( n⊗k even,k=l+1
Fρk,k+1Σk,k+1(ρ)
)].
Eq. (5.44) can only be satisfied if ρ = ρ1...l ⊗ ρl+1...n.
The above proposition captures the fact that, for a state of a 1D NN-coupled chainto be FFQLS, there are limitations to the correlations that the state can exhibit.Similar restrictions are, generically, sufficiently strong to prevent Gibbs states of 1D
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5.5 Illustrative Applications
commuting NN Hamiltonians to be FFQLS relative to NN neighborhoods. A simpleexample is a 1D Ising Hamiltonian:
H = −n∑j=1
ZjZj+1, n > 3, (5.45)
where periodic boundary conditions are assumed. The Schmidt span for each NNpair consists of the space of diagonal matrices, and is closed under generation of thedistorted algebra. The intersection of all the Schmidt spans is thus the 2n-dimensionalspace of diagonal matrices, implying that, for all temperatures and size n, the Gibbsstate is not FFQLS. In particular, no FF thermal dynamics subject to the NN QLconstraint can stabilize (or be primitive with respect to) this state.
The thermal dynamics of the Davies or heat-bath generators that stabilize com-muting Gibbs states are, in fact, both FF and QL, albeit relative to a differentneighborhood structure than the one solely determined by the system’s Hamiltonian[156]. This is most transparent in the weak-coupling derivation of the QDS, wherebythe evolution induced by this Hamiltonian, Ut ≡ e−itH , effectively “modulates” intime the bare system-bath neighborhood coupling operators, in turn determining therelevant Lindblad operators in frequency space [136]. The net effect is that Ut actsas a QL commuting circuit, resulting in a neighborhood structure which is expandedwith respect to the one associated to H or to the coupling operators alone (recallProposition 5.5.1). In the specific Ising example of Eq. (5.45), Davies generators areQL for three-body (next-to-NN, NNN for short) neighborhoods, Nj = j−1, j, j+ 1.Similarly, one can generalize the idea and define an enlarged “Davies QL notion”.With respect to this QL constraint, commuting Gibbs states may be shown to obeyour necessary and sufficient condition for FFQLS, as expected on physical grounds:
Proposition 5.5.4. (FFQLS commuting Gibbs states) Gibbs states of 1D NNcommuting Hamiltonians are FFQLS relative to the Davies (NNN) neighborhoodstructure.
Proof. Up to normalization and letting, for convenience, βH ≡∑n−1
j=1 Hj,j+1, 0 < β <
∞, the Gibbs state of Eq. (5.43) may be written as ρβ ≡ e−H12e−H23 . . . e−Hn−1,n =σ12σ23 . . . σn−1,n, where the σi,i+1 are pairwise-commuting, invertible matrices definedthat are different from the identity only in NN sites. Our strategy is to first computethe minimal fixed-point set Fρ234(Σ234(ρ)) and its intersection with Fρ123(Σ123(ρ)),and then, by iterating, to show that the resulting intersection is span(ρ). To computeFρ234(Σ234(ρ)), we first obtain the corresponding Schmidt span. Using Eq. (5.22),Σ234(ρ) = spanTr234 (ρM), for all M ∈ B(H1) ⊗ I234 ⊗ B(H5,...,n). Letting τ2 ≡
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
Direct calculation verifies that Fρ234(Σ234(ρ)) obeys the required properties of closureunder the distortion map Φρ234 and invariance under ρ234-modular action. Similarly,we have
Fρ123(Σ123(ρ)) = σ12σ23[I12 ⊗Fσ3(Σ3(σ34))].
Finally, we compute the intersection of these two adjacent fixed-point sets. Tohighlight the necessary structure, we write F i,i+1
σk≡ Fσk(Σk(σi,i+1)), where k = i or
i+ 1. The relevant intersection is then σ12σ23[I12⊗F34σ3⊗B(H4)]∩σ23σ34[B(H1)⊗
F12σ2⊗I3⊗F45
σ4]. Factoring out the common invertible multiple of σ23, this intersection
simplifies to
σ23[(σ12)⊗ (F34σ3⊗ B(H4)) ∩ (B(H1)⊗F12
σ2)⊗ (σ34F45
σ4)]]
= σ23[(σ12) ∩ (B(H1)⊗F12σ2
) ⊗ (F34σ3⊗ B(H4)) ∩ (σ34F45
σ4).
For each intersection, notice that one argument is contained in the other, giving
σ23[σ12 ⊗ σ34F45σ4
] = σ12σ23σ34[I123 ⊗F45σ4
].
By iterating, we find that subsequent intersections simplify to σ12 . . . σj−1,j[I1,...,j−1 ⊗F j,j+1σj
]. For the final intersection, we may take σn,n+1 = In⊗ 1n+1, where we take the(n+1)-th system to be trivial. This leads to Fσn(Σn(σn,n+1)) = span(In). Thus, after
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5.5 Illustrative Applications
taking the intersection over all fixed-point sets, we are left with⋂Fρj,j+1,j+2
So far, the identification of a commuting structure has played an important rolein the verification of the FFQLS property. Thus, it is an important question todetermine the extent to which “lack of commutativity” may hinder FFQLS. The issueis simpler and better (albeit still only partially) understood for pure target states,in which case families of QL stabilizable states not stemming from a commutingstructure have been identified for arbitrary system size and complex multi-partiteentanglement patterns. Notably, spin-1 AKLT states in 1D, which are the archetypalexample of a valence-bond-solid state in condensed-matter physics [185], as well asa spin-3/2 (or higher) AKLT states in 2D, which provide a resource for universalquantum computation [186, 187], are unique ground states of FF anti-ferromagnetHamiltonians. As such, they are FFQLS using NN, two-body dissipative dynamics[32]. Perhaps even more surprisingly, the FFQLS property still holds for long-rangeentangled states known as Motzkin states [188], which are also unique ground statesof FF NN spin-1 Hamiltonians and have been proved to (logarithmically) violate thearea law.
In what follows, we first exhibit a family of “non-commuting” FFQLS multi-quditgeneralized Dicke states, by also including a general result linking QL stabilizabilityof a pure state to its ability to be uniquely determined by its neighborhood marginals.Focusing then on mixed target states, we construct and analyze two explicit (non-scalable) examples showing that commutativity of the parent Hamiltonian or thegenerating QL circuit is, as for pure states, not necessary for FFQLS in general.
Pure Dicke states on qudits
Since the main emphasis of this chapter is on mixed states, most of the technicalproofs of the results in this section are omitted and can be found in Appendix C of[172]. Similar to qubit Dicke states from quantum optics [189], qudit Dicke statesmay be constructed by symmetrizing an n-qudit product state in which k of the nsubsystems are “excited” to a given single-particle state, and the remaining (n − k)are in their “vacuum”. We may further naturally generalize by allowing multi-level
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
excitation. Specifically, let |`〉, ` = 0, . . . , (d − 1), denote an orthonormal basisin Cd and Sn ≡ π the symmetric (permutation) group on n objects. Then eachgeneralized Dicke state is in one-to-one correspondence with a vector of integers,~Λ = (k0, . . . , kd−1), where each k` specifies the occupation number (multiplicity) ofeach single-qudit state. That is6,
|~Λ〉 ≡ |(0 . . . 0︸ ︷︷ ︸k0
, 1 . . . 1︸ ︷︷ ︸k1
, . . . , d− 1 . . . d− 1︸ ︷︷ ︸kd−1
)〉 ≡ 1
n!
∑π∈Sn
Vπ| 0 . . . 0︸ ︷︷ ︸k0
1 . . . 1︸ ︷︷ ︸k1
. . . d− 1 . . . d− 1︸ ︷︷ ︸kd−1
〉,
(5.47)where
∑i ki = N and Vπ permutes the subsystems according to the permutation π.
A useful fact to our purpose is that generalized Dicke states admit a simple Schmidtdecomposition. Consider a partition of the system into two groups of nA and nBsubsystems, respectively. It is then easy to show that the Schmidt decomposition of|~Λ〉 is
|~Λ〉 =1√(n~Λ
) ∑~ΛA+~ΛB=~Λ
µ~ΛA,~ΛB |~ΛA〉 ⊗ |~ΛB〉, µ~ΛA,~ΛB =
√(nA~ΛA
)(nB~ΛB
). (5.48)
In order to specify the relevant class of FFQLS generalized Dicke states, the choiceof the neighborhood structure is crucial. The following definition captures the re-quired feature:
Definition 5.5.5. A neighborhood structure Nk is connected if for any bipartitionof the subsystems, there is some neighborhood containing subsystems from both parts.
Our main result is then contained in the following:
Proposition 5.5.6. (FFQLS Dicke states) Given n qudits and a connected neigh-borhood structure N , there exists a (non-factorized) FFQLS generalized Dicke staterelative to N if d(m− 1) ≥ n, where m is the size of the largest neighborhood in N .
The proof (found in Appendix C of [172]) is constructive, and yields, in particular,the state
Dn,m ≡ |(0 . . . 0︸ ︷︷ ︸m−1
, 1 . . . 1︸ ︷︷ ︸m−1
, . . . , d− 1 . . . d− 1︸ ︷︷ ︸r
)〉, r = n− (d− 1)(m− 1), (5.49)
as a non-factorized (entangled) DQLS state for given, arbitrary system size. Nonethe-less, note that the product of the neighborhood size and qudit dimension must be
6States of this form have been recently analyzed in [190], where additionally non-uniform super-positions are also considered. We maintain permutation symmetry as a hallmark our generalizationof standard Dicke states.
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5.5 Illustrative Applications
scaled accordingly. For example, if N is fixed to be two-body NN (hence m = 2),then the qudit dimension itself must be at least d = n. In this sense, the resultingfamily of states is non-scalable.Remark 6. As a particular case of Propositions 5.5.6, we recover the fact (establishedin [31]) that the state |(0011)〉 = D4,3, also previously defined in Eq. (5.39), is FFQLSwith respect to the neighborhood structure N1 = 1, 2, 3, N2 = 2, 3, 4.
Generalized Dicke states are non-trivially entangled. Their multiparty correlationshave the feature of being uniquely determined by a proper subset of all possiblemarginals [190], that is, of being “uniquely joined” [61]. Remarkably, an interestingconnection may be made between the extent to which an arbitrary pure target stateis uniquely determined by the set of its neighborhood-marginals and the FFQLSproperty. This is formalized in the following:
Proposition 5.5.7. (Unique joinability) If a pure state |ψ〉 is FFQLS relativeto N , then |ψ〉 is uniquely determined by its neighborhood reduced states ρNj =TrN j(|ψ〉〈ψ|).
Proof. By contradiction, if |ψ〉 is not uniquely joinable, then there exists a state τ , notnecessarily pure, such that τ 6= |ψ〉〈ψ| and TrN j(τ) = ρNj , for all Nj ∈ N . Clearly,supp(τ) ≤ supp(ρNj) ⊗ B(HN j), for all j, hence by recalling Theorem 5.2.12 it alsofollows that
supp(τ) ≤⋂j
supp(ρNj)⊗ B(HN j) = span(|ψ〉).
On the other hand, since by assumption supp(τ) 6= span(|ψ〉), the above inclusionmust be strict, which is impossible since span(|ψ〉) is one-dimensional.
We now establish that the class of states of Eq. (5.49) constitute genuinely non-commuting examples of FFQLS. To do so, we show that, with respect to a certainclass of neighborhood structures, these states are the unique ground states of non-commuting FF QL Hamiltonians, but they cannot be the unique ground states of anycommuting FF QL Hamiltonians.
First, since the ground-state space of a FF Hamiltonian is simply the intersectionof the ground-state spaces of the individual Hamiltonian terms in the sum, replacingeach such term with the projector onto its excited space preserves the ground spaceof the global FF Hamiltonian. Without loss of generality, we may then restrict atten-tion to QL Hamiltonians consisting of sums of neighborhood-acting projectors. Animportant consequence of this simplification is that for a given FFQL Hamiltonian,the only possible candidates for other FFQL Hamiltonians with the same ground-state space are sums of projectors with enlarged ground-state spaces, with respect to
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
those of the given Hamiltonian. However, the following Lemma demonstrates that,in the two-neighborhood case, a non-commuting FFQL Hamiltonian does not admita commuting enlargement with the same ground state space:
Lemma 5.5.8. (Commuting enlargements) Assume that Π01,Π
02 are two non-
commuting projections such that
limn→∞
(Π01 Π0
2)n ≡ Π0GS 6= 0.
If Π1,Π2 are enlarged commuting projectors such that
Πk Π0k = Π0
k Πk = Π0k, k = 1, 2, (5.50)
then Π1Π2 ≡ ΠGS, with ΠGS Π0GS.
Proof. Using Eq. (5.50), we have that
Π01 Π0
2 = Π01 Π1 Π2 Π0
2, Π02 Π0
1 = Π02 Π2 Π1 Π0
1.
Towards proof by contradiction, assume that ΠGS = Π1Π2 = Π0GS. It follows that
Π01 Π0
2 = Π01 Π0
GS Π02, Π0
2 Π01 = Π0
2 Π0GS Π0
1.
By using the defining property of Π01,Π
02, however, the right hand-side in each of the
above equalities simplifies to Π0k Π0
GS Π0k = Π0
GS, for k = 1, 2. This in turn yields
Π01 Π0
2 = Π0GS = Π2
0 Π10,
which contradicts the non-commuting assumption.
With this Lemma in place, we now verify the genuine non-commutativity of thesestates.
Proposition 5.5.9. (Non-commutativity of FFQLS Dicke states) For each(non-factorized) Dicke state Dn,m of Eq. (5.49), there exists a neighborhood struc-ture for which Dn,m cannot be the unique ground state of any commuting FF QLHamiltonian.
Proof. Consider the (non-factorized) Dicke state Dn,m. Correspondingly, we chooseany connected neighborhood structure N withm-body neighborhoods, whereby thereis at least one neighborhood (N1, say) containing a system that is not contained in any
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5.5 Illustrative Applications
other neighborhood. Such a neighborhood structure always exists, and Proposition5.5.6 ensures that Dn,m is FFQLS relative to that. From [31], any FFQLS state|ψ〉 is the unique ground state of some FF QL Hamiltonian. In particular, letting|ψ〉 ≡ Dn,m, one such parent Hamiltonian is
H =∑j
Hj ≡∑j
(I− ΠNj ⊗ IN j), (5.51)
where ΠNj is the projector onto the Schmidt span of Dn,m with respect to Nj. We firstshow that theHj do not commute with one another. From the Schmidt decompositionof Dn,m, given in Eq. (5.48), we have ΠNj =
∑Λ |Λ〉〈Λ|, where the sum extends over
all choices of m symbols from the symbols in Dn,m = |~Λ〉. A direct calculation theshows that the ΠNj , and therefore, the Hj, of overlapping neighborhoods do notcommute with one another.
In order to establish the desired result, note that enlarging the neighborhoodspreserves FFQLS. Hence, Dn,m is also FFQLS with respect to the two-neighborhoodneighborhood structure with one neighborhood being N1 and the other neighborhood,say, NU , being the union of the remaining neighborhoods of N . Furthermore, bybuilding a parent Hamiltonian out of these two projectors as in Eq. (5.51), thereasoning above shows that the projectors of Dn,m with respect to N1 and NU donot commute. Hence, by Lemma 5.5.8, no commuting enlargement exists for N1,NU. Assume now that a commuting QL parent Hamiltonian exists for the originalneighborhood structure. The projectors Πk onto the ground state spaces of theseHamiltonians also commute. Along with FF condition, this implies that
|ψ〉〈ψ| = Π1Π2 . . .Π|N | ≡ Π1ΠU ,
where ΠU ≡ Π2 . . .Π|N |. Hence, the QL Hamiltonian H = (I − Π1) + (I − ΠU)constitutes a commuting enlargement for N1, NU, which we showed cannot exist.This contradiction then implies that no commuting enlargement can exist for N .
Non-commuting Gibbs states
In order to demonstrate that genuinely mixed target states may also be FFQLSdespite not being obviously associated to a commuting structure, specific examplesmay be constructed in 1D by considering a generalization of the NN Ising Hamiltonian
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
considered in Eq. (5.45), obtained by adding a transverse (magnetic) field. That is7:
H = −n−1∑j=1
ZjZj+1 − gn∑j=1
Xj, n ≥ 4, g ∈ R+. (5.52)
In particular, we consider (full-rank) Gibbs states, constructed as in Eq. (5.43),as well as variants inspired by non-equilibrium quench protocols, wherein an initialthermal state of a Hamiltonian with given g is evolved under a Hamiltonian withg′ 6= g. In all cases, the relevant minimal distorted algebras and their intersectionhave been numerically constructed (in Matlab) for given QL constraints, and Theorem5.4.4 used to determine FFQLS. The results are found to depend sensitively on theneighborhood structure: Gibbs (and generalized Gibbs) states on n = 4 qubits arefound to be FFQLS for three-body neighborhoods (as in the corresponding commutingIsing case), however extending to n = 5, 6 qubits requires neighborhoods to be furtherenlarged to allow for four-body Liouvillians.
While a direct (numerical) verification is beyond reach, this points to the possibil-ity that the (maximal) neighborhood size will have to scale extensively as n increases,thereby preventing scalable FFQLS. An intuitive argument in support of this is theobservation that, as the size of the “neighborhood complements” increase, the dimen-sions of the extended Schmidt spans do as well; correspondingly, the uniqueness oftheir intersection, as is required for FFQLS, becomes less likely. Despite this limita-tion, these results show the general applicability of our framework.
Entangled mixed states
As a final application, we analyze QL stabilizability of a one-parameter family ofmixed entangled states on n = 4 qubits. Beside illustrating the full procedure neededto check if ρ is FFQLS and to construct the stabilizing maps, this example is useful fora number of reasons: first, it reinforces that genuinely multipartite entangled mixedstates can be FFQLS; second, it explicitly shows that the support condition underwhich we proved sufficiency for general target states in Theorem 5.4.5 is not necessaryin general; lastly, it shows how a non-FFQLS state may still admit arbitrarily close (inHilbert-Schmidt space) states that are FFQLS, that is, in control-theoretic language,it may still in principle allow for “practical stabilization”. The family of mixed stateswe analyze may be parametrized as follows:
ρε ≡ (1− ε) |(0011)〉〈(0011)|+ ε |GHZ42〉〈GHZ4
2|, ε ∈ (0, 1), (5.53)7Hamiltonians such as in Eq. (5.52) are exactly solvable upon mapping to a free-fermion problem,
and commuting once expressed in terms of appropriate quasi-particles. Indeed, thermal dynamicsassociated to free-fermion QDS is known to be hypercontractive [154], in the absence of a QL “real-space” constraint as we consider.
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5.5 Illustrative Applications
with neighborhoods N1 = 1, 2, 3, N2 = 2, 3, 4. Here, GHZ42 is the usual GHZ
state on qubits, that is, |GHZnd〉 = (|0〉⊗n + . . .+ |d− 1〉⊗n) /√d. As established
in [31], this state is not DQLS for any non-trivial neighborhood structure8, as onemay verify by seeing that the d-dimensional space span|0, 0, . . . , 0〉, . . . , |(d− 1, d−1, . . . , d−1〉 is contained in each extended Schmidt span, and hence their intersectionis greater than just span(|GHZnd〉).
Let us use the notation 123|4 to denote the partition of the index set 1, 2, 3, 4 inthe neighborhood 1, 2, 3 and the remaining index 4, and similarly for 1|234: the1|234-Schmidt decomposition means the Schmidt decomposition with respect to suchbipartition. In order to construct QL FF dynamics which render ρε GAS, the first stepis to compute the operator Schmidt span for each neighborhood. Two properties aidour analysis. First, both the Dicke and GHZ components are permutation symmetric,so that the analysis of the 1|234 partition carries over to that of 123|4. Second,they have compatible Schmidt decompositions, in the sense that we can find a singleoperator basis in which to Schmidt-decompose both of them and their mixtures. Usingthat
|(0011)〉 =√
1/2|0〉|(011)〉+√
1/2|1〉|(001)〉,|GHZ4
2〉 =√
1/2|0〉|000〉+√
1/2|1〉|111〉,
the desired operator Schmidt decomposition is
ρε =1
2|0〉〈0| ⊗ [(1− ε)|(011)〉〈(011)|+ ε|000〉〈000|]
+1
2|0〉〈1| ⊗ [(1− ε)|(011)〉〈(001)|+ ε|000〉〈111|]
+1
2|1〉〈0| ⊗ [(1− ε)|(001)〉〈(011)|+ ε|111〉〈000|]
+1
2|1〉〈1| ⊗ [(1− ε)|(001)〉〈(001)|+ ε|111〉〈111|].
Let us focus on the factors relative to subsystems 234 from each term above. Tocompute the minimal fixed-point set containing this Schmidt span, we first undo thedistortion of the elements of this space by conjugating with respect to
8 GHZ states are graph states, though with respect to a star graph (i.e., a central node connectedto (n− 1) surrounding nodes). The neighborhood structure induced by this graph is trivial, in thatit consists of a single neighborhood containing all n qubits. Hence, we treat GHZ states as separatefrom graph states, considering only non-trivial QL constraints.
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
where, as noted, the above inverses are taken as the Moore-Penrose inverse (with1√εor 1√
1−ε being replaced by 0 in the singular cases of ε = 0 or 1, respectively).
Conjugation of each Schmidt basis element with respect to this ρ−12
where |e1〉, |e2〉, |e3〉, |e4〉 are chosen to ensure orthonormality. This transformationreveals that the Schmidt-span operators share a common identity factor, as in thisbasis they read:
(|0〉〈0| ⊗ I)⊕ O, (|0〉〈1| ⊗ I)⊕ O, (|1〉〈0| ⊗ I)⊕ O, (|1〉〈1| ⊗ I)⊕ O,
where the sector on which the zeros act is span|e1〉, . . . , |e4〉. The span of theseoperators is closed under ∗-algebra operations and constitutes a representation ofthe Pauli algebra. Thus, the distorted Schmidt spans of ρε are already *-closedalgebras. A simple calculation verifies that each distorted-algebra basis element isalso unchanged by M 1
2. To find the minimal fixed-point sets, we need to apply
the distortion map again, by conjugating the generators with ρ12234. We can write the
Schmidt decomposition, with respect to the basis in Eq. (5.54), as
|0〉〈0| ⊗ τ ⊕ O, |0〉〈1| ⊗ τ ⊕ O, |1〉〈0| ⊗ τ ⊕ O, |1〉〈1| ⊗ τ ⊕ O,
where we have defined τ ≡ ε|0〉〈0|+ (1− ε)|1〉〈1|.The last step is to construct QL Liouvillians for each neighborhood. The require-
ment is that the kernels of each of these are the corresponding minimal fixed-pointsets. This can be obtained by considering the operators:
The first Lindblad operator L0 is responsible for asymptotically preparing the sub-spacespan|000〉, |(011)〉, |111〉, |(001)〉, while L+ and L− stabilize the τ factor. All threeLindblad operators must commute with the distorted algebra in order that it be pre-served. Using the standard definitions of ladder operators, σ+ ≡ |0〉〈1| = (σ−)†, we
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5.5 Illustrative Applications
rewrite the τ -preparing Lindblad operators back in the original basis, in terms ofstandard Pauli matrices, as
e2πi/3, and similar terms to denote symmetric basis elements for the four dimensionalspace orthogonal to the symmetric subspace, the third Lindblad operator reads
These Lindblad operators form the neighborhood Liouvillian on systems 234, namely,
L234(ρ) = L0ρL†0 + L+ρL
†+ + L−ρL
†− −
1
2L†0L0 + L†+L+ + L†−L−, ρ,
The global generator L is obtained by constructing L123 in an analogous way, and byletting L = L234 + L123.
Using Matlab, we have verified that these dynamics are FF and stabilize ρε; thekernel of L is equal to span(ρε), as desired. In the limiting cases of ε = 0, 1, theabove dynamics fail to have a unique fixed state. As we have already established, forthe Dicke-state case of ε = 0, FFQL stabilizing dynamics can be constructed by aseparate procedure, whereas for the GHZ case of ε = 1, no FFQL stabilizing dynamicsexists relative to the given neighborhoods.
Remark 7: Failure of support condition. In Theorem 5.4.5, in order to obtaina general sufficient condition for FFQLS, we supplemented the necessary conditionwith the “support condition” of Eq. (5.33). We now show that ρε fails the supportcondition despite being FFQLS. The support of ρε is spanned by just |(0011)〉 and|GHZ4
Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
Since this intersection properly contains the support of ρε, the claim follows.Remark 8. Since stabilization of ρε is possible for ε arbitrarily close to the GHZ-value of one, the above example demonstrates that there exist FFQLS mixed statesthat are arbitrarily close to a non-FFQLS pure state. While this allows for practi-cal stabilization [174] or approximate stabilization in principle, we expect that theLiouvillian spectral gap will close as the non-FFQLS state is approached – makingstabilization inefficient. If we normalize the Liouviallians to ||L||2 = 1, the corre-sponding gaps ∆ are found to behave as ∆ ≈ 0.049 (1− ε). It remains an interestingquestion for future investigation to determine whether similar conclusions about QLpractical stabilization and associated efficiency trade-offs may be drawn for more gen-eral target states. In the following subsection we make these notions more preciseand give a few examples.
5.5.5 Approximate FFQLS
So far we have considered FFQL stabilizability as a binary property of quantum states.In practice, however, it is only ever possible (and suffices for all intents and purposes)to stabilize an approximate version of the target state. Furthermore, with time-independent continuous-time dynamics that we have considered, under even idealconditions, for any finite wait time the target state is only reached approximately.Thus, as mentioned in the previous subsection, it is interesting to explore the notionof practical or approximate stabilization.
There are various ways to define approximate stabilization. In the context ofcontinuous-time stabilization that we have considered, it is useful to define a state tobe approximately FFQLS if there is an FFQLS state in its proximity.
Definition 5.5.10. A state ρ is said to be ε-FFQLS with respect to neighborhoodstructure N if there exists a state ρ′ which is FFQLS and satisfies 1
2||ρ − ρ′||1 ≤ ε,
where || · ||1 denotes the trace-norm.
We give two examples of states which are ε-FFQLS for arbitrarily small ε, despite notbeing FFQLS.
Consider the 1D Ising Hamiltonian example from Sec. 5.5.3. The Hamiltonian isH = −
∑ni=1 ZiZi+1 with Gibbs state ρβ = exp(−βH)/Tr (exp(−βH)). From either
the results of [156] or of Sec. 5.5.3, this Gibbs state can be shown to be FFQLSwith respect to the NNN neighborhood structure for any finite β. However, in thezero-temperature limit, the Gibbs state converges to the non-full rank state
ρ∞ ≡ limβ→∞
ρβ =1
2(|0〉〈0|⊗N + |1〉〈1|⊗N).
This classically correlated state coincides with that of Eq. (5.37), which was proven
120
5.5 Illustrative Applications
not to be FFQLS. As β increases, ρβ gets arbitrarily close to ρ∞, implying that ρ∞ isε-FFQLS for all ε. To make this rigorous, we fix an arbitrary ε and show that thereexists β such that ||ρ∞ − ρβ||1 ≤ ε.
To upper bound the trace distance, we use the inequality 12||ρ−σ||1 ≤
√1− F (ρ, σ)2,
where the fidelity of two states is F (ρ, σ) ≡ Tr(√√
ρσ√ρ). For commuting density
operators, as in our example, the fidelity simplifies to F (ρ, σ) =∑
i
√piqi, where the
pi and qi are corresponding eigenvalues in the common eigenbasis. Since ρ∞ is rank-2with eigenvalues 1
2, the fidelity sum involves just two terms
F (ρ∞, ρβ) =
√1
2〈0|⊗Nρβ|0〉⊗N +
√1
2〈1|⊗Nρβ|1〉⊗N =
√2〈0|⊗Nρβ|0〉⊗N . (5.55)
where the second equality is a simplification due to the symmetry of ρβ. The par-tition function of the N -spin periodic boundary condition Ising model is calculatedto be Z = Tr
(e−βH
)= 2N(coshN(β) + sinhN(β)). Then, after calculating that
〈0|⊗Nρβ|0〉⊗N = eβN
2N (coshN (β)+sinhN (β)), we can upper bound the trace-norm distance as
1
2||ρ∞−ρβ||1 ≤
√1− 2eβN
2N(coshN(β) + sinhN(β))=
√1− 2
(1 + e−2β)N + (1− e−2β)N.
(5.56)We must choose β such that ε is an upper bound for this expression. Set α = e−2β
and rewrite the ε bound inequality as
1
2((1 + α)N + (1− α)N) ≤ 1
1− ε2, (5.57)
On the left-hand side, we find that the odd terms in the binomial expansions cancel,leaving
By truncating the Taylor expansion of 11−ε2 , we obtain the lower bound 1 + ε2 + . . .+
(ε2)bN/2c ≤ 11−ε2 . Comparing the two finite sums, we establish that choosing β such
that Nα = Ne−2β = ε ensures the inequality of Eq. (5.57) to hold. Thus, for anyε, setting β = 1
2log(N/ε) ensures that 1
2||ρ∞ − ρβ||1 ≤ ε. For any ε, we can choose
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
an FFQLS state ρβ that is within distance ε from ρ∞. This establishes that ρ∞ isε-FFQLS for arbitrarily small ε.
For the last example, we show that the n-qubit GHZ state is ε-FFQLS with re-spect to the NNN neighborhood structure for arbitrary ε. This example is somewhatsurprising because, not only is GHZ not FFQLS, but it is not even QLS while allowingHamiltonian control or frustrated dynamics, as shown in [34]. Towards this, we provea result which connects FFQLS of pure states to a tensor network structure.
Proposition 5.5.11. If a state |ψ〉 ∈ H⊗n can be written as the product of invertible,mutually commuting NN-acting matrices Mi,i+1 applied to a product state |φ〉⊗n, then|ψ〉 is FFQLS with respect to the NNN neighborhood structure.
Proof. By hypothesis, |ψ〉 = M12M23 . . .Mn−1,n|φ〉⊗n. We check the pure state FFQLScondition (i.e. the DQLS condition of [34]),
span(|ψ〉) =n−2⋂j=1
Σj,j+1,j+2(|ψ〉). (5.60)
For any invertible matrixM , and vector subspaces U , V , andW , it is simple to provethat W = U ∩ V if and only if MW = (MU) ∩ (MV ). Let M ≡ M12 . . .Mn−1,n.Then the above condition is true if and only if
span(M−1|ψ〉) =n−2⋂j=1
M−1Σj,j+1,j+2(|ψ〉). (5.61)
Consider any intersection argument for 2 ≤ j ≤ n − 3 (i.e. excluding the boundaryarguments). We can simplify as
Since M−1|ψ〉 ∈ M−1Σj,j+1,j+2(|ψ〉) for all j, the FFQLS intersection condition issatisfied by |ψ〉.
We demonstrate that the n-qubit GHZ state can be δ-approximated by a state ofthe form above. Define the two-qubit invertible matrix
M =√
cosh(ε)(|00〉〈00|+ |11〉〈11|) +√
sinh(ε)(|01〉〈01|+ |10〉〈10|).
LetMi,i+1 = Mi,i+1⊗Ii,i+1. SinceMi,i+1 is diagonal in the computational basis, it com-mutes with allMi′,i′+1. The approximate GHZ state is |ψε〉 = 1√
NM12 . . .Mn−1,n|+〉⊗n,
where settingN = cosh[(n−1)ε]2n−1 gives the proper normalization. The trace-norm distance
between |ψε〉 and |GHZ〉 is
1
2|| |GHZ〉〈GHZ| − |ψε〉〈ψε| ||1 =
√1− |〈GHZ|ψε〉|2
=
[1−
(√1
2N〈0|⊗nM12 . . .Mn−1,n|ψε〉
−√
1
2N〈1|⊗nM12 . . .Mn−1,n|ψε〉
)21/2
=
1−
(2
√1
2N(cosh ε)(n−1)/2
(1√2
)n)21/2
=
(1− cosh(n−1) ε
cosh[(n− 1)ε]
)1/2
=
(1−
1 + (n− 1) ε2
2!+ . . .
1 + (n− 1)2 ε2
2!+ . . .
)1/2
≤ ε
√(n− 1)(n− 2)
2. (5.66)
Therefore, for any δ, by choosing ε =√
2(n−1)(n−2)
δ, we ensure that the FFQLS state|ψε〉〈ψε| is within trace-norm distance δ of the n-qubit GHZ state. This leads us to the
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Asymptotic stabilization of quantum states with continuous-timequasi-local dynamics
surprising conclusion that, even for a state which fails to be stabilizable in the weakestsense (i.e. GHZ is not even QLS with respect to frustrated QL dynamics), there mayexist an arbitrarily close proxy state, which is stabilizable in a much stronger sense.As with the GHZ-Dicke mixture from the previous chapter, we expect the scaling ofthe time needed to stabilize such an approximate (with respect to the proximity) tobe unfavorable.
124
Chapter 6
Finite-time stabilization of quantumstates with discrete-time quasi-localdynamics
125
Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
This chapter presents material based on a paper titled “Exact stabilization of en-tangled states in finite time by dissipative quantum circuits” that is currently beingfinalized for submission and is joint work with Francesco Ticozzi and Lorenza Viola.
6.1 IntroductionPreparation of a target quantum state is a ubiquitous task in quantum informationprocessing. In particular, measurement based quantum computing using graph states[191] or AKLT states [192] relies on initializing the system into a desired resource state.A more recent application of quantum state preparation is to quantum Gibbs samplers[156]. Determining the feasibility of preparing of a target state is only an interestingor relevant question when there are limitations to the control of the system. Theparticular control limitations from one implementation to the next can vary widely.Yet, a physically motivated and commonly assumed constraint is quasi-locality. Here,the dynamics are constrained to addressing only certain sets of subsystems. As anexample, in an ion trap, one might only have control over nearest-neighbor couplingsbetween adjacent ions, while being unable to directly couple distant ions.
The standard approach to preparing a target pure state on a quantum computerbegins with the initialization of the systems into a fiducial product state (e.g. |0〉⊗N)after which a circuit of unitary gates transforms the initial state into the desired state.Other approaches attempt to cool the quantum system to drive the system into anapproximation of the ground state of the Hamiltonian [19]. An alternative approach,known as sequential generation [193], exploits a matrix product state (MPS) or pro-jected entangled pair state (PEPS) representation of the target state. The tensorsin the matrix product determine a sequence of CPTP maps which act in a specifiedorder to transform any unspecified input state into the target state.
However, as emphasized in [194], such preparation schemes face a number of draw-backs. Most prominently, in these schemes, the target state is only available at theend of a “cycle”. The action of the intermediate steps alters the target state. Thepreparation scheme of target state stabilization addresses these drawbacks. With sta-bilizing dynamics, we demand that the CPTP maps which both drive the system tothe target state with all-to-one dynamics and leave the target state invariant in eachstep.
An important dichotomy among dynamical models is discrete vs continuous time.Much recent work has been devoted to analyzing how to engineer time-independentcontinuous-time Liouvillian dynamics for stabilizing a target state [31, 34, 172]. Withsuch continuous-time dynamics, an arbitrary input state is driven asymptotically to-wards the target state; the target state is only prepared in infinite-time.
In contrast, discrete-time dynamics, or a dissipative quantum circuit, have the po-tential for stabilizing a target state in finite time. In [194] we analyze the discrete-time
126
6.2 Preliminaries
analog of the above continuous-time cases; a fixed set of CPTP maps are alternatinglyapplied to drive the system towards the target state. Generally, the target state is ap-proached asymptotically, with stabilization in infinite-time. However, in cases whenthe CPTP maps commute with one another, the target state is stabilized in finitetime. When the target state is stabilized independent of the order of the maps, we re-fer to the implementation as robust. Both the finite-time and robust implementationproperties are favorable from a control-theoretic perspective.
Two questions naturally arise:
(a) What ensures that a target state can be stabilized in finite time?
(b) Furthermore, what properties enable a robust implementation of the stabiliza-tion?
In this chapter we explore aspects of discrete-time stabilization of a target pure statein finite-time, providing some existential results as well as some results the provide asynthesis for the stabilizing dynamics. In Section 6.3 we develop both necessary andsufficient conditions for determining if a target state can, in principle, be finite-timestabilized. In the case that a state is verified to be finite-time stabilizable (FTS), weprovide a scheme for its stabilization, although we lack an algorithm for synthesizinga part of the dynamics that is only shown to exist in principle. In Section 6.2 weprovide the necessary background and mathematical tools used in this chapter. InSection 6.4 we turn to the notion of robust finite-times stabilization (RFTS), giving anumber of examples and several necessary conditions which restrict the correlationson a target state if it is to be RFTS. In Section 6.5 we develop several sufficientconditions which ensure RFTS of a target state. Here, a useful structure that werely on is a virtual subsystem factorization of the target state which facilitates robustfinite-time stabilization. In Section 6.6 we explore the efficiency of the RFTS schemefor quasi-locality defined on a lattice and then make connections between RFTS andthe existence of quasi-local continuous-time dynamics which “rapidly” prepare anequilibrium state. In Section 6.7 we extend a number of results to the case of a mixedtarget state. Finally, we have reserved all of the technical proofs to a final section atthe end of the chapter.
H. The state of the system at each time is a density operator in the space of positive-semidefinite, trace-one operators, denoted D(H). The time evolution of the systemis modeled by non-homogeneous Markov dynamics. Such dynamics are representedby sequences of completely positive trace-preserving (CPTP) maps Et, whereby theevolution of the state ρt from step t to t+ 1 is
ρt+1 = Et(ρt). (6.1)
For any t > s ≥ 0, we denote the evolution map from s to t as
Et,s = Et Et−1 . . . Es. (6.2)
We assume that each step in the dynamics is engineered towards achieving stabiliza-tion of a target state. In a practical setting, such control of the quantum system canbe limited by a variety of constraints. Without any restrictions, stabilization of anarbitrary target state becomes trivial. We thus assume that each map Et is chosenfrom some set of “available” maps reflecting the limited control of the physical system.In particular, we assume that each map must act quasi-locally. Following [31, 34, 172],the notion of quasi-locality we consider is described by a neighborhood structure onthe multipartite Hilbert space. We define a neighborhood structure N to be specifiedby a list of subsets of subsystems, Nk ⊆ 1, . . . , N, for k = 1, . . . , T . As an example,the 1-D nearest-neighbor (NN) neighborhood structure is depicted in Fig. 6.1.
Definition 6.2.1. A CPTP map E is a neighborhood map with respect to a neigh-borhood Nj if
E = ENj ⊗ INj , (6.3)
where ENj is the restriction of E to operators on the subsystems in Nj and INj is theidentity map for operators on HNj . A QDS is quasi-local with respect to a neighbor-hood structure N if, for each map Et in the sequence, Et is a neighborhood map withrespect to some Nj ∈ N .
A useful tool for analyzing the neighborhood-wise features of a quantum state isthe Schmidt span of a linear object (vector, operator, or tensor) [172]:
128
6.2 Preliminaries
Definition 6.2.2. Given the tensor product of two finite-dimensional inner-productspaces W1 ⊗W2 and v ∈ W1 ⊗W2 with Schmidt decomposition v =
∑i siv
i1 ⊗ vi2, the
Schmidt span of v with respect to W1 is defined as Σ1(v) ≡ spanvi1.
We will often use the notion of an extended Schmidt span, which is denoted anddefined as Σ1(v) ≡ Σ1(v) ⊗W2. We will mostly make use of the extended Schmidtspan of the target state |ψ〉 with respect to neighborhood Hilbert spaces, ΣNk(|ψ〉) =ΣNk(|ψ〉)⊗HN k .
6.2.2 Convergence notions
The task that we analyze in this chapter is the design of dynamics which drive asystem towards a target state. The following definitions are used to make rigorousthe notion of target state stabilization as discussed in the introduction.
A state ρ ∈ D(H) is invariant with respect to the dynamics Es,tt≥s≥0 if Et,s(ρ) =ρ for all τ ∈ S and all t ≥ s ≥ 0. With this, we define a notion of convergence towardsthe target state:
Definition 6.2.3. With respect to a dynamical semigroup Es,tt≥s≥0, an invariantstate ρ is globally asymptotically stable (GAS) if, for any initial state σ,
limt→∞|Et,s(σ)− ρ| = 0, ∀s ≥ 0. (6.4)
Following [194], we define a notion of GAS with respect to a quasi-local dynamicalsemigroup.
Definition 6.2.4. A target pure state ρ = |ψ〉〈ψ| is said to be quasi-locally stabiliz-able (QLS) with respect to neighborhood structure N if there exists a sequence Ett≥0
of N -acting maps rendering ρ GAS.
A main result in [194] establishes the following necessary and sufficient conditionfor determining whether or not a target pure state is QLS. Adapting the notation toour purposes, we have:
Theorem 6.2.5 ([194]). A target pure state ρ = |ψ〉〈ψ| is QLS if and only if
span(|ψ〉) =⋂k
ΣNk(|ψ〉). (6.5)
For certain target states, the asymptotic scheme for QLS devised in [194] convergesto the target state exactly in a finite number of steps. In this work, our aim is tobuild off of the QLS results in [194] to determine further conditions on the targetstate which enable such finite-time quasi-local stabilization.
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
Definition 6.2.6. A target pure state ρ = |ψ〉〈ψ| is quasi-locally finite-time stabiliz-able (FTS) with respect to neighborhood structure N if there exists a finite sequenceEtTt=1 of N -acting maps satisfying
(a) Ei(ρ) = ρ, for all i = 1, . . . , T , and
(b) ET . . . E1(σ) = ρ for all σ ∈ D(H).
Furthermore, ρ is said to be robustly FTS (RFTS) if property 2 holds for any per-mutation of the T maps.
Although the focus of this chapter is on discrete-time dynamics, we briefly describethe analogous continuous-time dynamics in order to draw parallels between the twocases and to make contact with previous work. A continuous-time Markovian semi-group is generated by a Liouvillian master equation,
ρ = L(ρ) ≡∑j
LjρL†j −
1
2L†jLj, ρ, (6.6)
where L is a linear superoperator and Lj are Lindblad operators which may vary intime. Following [172], a Liouvillian generator L is QL with respect to neighborhoodN if L =
∑k LNk ⊗IN k , where I denotes the identity superoperator. Previous work
has explored quasi-local stabilization of a target state with respect to such continuous-time dynamics [158, 31, 34, 172]. We emphasize that, in contrast to the finite-timediscrete dynamics that we consider in this chapter, time-independent continuous-timedynamics can never stabilize a target state exactly for any finite wait time. One of themain motivations of this work, then, is to extend these investigations of continuous-time stabilization to the discrete-time case, where finite-time stabilization is possible.
6.2.3 Quasi-local parent Hamiltonians
In this subsection we introduce a useful tool for analyzing QLS, FTS, and RFTS.Remarkably, QLS can be related to the existence of a particular Hamiltonian. Aquasi-local Hamiltonian H ≡
∑kHk =
∑kHNk ⊗ IN k is frustration-free (FF) if the
ground state space of H is contained in the ground state space of each neighborhoodterm Hk. In particular, if a state |ψ〉 has minimal energy with respect to H, it hasminimal energy with respect to each Hk.
A corollary in [194] shows that a target pure state |ψ〉 is QLS with respect to Nif and only if |ψ〉 is the unique ground state of some frustration-free (FF) quasi-localHamiltonian H =
∑kHNk⊗ IN k . This result, then, provides a physical interpretation
for the QLS scheme: each neighborhood map serves to locally “cool” the system with
130
6.2 Preliminaries
respect to Hk, from which the FF condition on H ensures that these local coolingscollectively achieve the global cooling to |ψ〉, the unique ground state of H.
A Hamiltonian for which a state is a ground state is known as a parent Hamilto-nian. Among FF parent Hamiltonians of a given state, one is uniquely constructedfrom the state itself:
Definition 6.2.7. The canonical FF Hamiltonian is defined as
H|ψ〉 =∑k
(I− ΠNk ⊗ IN k), (6.7)
where ΠNk is the projector onto the Schmidt span ΣNk(|ψ〉).
This Hamiltonian satisfies the following universal property: if there exists a FFHamiltonian with |ψ〉 as the unique ground state, then |ψ〉 is the unique ground stateof the canonical FF Hamiltonian H|ψ〉. Thus, a target pure state |ψ〉 is QLS if andonly if it is the unique ground state of its canonical FF Hamiltonian.
As FTS and RFTS are special cases of QLS, a necessary condition for target state|ψ〉 to be FTS or RFTS is that it satisfies Eq. (6.5). We seek further conditionswhich ensure FTS or RFTS of a target state. Especially in the RFTS case, we areled to investigate further properties of the canonical FF Hamiltonian. In [194], thestates which were found to be RFTS had canonical FF Hamiltonians for which theneighborhood-acting terms commuted with one another, pairwise. Commutativity ofa Hamiltonian is a useful property which can be related to the concept of hypercon-tractive dynamics [154] and rapid thermalization of a Gibbs state [156].
Surprisingly, we show that commutativity of the canonical FF Hamiltonian is notnecessary for a state to be RFTS. Nevertheless, in developing sufficient conditions forRFTS in Section 6.5 we will use certain algebraic structures induced by the canonicalFF Hamiltonian to diagnose RFTS. Furthermore, although the canonical FF Hamil-tonian need not commute for a state to be RFTS, we conjecture that if |ψ〉 is RFTS,then there exists some commuting quasi-local Hamiltonian for which |ψ〉 is the uniqueground state.
Unlike the property of QLS, we lack a general necessary and sufficient conditionfor FTS or RFTS. However, in Section 6.5.2, by restricting to certain neighborhoodstructures we establish that |ψ〉 is RFTS if and only if the canonical FF Hamiltonianis commuting and |ψ〉 is its unique ground state.
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
6.3 Finite-time stabilization
6.3.1 Conditions for finite-time stabilization
We state a necessary condition for a target state to be finite time stabilizable. Asalready remarked, proofs are deferred to the final section.
Theorem 6.3.1. A pure state |ψ〉 is FTS with respect to N only if |ψ〉 satisfies Eq.(6.5) and there exists at least one neighborhood Nk ∈ N for which
dim(HNk) ≥ 2 · dim(ΣNk(|ψ〉)). (6.8)
We refer to the above condition as the small Schmidt span condition. In principle,the small Schmidt span could be implied by satisfaction of Eq. (6.5), and, hence, beredundant. We give an example which shows that this condition is truly necessary.This verifies the existence of states which are only QLS in infinite-time.
Example 6.3.2 (2D AKLT state). Consider a system of six spin-3/2 systems.We define the spin-3/2 AKLT state on this system as described in Fig. 6.2. Theneighborhood structure consists of pairs of systems that are connected by an edge inFig. 6.2. AKLT states are the unique ground states of particular non-commutingtwo-body FF Hamiltonians, where the two-body terms act on the edges of the graphwhich define the state (i.e. the neighborhoods). Thus, the AKLT state satisfies Eq.(6.5) (which also follows from analysis in [158]). Nevertheless, as verified numericallyin MATLAB, this state violates the small Schmidt span condition.
The necessity of the small Schmidt span condition can be understood intuitively.It must be the case that, for some step T in the sequence, a neighborhood map ETtransforms some mixture of states into the target state. This mixture must includea contribution from a state |ψ⊥〉〈ψ⊥| that is orthogonal to the target state so thatET (λ|ψ⊥〉〈ψ⊥| + (1 − λ)σ) = |ψ〉〈ψ| for some state σ. Hence, this neighborhoodmap must take |ψ⊥〉〈ψ⊥| into |ψ〉〈ψ|. This action can be viewed as correcting aneighborhood-acting error on |ψ〉. If the Schmidt span of |ψ〉 on the neighborhood NTof ET is too large, no neighborhood-acting errors can transform |ψ〉 into an orthogonalstate. This highlights an important difference between finite-time and asymptoticstabilization. From the view of error correction, asymptotic dynamics are able tocorrect errors which cannot be exactly corrected by discrete dynamics in finite time.
Next, we describe a checkable condition for determining if a state is FTS. To provethat any state satisfying this condition with respect to a neighborhood structure isFTS, we construct a sequence of CPTP maps which render the state FTS. A crucialcomponent of the scheme that we present is the use of neighborhood-acting unitarymaps. One might guess that unitary maps would not be useful for the task of state
132
6.3 Finite-time stabilization
Figure 6.2: Example of a non-FTS state: AKLT state on a bipartite cubic graph.The pairs of nodes connected by an edge are virtual spin-1/2 particles in a singletstate. The perforated circles contain the systems which are projected into the spin-3/2subspace. The resulting system describes 6 spin-3/2 particles. As verified numericallyin MATLAB, this AKLT state violates the small Schmidt span property since for eachtop-bottom pair of systems (i.e. each neighborhood), the Schmidt span dimension(= 9) exceeds half the Hilbert space dimension (= 16/2).
stabilization, as they are not entropy-decreasing. However, in the scheme we develop,unitary maps play a crucial role in stabilizing a target state in finite time. LetU(H) to denote the group of unitaries acting on H, and u(H) the corresponding Liealgebra. The unitary subgroups (subalgebras) that we define below are subgroups(subalgebras) of U(H) (u(H)).
Definition 6.3.3. The unitary stabilizer group of a vector |ψ〉 ∈ H is defined as
U|ψ〉 ≡ U ∈ U(H) |U |ψ〉〈ψ|U † = |ψ〉〈ψ|.
This is a Lie subgroup of U(H). The Lie algebra associated to U|ψ〉 is denoted u|ψ〉.The neighborhood unitary stabilizer group of a vector |ψ〉 ∈ H with respect to Nk
is defined as
UNk,|ψ〉 ≡ U ∈ U(HNk)⊗ IN |U |ψ〉〈ψ|U † = |ψ〉〈ψ|.
The Lie algebras associated to these Lie groups are denoted uNk,|ψ〉.
An important component of our FTS scheme is the decomposition of elements ofthe global stabilizer group U|ψ〉 into a finite product of elements from the neighborhoodstabilizer groups UNk,|ψ〉. The following proposition describes a checkable conditionfor determining whether or not such a decomposition is possible.
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
Proposition 6.3.4 (Unitary generation property). Given a state |ψ〉 and aneighborhood structure N , any element in U|ψ〉 can be decomposed into a finite productof elements in UNk,|ψ〉 if and only if
〈uNk,|ψ〉〉k = u|ψ〉, (6.9)
where 〈·〉k denotes the smallest Lie algebra which contains all Lie algebras from theset indexed by k.
The linear algebraic closure, 〈·〉k, may be computed numerically. Hence, for a givenstate, we may determine whether or not the unitary generation property holds usingsoftware such as MATLAB. However, we note that determining such a decompositionmight be difficult in practice.
Some care must be taken when we apply the unitary generation property. Considera neighborhood structure comprised of two disjoint sets of neighborhoods (i.e. noneighborhood from the first set and from the second set have non-trivial intersection),giving, say, H ' HL ⊗ HR. Towards checking FTS, Eq. (6.5) can only be satisfiedif the target state |ψ〉 = |ψ〉L ⊗ |ψ〉R is factorized with respect to HL and HR. Withthis product state, the neighborhood unitary stabilizers can, at most, generate U|ψ〉
L⊗
U|ψ〉R, which is strictly smaller than U|ψ〉. Disconnected neighborhood structures will
never allow the unitary generation property to hold. Trivially, any product state isFTS with respect to a disconnected neighborhood structure. What is needed then,is that the unitary generation property hold for each connected component of theneighborhood structure. For example, the unitary generation property should bechecked for |ψ〉L with respect to the corresponding set of neighborhoods and similarlyfor |ψ〉R. We thus simplify our analysis by only considering neighborhood structureswhich are connected, as disconnected neighborhood structures trivially limit many-body entanglement.
The following example demonstrates the general scheme that we will use in veri-fying that a state is FTS.
Example 6.3.5 (Dicke state). Consider a four qubit system with a connected neigh-borhood structure N = N1,N2, with N1 = 1, 2, 3 and N2 = 2, 3, 4. The Dickestate
|(0011)〉 ≡ 1√6
(|0011〉+ |0101〉+ |0110〉
+|1001〉+ |1010〉+ |1100〉) (6.10)
was shown to be stabilizable asymptotically in [194]. We show that this state is FTSwith respect to the same 3-body neighborhood structure. The Schmidt span of |(0011)〉with respect to N1 is ΣN1(|(0011)〉) = span|(001)〉, |(011)〉. Thus, the small Schmidt
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6.3 Finite-time stabilization
span condition is satisfied:
dim(HNk)dim(ΣNk(|ψ〉))
=8
2= 4 ≥ 2.
In our FTS scheme, entropy is removed from the system only by the action of adissipative map W123 acting on neighborhood N1. This contrasts the QLS scheme of[194], wherein dissipative maps alternatingly act on the two neighborhoods to asymp-totically drive the system towards the target state. There, in stabilizing |(0011)〉, theneed for infinite-time convergence reflects some degree of “competition” between thetwo dissipative maps. For states satisfying certain sufficient conditions, we achievefinite-time convergence by designing “collaboration” among the CPTP maps.
In our FTS scheme, W123 is used just twice. Between these, in a finite sequence,unitaries alternatingly act on the two neighborhoods. As we show, W123 maps anydensity operator with support in a particular four-dimensional subspace into the targetstate. The unitaries serve to transform the state of the system into this space. Then,the final action ofW123 maps this state to the target. The general scheme also employsthis approach of interspersing a single-neighborhood dissipative map with sequences ofunitaries, the maps collaborating to achieve FTS. However, in general, we will need tocall on more than just two uses of the dissipative map and will need to design distinctsequences of unitaries for each step.
The dissipative map W123 =∑
iKi ·K†i is defined by its Kraus operators
K0 ≡ (|(001)〉〈(001)|+ |(011)〉〈(011)|)⊗ I,
K1 ≡ (|(001)〉〈000|+ |(011)〉〈111|)⊗ I,
K2 ≡ (|(001)〉〈(001)ω|+ |(011)〉〈(011)ω|)⊗ I,
K3 ≡ (|(001)〉〈(001)ω2 |+ |(011)〉〈(011)ω2|)⊗ I,
where |(abc)ν〉 ≡ 1√3(|abc〉+ ν|bca〉+ ν2|cab〉) and ω ≡ e
2πi3 . W123 maps the following
four orthogonal states (including the target state, itself) into |(0011)〉:
|ψ0〉 ≡ |(0011)〉,|ψ1〉 ≡ (|000〉|1〉+ |111〉|0〉)/
√2,
|ψ2〉 ≡ (|(001)ω〉|1〉+ |(011)ω〉|0〉)/√
2,
|ψ3〉 ≡ (|(001)ω2〉|1〉+ |(011)ω2〉|0〉)/√
2.
The range ofW123 is the set of operators with support on the Schmidt span ΣN1(|(0011)〉).Thus, we design a sequence of neighborhood unitaries Ui which maps ΣN1(|(0011)〉)
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
where UR is any matrix which ensures that U is unitary. The fact that U can bedecomposed into such a product is ensured by the fact that |(0011)〉 satisfies the Liealgebraic generation property of Eq. (6.9). This condition was checked using MatLab.We do not determine the actual form of the Ui, but this check ensures us that such afinite sequence of unitaries exists.
Finally, a simple calculation shows that
W123 UT . . . U2 U1 W123(I/16) = |(0011)〉〈(0011)|.
Hence, |(0011)〉 is FTS, as desired.This Dicke state is provably not RFTS (see Prop. 6.4.6 in Sec. 6.5). With
the simplicity of this neighborhood structure, in order for a state to be RFTS, theSchmidt span projectors must commute with one another. The lack of commutativityin the Dicke state example, then, renders this state not RFTS. This demonstrates thatthe condition of being RFTS is strictly stronger than the condition of being FTS, asexpected.
We now state our general sufficient condition for FTS.
Theorem 6.3.6. A state |ψ〉 is FTS relative to a connected neighborhood structureN if there exists at least one neighborhood satisfying the small Schmidt span conditionand
〈uNk,|ψ〉〉k = u|ψ〉. (6.11)
Satisfaction of these sufficient conditions certainly implies satisfaction of the nec-essary conditions in Thm. 6.3.1. However, it is interesting to directly prove a connec-tion between the linear intersection property of Eq. 6.5 and the Lie algebraic unitarygeneration property:
Proposition 6.3.7. If |ψ〉 satisfies 〈uNk,|ψ〉〉k = u|ψ〉 with respect to neighborhoodstructure N , then |ψ〉 satisfies Eq. (6.5) with respect to N .
We conjecture that satisfaction of Eq. (6.5) along with the small Schmidt spancondition (i.e. the necessary conditions in Thm. 6.3.1) is sufficient to ensure FTS.One avenue to proving this would be to establish the converse of Prop. 6.3.7.
To conclude this section we describe an example of a state which is FTS withrespect to a more complicated neighborhood structure.
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6.3 Finite-time stabilization
Example 6.3.8 (1D AKLT state). Consider a chain of N spin-1 particles alongwith the NN neighborhood structure. The 1D AKLT state can be defined as
|AKLTN〉 ≡ P1
( dN/2e∏i=1
P2i,2i+1
)PN |ψ−〉⊗N−1, (6.12)
where P2i,2i+1 : C2⊗C2 → C3 projects the two spin-1/2 systems into the spin-1 tripletspace, P1 and PN are isometries mapping |0〉 → |m = 1〉, |1〉 → |m = −1〉, and|ψ−〉 = 1√
2(|01〉 − |10〉). With respect to the “boundary neighborhoods” (1, 2) and
(N − 1, N), the Schmidt spans have dimension 2. With respect to the remaining,“bulk neighborhoods”, the Schmidt spans have dimension 4. The neighborhood Hilbertspaces have dimension 3 · 3 = 9, so the small Schmidt span condition is satisfied.
It remains to show that the AKLT states satisfy the unitary generation property.For small values of N this may be checked numerically. Indeed, we have verifiednumerically that the unitary generation property is satisfied for N = 3 and N = 4.We conjecture that for all N , the 1-D AKLT state is FTS with respect to the NNneighborhood structure.
6.3.2 Efficiency of finite-time stabilization
In the Dicke state example of Section 6.3.1, the sequence of stabilizing CPTP maps isof the formWU W , whereW is a dissipative map acting on N1, and U is a unitarycomposed of a sequence of neighborhood unitaries. At most, W is able to reducethe rank of the input density operator by a factor of four. Acting on the completelymixed state, W outputs a density operator of rank = 16/4 = 4. Then, a properchoice for U shifts the support of this rank-four density operator in such a way thata second application of W reduces the rank to one (again, a reduction by a factor offour). In the general case, more than one iteration of the the unitary followed by thedissipative map is needed to achieve FTS.
Two principles guide the general FTS scheme:
(i) Choose the dissipative map W to maximally reduce the rank of the completelymixed state.
(ii) Choose the unitaries Ui so that the subsequent action of W maximally reducesthe rank of its input.
Consider a general target state |ψ〉 and neighborhood structure N , where |ψ〉satisfies the conditions of Thm. 6.3.6. For ease of notation, let Σ0 ≡ ΣN (|ψ〉).Decompose HN '
⊕s−1i=0 Σi ⊕ R, where Σi are orthogonal isomorphic copies of Σ0
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
Figure 6.3: Finite-time stabilization scheme for the N = 3 particle spin-1 AKLTstate. In each numbered panel, each square represents one of the 27 dimensions inthe C3⊗C3⊗C3 state space, while the dots represent the probabilistic weight of eachbasis vector for the current state. The task is to move all the probabilistic weightfrom the initial flat distribution (completely mixed state) into the box in the upperleft-hand corner, corresponding to the target state. The Schmidt span on the first twosystems is 2-dimensional, leading to the 6-dimensional extended Schmidt span Σ0 asrepresented by the first row of boxes. The remaining rows, labeled Σi, are isometric“copies” of this subspace, leaving the 3-dimensional remainder space R ⊗ HN . Thedissipative map W acts only on the first two qutrits, cooling each Σi to Σ0. Theunitaries U1 and U2 leave the target state invariant while preparing probabilisticweight to be cooled by the neighborhood map W . Since the AKLT state satisfiesthe unitary generation property with respect to this neighborhood structure, each Uican be decomposed into a finite sequence of neighborhood-acting invariance-satisfyingmaps.
and R is the remainder space of minimal dimension. Factoring out the index i, theglobal space decomposes as( s−1⊕
i=0
Σi ⊗HN)⊕R⊗HN ' Cs ⊗ (Σ⊗HN )⊕R⊗HN .
Now let the dissipative map W take each isomorphic copy Σi into Σ0. Using theabove decomposition, the dissipative map is defined as
W ≡ (|0〉〈0|Tr)⊗ I ⊕ I. (6.13)
Having fixed W , the interspersed unitary maps Ui are constructed so as to maximizethe rank-reduction achieved by each W . This is done using the following algorithm:
(a) Choose an orthonormal basis |ψ0α〉 for Σ0 ⊗ HN where |ψ0
0〉 ≡ |ψ〉. This
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6.3 Finite-time stabilization
determines isomorphic orthonormal bases |ψiα〉 for the copies Σi ⊗HN .
(b) Choose an orthonormal basis |λβ〉 for R⊗HN .
(c) Order the basis vectors as
|ψ00〉, |ψ1
0〉, . . . , |ψs−10 〉, |ψ0
1〉, |ψ11〉, . . . , |ψs−1
1 〉,...|ψ0l 〉, |ψ1
l 〉, . . . , |ψs−1l 〉, |λ0〉, . . . , |λw〉. (6.14)
(d) The choice of each Ui depends recursively on the input density matrix ρi =W(Ui−1(ρi−1)), beginning with ρ1 ≡ W(I).
(e) In each step, Ui is a permutation of the basis vectors, chosen so that the targetstate is fixed and, iteratively, each basis vector in the support of ρi is mappedto the first basis vector outside of the support of ρi according to the orderingof Eq. (6.14). The aim is for the output density matrix to have contiguoussupport starting from |ψ0
0〉.
Note that, by taking the completely mixed state the initial state, the output of eachmap W and Ui is diagonal in the chosen ordered basis. Furthermore, the particularchoice of each Ui ensures that, in the sequence of CPTP maps applied to the com-pletely mixed state, each W maximally reduces the rank of the input density matrix.Finally, since |ψ〉 satisfies the unitary generation property with respect to N , eachglobal stabilizer Ui can be decomposed into a finite number of neighborhood stabiliz-ers. The sequence of CPTP maps terminates after a finite number of steps becausethe rank of the input density matrix is necessarily reduced in each step (assumingthe initial input is the completely mixed state). Having verified that the sequenceconverges to the target for the initial state being completely mixed, we are guaranteedthat the sequence converges to the target for arbitrary input.
The scheme is illustrated for the three spin-1/2 AKLT target state in Fig. 6.3.Each global unitary stabilizer map is decomposable into no more than 2 · 262 (twicethe dimension of the Lie group, cf. Thm. 6.8.1) neighborhood stabilizer unitaries.The total number of neighborhood maps used in the sequence is thus at most 1 + 2 ·262 + 1 + 2 · 262 + 1 = 2707.
More generally, consider a system of N qudits with target state |ψ〉 and neighbor-hood structure N . We define the cooling rate of the neighborhood Nk with respect to|ψ〉 to be rk ≡ logd (bdim(HNk)/dim(ΣNk [|ψ〉])c). From this, we define the maximumcooling rate r = maxk rk. The small Schmidt span condition ensures that the max-imum cooling rate is greater than zero. A larger cooling rate affords the dissipativemap W to more greatly reduce the rank of the input density matrix. In Fig. 6.3, the
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
cooling rate is log3(4), as W maps four such copies of Σ0 (including itself) into Σ0.Take r to be the largest cooling rate of |ψ〉 among the neighborhoods. For large dand N , the circuit size is dominated by the number of neighborhood unitaries, whilethe number of dissipative maps is negligible. The maximum number of neighborhoodunitaries comprising each Ui is 2(dN − 1)2 ∼ O(d2N). The number of Ui needed de-pends on the rate at which W can reduce the rank of the input density matrix. Bychoosing Ui according to the above scheme, W is able to reduce the rank of the inputdensity matrix by at least a factor of dr−1 ∼ O(dr). Thus, the number of W thatmust act to reduce the rank to 1 is at most O(N/r). Putting these together, thecircuit size scales as O(d2N2/r).
For certain neighborhood structures, this circuit size scaling can be reduced. Con-sider a neighborhood structure, where the neighborhoods admit an L-layering. Thatis, the neighborhoods can be partitioned into L sets, such that all neighborhoodsin a given set are mutually disjoint. The 1-D NN neighborhood structure admits a2-layering. Assume that the cooling rate of all neighborhoods in a particular layer isr. In this case, instead of defining a single neighborhood-acting dissipative map W ,we define a dissipative map Wi for each neighborhood in the particular layer, andthen let W =
∏iWi. While a given Wi reduces the rank of the input density matrix
by a factor of O(dr), W reduces the rank by a factor of O((dr)|N |/L), assuming thateach layer contains about |N |/L neighborhoods. If we take |N | ∼ N , then the circuitsize of the L-layered neighborhood case scales as O(d2LN/r). Hence, in the 1-D NNcase, the circuit size scales as O(d4N/r).
This exponential scaling with respect to the system size is quite unfavorable. Wenote however, that the exponential scaling is completely due to the compilation of∼ d2N neighborhood stabilizer unitaries making up the global stabilizer unitaries.This is a worst-case scaling, which, in principle, could be drastically reduced forparticular cases.
The next section explores the far more efficient stabilization scheme of robustFTS. This scheme only uses dissipative maps, with a circuit size that is proportionalto the number of neighborhoods. Furthermore, the stabilization does not depend onorder of maps, leading to a favorable scaling of circuit depth. We will demonstratethat RFTS states are a proper subset of FTS states. In spite of this, we show thatimportant classes of states, such as graph states used in measurement based quantumcomputing, are RFTS.
i=1Hi, consider a strictly localneighborhood structure Ni, with Ni = i, and an arbitrary product state ρ =⊗N
i=1 ρi. To each neighborhood let us associate the map Ei ≡ (ρiTri)⊗ Ii. Then, anycomplete sequence of such maps gives
EN . . . E2 E1 =N⊗i=1
(ρiTri) =
(N⊗i=1
ρi
)(N⊗i=1
Tri
)= ρTr, (6.15)
demonstrating that ρ is RFTS, as intuitively expected. Of course, by considering aneighborhood structure with enlarged neighborhoods (relative the the strictly local oneabove), a product state remains RFTS. Hence, any product state is RFTS with respectto any neighborhood structure which covers all systems.
Although the above example is trivial, its structure is important. In much ofour analysis of RFTS, we seek to translate more complicated examples into the formof the product state example. Our next example demonstrates this translation andshows that RFT stabilizability is not limited to product states.
Example 6.4.2 (Graph states). Graph states are many-body entangled states whichare known to be resources for universal measurement based quantum computing [191].This example is valuable in that it demonstrates that the RFTS of a non-trivial targetstate can be translated into the task of stabilizing a product state with respect to “virtualdegrees of freedom”.
Following [179, 172], a graph state on N qudits is defined by a graph G = (V,E)with N vertices and a choice of Hadamard matrix H. The Hadamard matrix mustsatisfy H†H = dI, H = HT , and |[H]ij| = 1 for all i, j. The edge-wise action CH
is defined, according to the choice of Hadamard matrix, by CH |ij〉 = [H]ij|ij〉. Thestandard choice in the qubit case is that CH equals a controlled-Z transformation.Note that CH is diagonal in the computational basis and symmetric under swap ofthe two systems it acts on. We define the global graph unitary transformation asUG ≡
∏(i,j)∈E C
Hi,j, with the corresponding CPTP map UG(·) ≡ UG · U †G. Then, the
graph state associated to G is
|G〉 ≡ UG|+〉⊗N , |+〉 = H|0〉. (6.16)
To each physical system i we associate a neighborhood Ni defined by that systemalong with the graph-adjacent systems (i.e. the set of j connected to i by some edge(i, j) ∈ E). For any given |G〉, we may construct a set of neighborhood maps whichrobustly stabilize |G〉 relative to N . Let E : B(Cd) → B(Cd) be defined by E(·) =|+〉〈+|Tr (·). Let Ei indicate the map E acting on system i with trivial action on
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
i. To each neighborhood Ni, we associate the map Ei = UG Ei U−1G . The Kraus
operators of Ei are of the form Xαi ⊗ Ii. The unitary conjugation of Ei transforms
its Kraus operators into those of Ei as UG(Xαi ⊗ Ii) = X ′αi . Crucially, each X ′αi acts
non-trivially only on Ni. This is seen as follows:
X ′αk = UG(Xα
k ⊗ Ik)
= UG(Xαk ⊗ Ik)U
†G
=
(∏j∼k
CHk,j
)(HXα
kH† ⊗ Ik)
(∏j∼k
CHk,j
)†= (X ′
αk )Nk ⊗ IN k .
Hence, each Ei is a valid neighborhood map. Finally, we show that each Ei leaves |G〉invariant and that the composition of an arbitrary complete sequence of these mapsprepares |G〉. Invariance is demonstrated by Ei(|G〉〈G|) = UG(Ei(U †G(|G〉〈G|))) =UG(|+〉〈+|⊗N) = |G〉〈G|. Preparation is seen from,
EN . . . E2 E1 = UG EN . . . E2 E1 U−1G
= UG (|+〉〈+|Tr)⊗N U−1G
= UG(|+〉〈+|⊗N)Tr= |G〉〈G|Tr.
Surprisingly, Example 6.4.2 shows that there exist “resourceful” many-body en-tangled states which are RFTS. Note that the scaling of the size of the dissipativecircuit is O(N), a drastic improvement on that of the general FTS scheme. In asense, this improvement comes along with a drawback. Although graph states areresourceful and many-body entangled, the correlation among their physical systemsis limited in a particular sense. Namely, as with product states, graph states exhibita finite correlation length with respect to the geometry imposed by the graph. Next,we describe a few necessary conditions for RFTS. These characterize the limits onthe target state’s correlations with respect to the “geometry” that the QL constraintinduced.
For a givenN , let the neighborhood expansion ofA be defined asAN =⋃Ni∩A 6=∅Ni.
Intuitively, AN is the set of subsystems which are connected to A by some neighbor-hood. Using this notion, we show how, in a sense, RFTS states cannot support“long-range” correlations.
Proposition 6.4.3 (Finite correlation length). A target pure state ρ = |ψ〉〈ψ|is RFTS with respect to N only if, for any two subsystems A and B having dis-joint neighborhood expansions (AN ∩ BN = ∅), arbitrary observables XA and YB are
uncorrelated, that is, Tr (XAYBρ) = Tr (XAρ)Tr (YBρ).
The above result can be modified to address the case where the neighborhoodexpansions are overlapping. Towards this, we give a lemma which describes the “QLrecoverability” property satisfied by RFTS states.
Lemma 6.4.4 (Recoverability property). Let target pure state ρ = |ψ〉〈ψ| beRFTS with respect to N . If a mapM acts on a subsystem A,M≡ MA ⊗ IA, thenthere exists a sequence of CPTP neighborhood maps EAN each acting only on AN ,such that ρ = El . . . E1 M(ρ).
As a physical interpretation, considerM to be an error map such as a bit-flip. Theabove result shows that, for any RFTS state, such errors can be corrected with arecovery map that acts on a confined region of the system, determined by the neigh-borhood structure.
Proposition 6.4.5 (Zero CMI). A target pure state ρ = |ψ〉〈ψ| is RFTS withrespect to N only if, for any two regions A and B, with AN ∩ B = ∅, the quantumconditional mutual information (CMI)
I(A : B|C)ρ ≡ S(A,C) + S(B,C)− S(A,B,C)− S(C)
satisfies I(A : B|C)ρ = 0, where C ≡ AN\A.
A common feature of product states and graph states is that the neighborhood-acting terms of their canonical FF Hamiltonians commute with one another. Whilewe will show that this commutativity is not necessary for RFTS, a weaker version ofthis property is necessary, nevertheless.
Proposition 6.4.6 (Commuting FF Hamiltonian). If a target pure state |ψ〉is RFTS with respect to neighborhood structure N , then [Πk,Πk] = 0 for all neigh-borhoods Nk, where Πk and Πk are the projectors onto ΣNk(|ψ〉) and ∩j 6=kΣNj(|ψ〉),respectively.
With this proposition, we can verify that neither the Dicke nor the AKLT states areRFTS on account of the lack of commutativity among the terms in their canonicalFF Hamiltonians. In the next section we build from the graph state example and theconnection to commuting Hamiltonians to develop conditions for verifying RFTS ofmore general states.
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
In this section, we describe three distinct sufficient conditions for ensuring RFTS of atarget state. The first condition incorporates all examples that we know of, althoughit is not a computable condition. The next condition is computable, applies to ar-bitrary neighborhood geometries, and incorporates a number of important examples,including graph states. However, there are some examples of RFTS states which itdoes not include. The final condition is applicable to a certain class of neighborhoodstructures, and target states satisfying this condition admit a simple tensor networkdescription [195].
6.5.1 Non-operational sufficiency criteria
To understand what features ensure RFTS of a general target state, we take a closerlook at the graph states of Example 6.4.2. An important feature here is that theCPTP maps Ei rendering the graph state RFTS do not interfere with one another.Although the maps on adjacent neighborhoods may overlap, the degrees of freedomthey transform are independent. Each map Ei prepares a single degree of freedominto the state |+〉. However, these degrees of freedom do not correspond to physicalqubits. Rather, they correspond to virtual subsystems [196, 197]. These virtualsubsystems are analogous to the quasi-particles of condensed matter physics in thattheir observables are linear combinations of the physical observables.
With respect to the virtual subsystems, the graph state is factorized. Using thelanguage of [132], the graph state is generalized unentangled with respect to thevirtual subsystem observables. However, this feature alone is not sufficient for RFTS.Additionally, each virtual subsystem must be “contained” in a neighborhood. Thisenables each neighborhood map to subject a contained virtual subsystem to anytransformation, while leaving the remaining virtual subsystems unaffected. For graphstates, each neighborhood map prepares a virtual subsystem into the state |+〉.
The graph state example is special with respect to the virtual subsystems. Thereis a 1-1 correspondence between physical subsystems and virtual subsystems. As wewill find, this feature is not necessary for RFTS. But, in the case that there is sucha correspondence, we are granted a unitary transformation which maps the physicalsubsystems into the virtual subsystems. In the graph state example, this transfor-mation was achieved by UG. This 1-1 correspondence makes possible the standardconstruction of graph states, whereby the physical product state |+〉⊗N is transformedinto the virtual product state |G〉 by the unitary transformation UG. Accordingly,each local physical observable σi is transformed into the virtual subsystem observableσi = U †GσiUG, which, like the Kraus operators of Ei, acts only on the neighborhood
Ni. Finally, we emphasize that, when the maps Ei are represented in the appropriatevirtual subsystem description, the sequence EN . . . E1 is seen as a product of mapsacting on distinct (virtual) subsystems analogous to the product state stabilizationin Eq. (6.15).
The essential feature of this structure is the identification of the physical subsystemHilbert space with a virtual subsystem Hilbert space,
W :⊗i
Hi →⊗j
Hj, (6.17)
where, in general, we need not require any pair Hi and Hj to be isomorphic (e.g. thephysical systems could be qubits, while the virtual subsystems are four-dimensional).We denote this identification, or isomorphism, by
⊗iHi '
⊗j Hj, where the “'”
symbol serves to remind that the tensor product structure on the left and right-handsides may not correspond.
This relabeling of the degrees of freedom leads to two conditions which ensureRFTS of a target state. First, the target state should be factorized with respect tothe virtual degrees of freedom,
|ψ〉 =⊗j
|ψj〉.
Second, the operators associated to any given virtual subsystem should, themselves,be neighborhood operators; that is, for every j, there exists k such that for any virtualsubsystem operator Xj ∈ B(Hj),
Xj ⊗ Ij ∈ B(HNk)⊗ IN k .
We describe how these two features ensure RFTS. The logic closely parallels ourprevious demonstration of RFTS for graph states.
Assume that the conditions above hold for some |ψ〉. We construct a finite se-quence of commuting QL CPTP maps which robustly stabilize |ψ〉. On the virtualsystems, define the maps Ej such that
Ej : B(Hj)⊗ B(Hj)→ B(Hj)⊗ B(Hj),
Ej(·) = (|ψj〉〈ψj|Tr)j ⊗ Ij.
The Kraus operators of Ej are contained in B(Hj) ⊗ Ij. Hence, by the inclusionB(Hj) ⊗ Ij ≤ B(HNk) ⊗ INk , the Kraus operators of Ej act non-trivially only onneighborhood Nk. Thus, each map Ej is a valid neighborhood map. Finally, we mustshow that an arbitrary sequence of these maps prepares |ψ〉 while leaving it invariant.
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
For invariance, we have
El(|ψ〉〈ψ|) = El( M⊗
j=1
|ψj〉〈ψj|)
= |ψl〉〈ψl|Tr(|ψl〉〈ψl|
)⊗⊗j 6=l
|ψj〉〈ψj| = |ψ〉〈ψ|. (6.18)
To show that an arbitrary complete sequence of the neighborhood maps prepares |ψ〉,note that
ET . . . E2 E1 =T⊗j=1
|ψj〉〈ψj|Trj
=
( T⊗j=1
|ψj〉〈ψj|)( T⊗
j=1
Trj)
= |ψ〉〈ψ|Tr. (6.19)
The product state and graph state examples admit such a factorization structure. Inthe following example we demonstrate another class of RFTS states by establishinga similar factorization structure. Qubit graph states are an example of so-calledstabilizer states; they can be written as the unique eigenvalue-1 state with respect toa commuting set of operators from the N -qubit Pauli group. We note that the classof states presented in the following example are not standard stabilizer states [198].
Example 6.5.1 (CCZ states). In [198] the authors introduce the class of so-called(Z2)3-states, which exhibit genuine two-dimensional symmetry-protected topologicalorder and for which the construction parallels that of the graph states. While this classof states may be defined for any 3-uniform hypergraph (i.e. one with only 3-elementedges), we restrict to the triangular lattice, which allows for scaling the system. Aswith graph states, each qubit in the lattice is initialized in |+〉 = (|0〉+ |1〉)/
√2. Then,
on each triangular cell a controlled-controlled-Z (CCZ) gate is applied, where
in the computational basis. Noting that all CCZ gates commute with one another, wedefine U∆ ≡
∏(i,j,k)∈T CCZijk, where T is the set of triangular cells on the lattice.
The target CCZ-state is|∆〉 ≡ U∆|+〉⊗N , (6.20)
with N the number of lattice sites. To each site we associate a neighborhood definedby that qubit along with the six adjacent qubits. We verify that |∆〉 is RFTS withrespect to this neighborhood structure by identifying a virtual subsystem decompositionsatisfying the needed properties. As with the graph state example, we can identify eachphysical subsystem to a virtual subsystem. The unitary transformation U∆ takes thephysical subsystem observables into the virtual subsystem observables. easily verifythat Then, each virtual subsystem algebra corresponds to a neighborhood-containedalgebra thanks to the commutativity of the CCZ gates,
B(Hi)⊗ Ii = U∆(B(Hi)⊗ Ii)U−1∆
=[UNi(B(Hi)⊗ INi\i)U
−1Ni
]⊗ INi
≤ B(HNi)⊗ INi ,
where UNi ≡∏
k,l∈Ni\iCCZikl acts only on the physical systems in neighborhood Ni.Furthermore, by construction, the CCZ state is a virtual product state. Considering|∆〉 = U∆|+〉⊗N , U∆ maps each physical factor into a corresponding virtual subsystemfactor, giving |∆〉 ' |+〉⊗N with respect to
⊗Ni=1 Hi. As the neighborhood containment
property of the virtual subsystems and the factorization of |∆〉 are satisfied, the CCZstate is verified to be RFTS. To be concrete, we can construct commuting neighborhoodacting CPTP maps which each prepare a factor of |∆〉 viewed as a virtual productstate.
Next, we describe an important generalization of the above neighborhood factor-ization scheme. This generalization is motivated by the following example of a statewhich does not admit a simple neighborhood factorization, yet is RFTS.
Example 6.5.2 (Non-factorizable RFTS state). Consider the system H ' HA⊗HB ⊗HC ' C2 ⊗ C5 ⊗ C2. We decompose the middle system as
HB ' C5 ' (C2 ⊗ C2)⊕ C1 ' Hb ⊗Hb′ ⊕H0B ' HB ⊕H0
B,
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
Figure 6.4: Example of a non-factorizable RFTS state.
and similarly for E2. Hence, the product of either order of the maps is
E1 E2 = (E1 ⊗ IC) (IA ⊗ E0 ⊗ IC) E2
= (E1 ⊗ IC) E2
= (E1 ⊗ IC) (IA ⊗ E2) (IA ⊗ E0 ⊗ IC)
= |ψ〉〈ψ|Tr.
The key feature of this state that enables it to be RFTS is that, once part ofthe Hilbert space is removed locally (in this example, H0
B ≤ HB), there exists aneighborhood factorization of |ψ〉 with respect to the remaining space, whereby thealgebra of each factor is contained in a corresponding neighborhood.
In the above example, the Hilbert space reduces into two parts; one containinga factorization of the target state and the other being orthogonal to the reducedstates of the target state. In contrast to such cases where the Hilbert space can bereduced into sectors and following [199], we will refer to a factorization such as in Eq.(6.17) as an irreducible factorization. Next, we show that, in general, an irreduciblefactorization is not required for RFTS and describe the reducible factorizations whichstill ensure RFTS. Towards this, we introduce two useful concepts.
First, we refer to a generalization of the reduction in Example 6.5.2 as a localrestriction.
Definition 6.5.3. Given H '⊗
iHi and a set of spaces Hi ≤ Hi, we define a locallyrestricted Hilbert space as H '
⊗i Hi.
Second, with respect to a given neighborhood structure, the physical locality of sub-systems may be more fine-grained than the locality determined by the neighborhoodstructure. For example, consider systems A,B,C,D, with neighborhoods A,B,Cand B,C,D. The physical locality describes four subsystems. But, as far as the
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
neighborhood structure is concerned, the separation of system B from C is artificial.In such a scenario we are afforded a more-reductive local restriction if we reducestarting from the coarse-grained subsystem decomposition, HA ⊗HBC ⊗HD.
Definition 6.5.4. Given H '⊗N
i=1Hi and a neighborhood structure N , we definethe coarse-grained subsystems to be the equivalence classes of the subsystems Hi withrespect to the relation “is contained in the same set of neighborhoods as”.
Though usually stated explicitly, in the remainder of the chapter the decomposi-tion of the Hilbert space H '
⊗iHi denotes the coarse-grained subsystems, and N
incorporates the coarse-graining.
Theorem 6.5.5 (Neighborhood factorization on local restriction). A state |ψ〉described on the coarse-grained subsystems H '
⊗Ni=1Hi with respect to the neigh-
borhood structure N is RFTS if there exists a locally restricted space H '⊗N
i=1 Hi,with a complement H0 ' H⊥, whereby a factorization H =
⊗Mj=1 Hj gives
|ψ〉 =M⊗j=1
|ψj〉 ⊕ 0 ∈M⊗j=1
Hj ⊕H0 (6.21)
and for each virtual subsystem Hj there exists a neighborhood Nk such that
B(Hj)⊗ Ij ⊕ I0 ≤ B(HNk)⊗ INk . (6.22)
The following example details a construction whereby such a factorization arises.
Example 6.5.6 (Generalized Bravyi-Vyalyi states). We introduce a class ofstates inspired by the work of Bravyi and Vyalyi in [199]. These authors studied thecomplexity of the problem “Common Eigenspace”, whereby one is to determine whetheror not there exists a state |ψ〉 that is a common eigenstate of the commuting Hamil-tonians H1, . . . , Hr with respect to eigenvalues λ1, . . . , λr. An important example thatthey consider is the case where the given Hamiltonians are each 2-body operatorswith respect to H '
⊗Ni=1Hi. Bravyi and Vyalyi show that the Hamiltonian terms
being 2-body and commuting induces a tensor factor decomposition of each physicalsubsystem.
Consider a set of commuting two-body projectors Πjk, acting with respect to theedges of a graph G = (V,E), with vertices corresponding to the factors of the Hilbertspace H '
⊗Ni=1Hi. Adapting the notation from Lemma 8 of [199], the commuting
two-body projectors can be shown to induce a decomposition of H,
H '⊕α
Hα
'⊕α
( N⊗i=1
Hαi
)⊗( N⊗
i=1
⊗j|(i,j)∈E
Hαij
), (6.23)
by which each projector Πij simplifies to the form
Πij =⊕α
Irest ⊗ Παij, (6.24)
where Παij acts only on Hα
ij⊗Hαji. This construction makes manifest the commutativity
among the Πij since, within each sector Hα, the non-trivial parts of each projector (i.e.the Πα
ij) act on disjoint factors. This simplification in the structure of the projectorsenables Bravyi and Vyalyi to show that the 2-local version of the Common Eigenspaceproblem is in NP.
Bravyi and Vyalyi then describe the construction of states which are “simple” withrespect to the induced virtual subsystem decomposition. Within a given sector Hα,they first define
|φ〉 =n⊗i=1
|φi〉 ⊗( ⊗
(i,j)∈E
|φij〉), (6.25)
which is a product of single-factor and bipartite states. Then, defining a set of isome-tries
Vi : Hαi ⊗
⊗j
Hαij → Hi,
they construct the state |φ′〉 = (V1 ⊗ . . .⊗ Vn)|φ〉 ∈ H. We call any state constructedas such a Bravyi-Vyalyi (BV) state. We remark that such a form can be recast as atensor network state [195].
A BV state is RFTS with respect to the neighborhood structure determined by itsinteraction graph: Compare the factorization of Eq. (6.25) with that of Eq. (6.21),noting that the virtual subsystems are of the form Hα
i and Hαj ⊗Hα
ij ⊗Hαji. 2) Each
virtual subsystem Hαi or Hα
ij ⊗Hαji is contained in Nk = (i, j):
B(Hαi )⊗ Irest ⊕ 0 ≤ B(HNk)⊗ IN k
B(Hαij ⊗Hα
ji)⊗ Irest ⊕ 0 ≤ B(HNk)⊗ IN k , (6.26)
where the sector with zero is⊕
β 6=αHβ.Building off of this structure, we define a generalization of such states and show
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
that they are RFTS. Consider a neighborhood structure N and a virtual subsystemdecomposition of each particle
Hi ≡ H0i ⊕ Hi ' H0
i ⊕fi⊗j=1
Hij,
and, correspondingly, H ≡⊗
i Hi. To each neighborhood Nk, we associate HNk '⊗i∈Nk Hi '
⊗i∈N
⊗fij=1Hij. Then, for each neighborhood Nk, consider a subset Sk
of the ij (i.e. associated to Hij) contained in Nk, defining Hk ≡⊗
ij∈Sk Hij. Assumethat the Sk have been chosen such that each ij is accounted for exactly once. Thisensures that
H 'N⊗i=1
Hi '|N |⊗k=1
Hk, (6.27)
where we emphasize that the tensor products over the i index are not equivalent tothe tensor products over the k index. Let Vi : Hi → Hi be isometries from the virtualto the physical particles, with V ≡ (V1 ⊗ . . . ⊗ VN), and consider any product state|ψ〉 =
⊗k |ψk〉 ∈ H. We define a generalized BV state to be any of the form
|ψ′〉 ≡ V |ψ〉 = (V1 ⊗ . . .⊗ VN)
(⊗k
|ψk〉). (6.28)
Again, the tensor product structure among the Vi is not the same as the tensor productstructure among the |ψk〉. This mismatch of tensor product structures is precisely whatallows for |ψ′〉 to exhibit entanglement among the physical particles. An example ofa generalized BV state is depicted in Fig. 6.5 and the RFTS scheme for such statesis sketched in the caption therein.
6.5.2 Operational sufficiency criteria
Algebraic factorization
In the BV scheme of [199], the factorization is induced by a set of commuting Hamil-tonians. In this section we draw inspiration from their construction to investigateways to induce a factorization of the Hilbert space, amenable to RFTS, using, asopposed to a set of commuting Hamiltonians, the target state and the neighborhoodstructure.
In the BV decomposition, each virtual subsystem corresponds to, at most, twophysical subsystems. Hence, this scheme can only induce a factorization when theneighborhoods are two-body. From Example 6.4.2, the graph state on a 2-D square
Figure 6.5: Example structure of a generalized BV state. Perforated circles denotephysical particles, nodes correspond to virtual subsystems, and solid lines connect vir-tual subsystems which are entangled. The hatched semicircles indicate the subspacesH0
2 and H05 (H0
i ⊕Hi = Hi) where the respective reduced states do not have support.Solid curves delineate neighborhoods. The tensor product of isometries V1 ⊗ . . .⊗ V7
is applied to the virtual product state |ψ〉 = |ψ1〉 ⊗ |ψ2〉 ⊗ |ψ3〉 ⊗ |ψ4〉 to obtain thegeneralized BV state. Entanglement among the virtual subsystems is translated intomany-body entanglement among the physical particles. Despite this many-body en-tanglement, the state is RFTS. In the RFTS scheme, the job of each neighborhoodmap Ek is to transfer probabilistic weight into the spaces Hi and then prepare the cor-responding virtual factor |ψk〉 ∈ Hk, while acting trivially on the remaining degreesof freedom.
lattice is RFTS and admits a virtual subsystem factorization. Yet, because the neigh-borhoods are five-body, the BV scheme does not apply to this case. Drawing fromthe graph state example, we develop a scheme to induce a virtual subsystem decom-position as in Thm. 6.5.5 for an arbitrary state and neighborhood structure.
An important feature of graph states is the fact that the physical representationof the virtual subsystems B(Hi) ⊗ Ii, acting on neighborhood Ni, commutes withall Schmidt span projectors Πk for k 6= i. Furthermore, this algebra of operatorsis singled out as the algebra of operators acting on HNi which commute with theremaining neighborhood projectors (Πk, for k 6= i). Lastly, these neighborhood-acting algebras commute with one another and generate the full algebra B(H). Thisis seen by expressing these algebras in the virtual subsystem basis. In general, aset of associative algebras which is complete (has trivial commutant) and for which
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
the individual algebras mutually commute induces an irreducible factorization of thespace they act on. This fact is detailed in the following proposition.
Proposition 6.5.7 (Algebraically induced factorization). If a set of algebrasAj, Ai ≤ B(H), is complete and commuting, then each Aj has a trivial centerand there exists a decomposition of the Hilbert space H '
⊗Tj=1 Hj for which Aj '
B(Hj)⊗ Ij for each j.
In the case of graph states, the factorization of these algebras is given. But, for anarbitrary state, the above proposition points to the possibility that, by constructinga set of commuting neighborhood-contained algebras, we may obtain a Hilbert spacefactorization which factorizes the target state. We will aim to induce the factoriza-tion from the neighborhood-acting terms of a QL Hamiltonian, much like the BVdecomposition. One important difference, however, is that the Hamiltonians we useare derived from the target state itself.
Recall that Prop. 6.5.5 describes a factorization of the target state on a locallyrestricted space. Accordingly, we will define neighborhood acting algebras with re-spect to a locally restricted space. Towards this, we provide a means of constructing alocally restricted space from a positive-semidefinite operator such as the target state.
Definition 6.5.8. Given an operator M ≥ 0 acting on coarse-grained subsystems⊗Ni=1Hi with respect to the neighborhood structure N , the subsystem support and
subsystem kernel of M on p are supp(Trp (M)) and ker(Trp (M)), respectively. Thelocal support of M is H ≡
⊗Ni=1 supp(Tri (M)), with H ' H0 ⊕ H.
We note that, for a pure state |ψ〉, the subsystem support of M = |ψ〉〈ψ| onp is simply the Schmidt span Σp(|ψ〉). Furthermore, the support of each neigh-borhood projector Πj is contained in the subsystem support of the target stateH =
⊗Ni=1 supp(Tri (|ψ〉〈ψ|)). This allows us to define projectors Πk ≡ Πk|H re-
stricted to the subsystem support of |ψ〉〈ψ|. We also give a label for the sub-system support on a given neighborhood, HNj ≡
⊗i∈Nj supp(Tri (|ψ〉〈ψ|)). Also,
HN j ≡⊗
i/∈Nj supp(Tri (|ψ〉〈ψ|)) denotes the subsystem support restricted to thecomplement of Nj. With these, we consider the following candidate for the algebrasthat are to induce a factorization of the target state.
Definition 6.5.9. Given a target state |ψ〉 and a neighborhood structure N , for eachneighborhood Nj, we first define
ANj ≡ X ∈ B(HNj)|[XHNj ⊗ IHNj, Πk] = 0, ∀ k 6= j. (6.29)
with respect to the decomposition H ' (HNj ,0 ⊕ HNj) ⊗ HN j , where HN ,0 is thecomplement of HNj in HNj .
Each neighborhood algebra Aj is verified to be an associative algebra by writing itas a commutant,
Aj = Y ∈ B(H)|[Y,Πk] = 0 ∀ k 6= j
and [Y, Z] = 0 ∀ Z ∈ INj ⊗ B(HN j). (6.31)
If a set is closed under adjoint, as in our case, then its commutant is closed under ad-joint and therefore is a C*-algebra. Intuitively, each Aj is the largest C*-algebra of Njneighborhood operators which commute with the remaining neighborhood projectorsΠk.
We now give the main result of this section which states how the structure of theneighborhood algebras can ensure a particular factorization of the target state and,hence, ensure RFTS.
Theorem 6.5.10. A state |ψ〉 described on the coarse-grained subsystems H '⊗Ni=1Hi with respect to the neighborhood structure N admits a decomposition |ψ〉 =
0 ⊕⊗
j |ψj〉 induced by the neighborhood algebra-induced factorization H ' H0 ⊕⊗j Hj, and hence, is RFTS with respect to N if i) |ψ〉 satisfies Eq. (6.5) with respect
to N ; and ii) the neighborhood algebras Aj are commuting and complete on the localsupport space H.
The key feature of this sufficient condition for RFTS is that it is operationallycheckable; satisfaction of Eq. (6.5) is determined by an intersection of vector spaces,and the neighborhood algebras, and their commutativity properties can, in principle,be computationally determined.
This sufficient condition does not incorporate all examples of RFTS that we knowof. However, a simple pre-processing of the neighborhood structure and local supportspace allows for the inclusion of RFTS states that are otherwise excluded. In somecases, the reduced states of the target state on a particular neighborhood will containphysical factors which are full rank: TrN k (|ψ〉〈ψ|) = ρNk = ρNk\i⊗ρi, with ρi > 0. Insuch cases, invariance requires that any neighborhood map Ek act trivially on systemi. Thus, if |ψ〉 were RFTS with respect to the initial neighborhood structure, it willbe RFTS with respect to a neighborhood structure where Nk is replaced with N ′k ≡Nk\i. We have found cases in which the sufficient conditions of Thm. 6.5.10, whilenot initially satisfied, become satisfied after updating the neighborhood structure asabove.
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
Matching overlap
A simplification of the above condition is possible if the neighborhoods satisfy acertain geometry.
Definition 6.5.11. A neighborhood structure is said to satisfy the matching over-lap condition if, for any set of neighborhoods that have a common intersection, thiscommon intersection is the intersection of any pair of the neighborhoods in the set.
Two-body neighborhoods necessarily satisfy the matching overlap condition, whereasneighborhood structures of graph states or those in Fig. 6.5 do not. The matchingoverlap condition ensures a simple relationship between the intersection of neigh-borhoods and the coarse-grained particles: the intersection of any two non-disjointneighborhoods is a coarse-grained particle. This fact is used in the proof of thefollowing proposition.
Proposition 6.5.12 (Matching overlap RFTS). Let⊗N
p=1Hp be a Hilbert spaceof coarse-grained subsystems with respect to a neighborhood structure N that satisfiesthe matching overlap condition. If |ψ〉 satisfies Eq. (6.5) with respect to N and[Πj,Πk] = 0 for all pairs of neighborhood projectors, then |ψ〉 is RFTS.
As noted, the BV decomposition uses commuting 2-body Hamiltonians to decom-pose the Hilbert space. The matching overlap condition, in a sense, ensures whenneighborhood structures beyond 2-body are amenable to a similar type of decompo-sition. The above proposition then describes how, for a QLS state with commutingSchmidt span projectors, matching overlap ensures a factorization of the target state.
We remark that, by further restricting the neighborhood structure, it is possibleto establish that, for any state satisfying Eq. (6.5), commutativity of the neigh-borhood projectors is necessary and sufficient for RFTS. The restriction is that theneighborhood hypergraph should be connected and not have any cycles. This restric-tion includes the 1-D NN neighborhood structure, but excludes, for instance, the 2-Dlattice NN neighborhood structure.
It is tempting to conjecture that commuting Schmidt span projectors are necessaryfor a QLS state to, further, be RFTS. The following example provides a case wherethis is not true.
Example 6.5.13 (RFTS ground state of non-commuting FF parent Hamil-tonian). Consider nine qubits labeled 1-9 described by the state
|ψ〉W ≡ |W 〉123 ⊗ |W 〉456 ⊗ |W 〉789,
where |W 〉 = 1√3(|001〉 + |010〉 + |100〉). The neighborhood structure is depicted in
Figure 6.6. First, we demonstrate that |ψ〉W is RFTS, then we show that the neigh-borhood projectors Πk do not commute with one another. That |ψ〉W is RFTS follows
Figure 6.6: The ovals denote the neighborhoods of the neighborhood structure forwhich |ψ〉W is RFTS. The three neighborhoods are most easily described by theirrespective complements. Letting S ≡ 1, . . . , 9, we define NA ≡ S\6, 7, NB ≡S\1, 9, and NC ≡ S\3, 4.
simply from the fact that it can be factorized such that each factor is contained ina neighborhood. In this way, the three maps which compose to stabilize |ψ〉W areE123 ≡ (|W 〉〈W |123Tr) ⊗ I123 and similary for E456 and E789. To show that the Πk
do not commute, consider ΠA and ΠB. On systems 7, 8, and 9, these, respectivelyproject onto supp(I7 ⊗ Tr7 (|W 〉〈W |789)) and supp(Tr9 (|W 〉〈W |789) ⊗ I9). A directcalculation shows that these two projections onto systems 7, 8, and 9 do not commutewith one another. Hence, [ΠA,ΠB] 6= 0, and, by symmetry, this holds for any pair ofΠk.
Despite the fact that the canonical FF Hamiltonian does not have commutingterms, we can still construct a commuting FF Hamiltonian for which |ψ〉W is theunique ground state, namely
We conjecture that, if a state is RFTS, there always exists some FF commuting parentHamiltonian for which it is the unique ground state.
This example suggests that, in general, the canonical Hamiltonian∑
k Πk is notnecessarily a useful object for identifying whether a QLS state can be stabilized ro-bustly. We leave open the question of identifying a scheme for discovering neighbor-
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
Figure 6.7: Given 1-D lattice of nine systems and a NNN neighborhood structure,a state which is RFTS can be stabilized with a depth = 3 dissipative circuit byorganizing the application of maps into layers as shown in the schematic above.
hood factorizations of a general target state.
6.6 Efficiency of robust finite-time stabilization
6.6.1 Circuit complexity
In this section we analyze the circuit complexity of the RFTS scheme. We investigatehow the number of CPTP neighborhood maps (circuit size) and degree of paralleliza-tion (circuit depth) scale with N , the number of systems. We focus on systems andneighborhood structures defined with respect to a finite d-dimensional lattice. Byfixing a type of neighborhood structure (e.g. NNN as in Fig. 6.7), we show thatthe depth of the dissipative circuit rendering a state RFTS is upper-bounded by aconstant.
The lattice structure of the subsystems and neighborhoods affords a high degreeof parallelization for the neighborhood maps. As demonstrated in Figure 6.7, this isachieved by partitioning the set of neighborhood maps into “layers” wherein the mapsof a given layer are mutually disjoint. To appreciate the role played by the latticestructure, consider the following example of a scalable neighborhood structure forwhich constant depth is not achievable. Let the neighborhood structure N be givenby the set of all pairs of subsystems, giving |N | =
(N2
)= N(N−1)
2. The largest number
of neighborhood maps which may act in parallel is bN/2c. Hence, the best possibleparallelization will still require at least |N |/bN/2c = N − 1 layers of maps.
We develop a scheme for parallelizing the RFT stabilizing neighborhood maps fora lattice system which ensures finite depth. First, consider the following example.
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6.6 Efficiency of robust finite-time stabilization
Example 6.6.1 (CCZ states on Kagome lattice). In Example 6.5.1 we showedthat the CCZ state defined on the triangular lattice is RFTS. The CCZ state canbe defined similarly on the kagome lattice with CCZ gates acting on each triangle ofsystems. As depicted in Fig. 6.8, to each physical system we associate the five-bodyneighborhood made of that system along with its four nearest neighbors. By followingthe reasoning of Example 6.5.1, it is simple to show that the CCZ state defined onthis lattice is RFTS with respect to such neighborhoods.
Here we show that, for a lattice of any size, this stabilization can be achieved bya dissipative circuit with depth = 12. The unit cell of the kagome lattice consists ofthree physical systems, and, therefore, to three neighborhoods as shown in Fig. 6.8.By translating these three physical systems and three neighborhoods by the group oflattice translations (generated by unit lattice vectors e1 and e2), we generate the setof all systems and all neighborhoods.
With an RFT stabilization scheme, the irrelevance of the map ordering allows usto organize the neighborhood maps into layers. To construct a layer, consider the setof neighborhoods N 0 in the unit cell labeled N 0
1 , N 02 , and N 0
3 in Fig. 6.8. For eachdirection, translate this set until it becomes disjoint with respect to the un-translatedset. The diameter of the set, the maximum number of such translations needed overall directions, is found to be two. By translating any neighborhood in the unit cell bythis diameter, the resulting neighborhood is ensured to be disjoint from the former.We can generate a layer of disjoint neighborhoods by repeatedly translating a unit cellneighborhood by multiples of the diameter (i.e. an even number of translations) ineach direction. Three of the layers will correspond to the three neighborhoods in theunit cell. We still need to account for the neighborhoods translated by an odd numberof lattice vectors in either direction. These nine remaining layers are obtained bytranslating each of the previous three layers by lattice translations (0, 1), (1, 0) or(1, 1). Thus, we have partitioned the neighborhood maps into twelve layers. In eachlayer the neighborhood maps act in parallel ensuring that, for any lattice size, theCCZ state is RFTS with respect to a depth = 12 dissipative circuit.
We generalize this scheme to neighborhood structures defined on an arbitrarylattice. A lattice system is characterized by a unit cell containing an arrangementof c physical systems. The global system is generated by replicating this unit cell
by translations from the group Ld =
d︷ ︸︸ ︷Z× . . .× Z = Zd. As in Example 6.6.1, we
take the neighborhood structure N to be translationally invariant with respect to Ld.Hence, to each unit cell, there corresponds a set of neighborhoods N 0 from which thetranslation group generates the global neighborhood structure N . In defining a unitcell of neighborhoods, the neighborhoods may “spill over” into adjacent unit cells. Asdefined in Ex. 6.6.1, we denote the diameter of this set diam(N 0).
In order to describe how circuit size and depth scales with system size, we consider
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
Figure 6.8: Definition and features of the Kagome lattice, its unit cell, and corre-sponding neighborhood structure for the CCZ-state. In constructing the CCZ-state,the system is initialized in |+〉⊗N and a CCZ-gate is applied to each triangle of adja-cent systems (cf. Eq. (6.20)).
a sequence of finite-sized subsets of the infinite lattice. We take the system to becomprised of a width-L hypercube of Ld unit cells, totaling N = cLd subsystemsand |N 0|Ld neighborhoods. For each N we denote the corresponding neighborhoodstructure as NN . Using a generalization of the scheme described in Example 6.6.1,we bound the circuit complexity of stabilizing an RFTS state with respect to a latticeneighborhood structure (see Sec. 6.8 for a proof detailing the scheme).
Proposition 6.6.2 (Lattice circuit size scaling). Consider a scalable lattice neigh-borhood structure NN as defined above. If |ψ〉 is RFTS with respect to this neigh-borhood structure, then |ψ〉 can be stabilized by a dissipative circuit of size |N 0|(N/c)and depth D = |N 0|diam(N 0)d.
We remark that the scheme we use is not guaranteed to give an optimal depth. Itmerely captures the essential features of the lattice for the purposes of ensuring finitedepth. The following example gives a case in which a partition of neighborhoodsdeviates from the above scheme to give an improved circuit depth.
Example 6.6.3 (Optimal depth for 2D graph states). Consider the 2D squarelattice on which we define the 2D graph states. The group of lattice translations is
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6.6 Efficiency of robust finite-time stabilization
isomorphic to G ' Z × Z. Define a single neighborhood on site (0, 0) as that sitealong with the four adjacent sites, (1, 0), (0, 1), (−1, 0), and (0,−1). We generate theneighborhood structure by translating this neighborhood with respect to G = Z × Z.Hence, there is one neighborhood per physical system and each neighborhood is labeledby an element of G. There is one neighborhood per unit cell, and the diameter of theneighborhoods in a unit cell is diam(N 0) = 3. Therefore, using the above scheme, wemay stabilize the graph state with a circuit of depth D = |N 0| diam(N )d = 1 · 32 = 9.However, we can choose a different parallelizing scheme which results in a depth-five circuit. By translating the neighborhood on site (0, 0) with just the subgroupH ' 〈(1, 2), (2,−1)〉 ≤ G, the generated neighborhoods are disjoint. The size ofthe coset group is |G/H| = 5. Each coset gH corresponds to a layer of disjointneighborhood maps which may act in parallel. The number of layers needed so thatthe resulting circuit includes all neighborhood maps is |G/H| = 5. This shows thatthe 2-D graph states on N systems can be stabilized with a circuit of size N and depth5.
6.6.2 Connection to rapid mixing
In this section we aim to relate RFTS to the existence of continuous-time QL dynamicswhich “efficiently” stabilize a target state. For continuous-time dynamics, if the targetstate is an equilibrium point of the dynamics, then, for any finite time, the state of thesystem can only approximate the target state. We show how RFT stabilizability of ascalable family of states ensures the existence of QL Liouvillian dynamics such thatthe time needed to approximate the target state scales favorably with system size. Tomake this connection rigorous, we will restrict our considerations to RFTS scenarioswhere the CPTP maps in the sequence commute with one another. Although we havenot shown this feature to be necessary, each of the sufficient conditions in Sec. 6.5ensure the existence of such commuting maps.
Two special CPTP maps derived from the Liouvillian L are used in defining con-vergence to the target state [161]:
• Eφ is the CPTP map projecting onto the operators for which L has eigenvalueRe(λ) = 0.
• E∞ is the CPTP map projecting onto the operators for which L has eigenvalueλ = 0.
With these, the following definition provides a measure of how far the “worst-case”evolution is from an equilibrium state of the continuous-time dynamics.
Definition 6.6.4. Given a one-parameter semigroup of CPTP maps Et, define the
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
contraction of Et to be
η(Et) ≡1
2sup
ρ≥0,Tr(ρ)
||(I − Eφ)(Et(ρ))||1. (6.32)
In the case that the continuous-time dynamics is generated by a Liouvillian L, ηcharacterizes the slowest decay into the space corresponding to the peripheral spec-trum of L. We note that, in the cases we are interested in, the Liouvillian has nopurely imaginary eigenvalues, ensuring Eφ = E∞.
From the contraction, we define the mixing time of Et to be the minimal time suchthat η(Et) = 1
2. Since we are interested in the scaling of the mixing time, the choice
of 12as the “sufficient distance to the peripheral spectrum” is unimportant; the type
of scaling of this time does not depend on this distance.The contraction η generated by a Liouvillian may be bounded using the spectral
gap.
Definition 6.6.5. The spectral gap λ of a Liouvillian L is defined as
λ ≡ infabs(Re(λ))|Re(λ) < 0, λ ∈ spec(L). (6.33)
However, the gap of the Liouvillian alone is not always sufficient to ensure afavorable scaling of η as the system size is increased [182]. In light of this, we willmake use of the gaps of neighborhood Liouvillians as opposed to the gap of the globalLiouvillian. The authors of [182] proved the following two results.
Theorem 6.6.6. [182] (Contraction for commuting Liouvillians) Let Lj bea set of Liouvillians which commute with one another. Then,
η(e∑j Ljt)≤∑j
η(eLjt). (6.34)
(Contraction theorem) Let L be a Liouvillian with gap λ. Then, there existsL > 0 and for any ν < λ there exists R > 0 such that
Le−λt ≤ η(eLt)≤ Re−νt. (6.35)
The mixing time of a Liouvillian is inversely proportional to the scale of that Li-ouvillian. Thus, to make our results non-trivial, we fix our neighborhood Liouvilliansto have a bounded norm: c > ||Lj|| for all j for some constant c.
With these considerations, we can easily place an exponentially decaying upperbound on the contraction of sums of commuting Liouvillians which linearly scaleswith the number of Liouvillians.
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6.7 Extension of results to mixed target states
Proposition 6.6.7 (Commuting Liouvillian contraction bound). Let Lj bea, possibly infinite, set of bounded norm Liouvillians each acting on a subsystem ofuniformly-bound dimension. Assume that the spectral gaps λj are strictly boundedbelow by ν > 0. Then, there exists R > 0 such that for any subset S ⊆ Lj ofmutually commuting Liouvillians, defining L =
∑S Lj, the contraction is bounded by
η(eLt) ≤ |S|Re−νt. (6.36)
Such bounds are more generally related to the notion of rapid mixing.
Definition 6.6.8. For a family of one-parameter semigroups of CPTP maps E (s)t ,
indexed by s and corresponding to the finite dimensional Hilbert space H(s), E (s)t
satisfies rapid mixing if there exists constants c, γ, δ > 0 such that
η(E (s)t ) ≤ c logδ(dimH(s))e−γt. (6.37)
We relate commuting RFTS scenarios to rapid mixing.
Proposition 6.6.9 (Rapid mixing for commuting RFTS). Consider a family offinite systems H(s) indexed by s such that for each s there is a neighborhood structureN (s) with dim(HN (s)
k) < D. Furthermore, assume the number of neighborhoods scales
no more than polynomially in system size, N (s)k ∈ N (s) and |N (s)| ≤ b log(dim(H(s))),
for some constant b > 0. For each s, let ρs be RFTS with respect to N (s) by a setof commuting neighborhood maps E (s)
k , where there exists some ν > 0 such that forany eigenvalue λ ∈ eig(E (s)
k ), λ = 1 or λ < 1 − ν. Then, there exists a family ofbounded norm QL Liouvillians satisfying rapid mixing with respect to ρs.
Such rapid mixing ensures that these dynamics are “stable” with respect to localperturbations. The main result in [171] was to show that rapid-mixing of a family ofCPTP semigroups ensures such stability of the dynamics. Their result applies to thedynamics we’ve considered above since the QL Liouvillians defined with respect to alattice neighborhood structure fit the definition of a “uniform family with finite rangeinteraction”.
6.7 Extension of results to mixed target states
Although the focus of this chapter has been on target pure states, the notions of FTSand RFTS apply equally well to target mixed states. In this section we highlight afew finite-time stabilization results regarding target mixed states and also state a fewconjectures.
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
In our analysis of FTS in Sec. 6.3 (as opposed to RFTS) the purity of the targetstate played a crucial role. Even the necessary condition of Thm. 6.3.1 involvedcriteria that only apply to target pure states. For this reason, analysis of (strict) FTSfor target mixed state states remains unexplored and left to future work.
In contrast, a number of the RFTS results of Sections 6.4 and 6.5 are directlyapplicable to, or admit analogs for, the mixed state case. Proposition 6.4.3, Lemma6.4.4, and Proposition 6.4.5 each constrain the correlations of a state that is to beRFTS. The statements and proofs of these results generalize directly to the case of atarget mixed state.
In Sec. 6.5, we described how the existence of a particular virtual subsystemdecomposition of the Hilbert space can ensure that a target state is RFTS. Thisresult, too, extends directly to the case of a mixed target state. Here, instead of thepure state being factorized with respect to
⊗j Hj, the mixed state must be of the
form ρ =⊗
j ρj. Accordingly, the RFTS scheme employs neighborhood maps whichprepare the mixed state factors among the virtual subsystems Ej = (ρjTr)j ⊗ Ij.This observation can be applied to certain types of thermal states, as shown in thefollowing example.
Example 6.7.1 (Gibbs states of factorizable QL Hamiltonians). Consider aQL Hamiltonian H =
∑kHk acting on H '
⊗Ni=1Hi for which there exists a virtual
factorization H '⊗M
j=1 Hj satisfying 1) for all k there exists a j = jk such thatHk ' Hk
j ⊗ Ij and 2) for each j there exists a k such that B(Hj)⊗ Ij ≤ B(HNk)⊗ INk .Then, the Gibbs state
ρ(H) = exp(−βH)/Tr (exp(−βH)) (6.38)
is RFTS. This follows from the fact that each virtual subsystem algebra is containedin a neighborhood algebra and that the target state is factorized with respect to thevirtual subsystems,
ρ(H) = exp(−βH)/Tr (exp(−βH))
= exp(−β∑k
(Hk)jk ⊗ Ijk)/Tr (exp(−βH))
=1
Tr (exp(−βH))
M⊗j=1
exp(−β∑
k s.t. jk=j
Hk).
We expect that, in general, the algebraic dependence among the Hamiltonian termsshould play a significant role in determining whether or not such a factorization exists.We leave this to be studied in future work.
Theorem 6.5.5, involving a proper virtual subsystem decomposition, can also be
164
6.8 Proofs
generalized. Here, the local restriction is defined by the mixed state’s subsystemsupport, as in Def. 6.5.8. The construction, then, is completely analogous to that ofthe pure state case. The remaining results regarding RFTS involve the Schmidt spanprojectors derived from the target pure state. As we do not know of the analogousstructure for the mixed state case, we do not yet know how these results could beextended.
In Section 6.6.2 we showed that certain commuting RFTS scenarios implied theexistence of rapidly mixing dynamics. Considering the converse, we show that thereexist classes of mixed states which admit rapidly mixing dynamics, yet support cor-relations beyond what is allowed for RFTS.
Example 6.7.2 (Non-RFTS commuting Gibbs state). In [156] it is shown thatfor 1D lattice systems, the Davies generator derived from a commuting Hamiltonianconstitutes rapidly mixing dynamics with respect to the corresponding Gibbs state.Consider the 1D Ising model HN = −J
∑N−1i=1 σiz⊗σi+1
z , with coupling strength J > 0.It is well known that, in the thermodynamic limit, for any finite temperature, thetwo-point correlations of this Gibbs state are exponentially decaying with distance:Tr(σiz ⊗ σi+Lz ρ
)∼ e−ξL, see e.g. []. Therefore, spins with disjoint neighborhood
expansions are correlated, which violates the necessary condition for RFTS given inProp. 6.4.3.
6.8 Proofs
We present here complete proofs of all the technical results stated in the previoussections.• Necessary conditions for FTS.–
Theorem 6.3.1 |ψ〉 is FTS with respect to N only if |ψ〉 satisfies Eq. (6.5) and thereexists at least one neighborhood for which
dim(HN ) ≥ 2 · dim(ΣN (|ψ〉)).
Proof. We prove, by its contrapositive, that |ψ〉 being FTS implies |ψ〉 satisfies Eq.(6.5). Assume |ψ〉 does not satisfy Eq. (6.5). Then there exists some |φ〉 /∈ span(|ψ〉)for which |φ〉 ∈
⋂k ΣNk(|ψ〉). Any |ψ〉-preserving neighborhood map Ek must fix all
states in ΣNk(|ψ〉). Any sequence of such maps fixes |φ〉〈φ| and, hence, cannot map|φ〉〈φ| into |ψ〉〈ψ| as is required for FTS. We continue by showing that the remainingcondition is also necessary.
Assuming that |ψ〉 is FTS, let ET . . . E1(·) = |ψ〉〈ψ|Tr (·) be a sequence of CPTPmaps which stabilizes |ψ〉. Then, there must exist some map Ek for which Ek(ρ) =|ψ〉〈ψ| for some ρ /∈ span(|ψ〉). Using the locality and |ψ〉-invariance of Ek, we show
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
that the equation Ek(ρ) = |ψ〉〈ψ| places an upper bound on the Schmidt rank of|ψ〉 with respect to the N|N bipartition. The analysis is made easier considering apurification of the equation Ek(ρ) = |ψ〉〈ψ|, the purified equation being first-order in|ψ〉. In purifying, ancilla systems must be introduced for ρ and Ek, though not for|ψ〉〈ψ|, as it is already pure. Letting HA be the ancilla purifying ρ, we have
ρ→ |φ〉 ∈ HA ⊗HN k ⊗HNk ,s.t. TrA (|φ〉〈φ|) = ρ. (6.39)
Letting HB be the ancilla for Ek, we obtain an isometry representation,
Ek → V : HN k ⊗HNk → HN k ⊗HNk ⊗HB,
V = IN k ⊗ VN→NB,s.t. TrB
(V · V †
)= Ek(·). (6.40)
(Note: we drop the k in Nk and continue by using just N to label the neighborhoodsystem.) With respect to the isometry representation, Ek(ρ) = |ψ〉〈ψ| becomes
|ψ〉〈ψ|NN = TrB(V ρV †
)= TrAB
((IA ⊗ V )|φ〉〈φ|(IA ⊗ V )†
). (6.41)
Hence, (IA ⊗ V )|φ〉 is some pure state, which, upon tracing out AB, leaves the purestate |ψ〉. Therefore,
(IA ⊗ V )|φ〉 = |λ〉AB|ψ〉NN , (6.42)
where |λ〉AB is some pure state on ancillary systems HA ⊗HB.
The invariance condition, Ek(|ψ〉〈ψ|) = |ψ〉〈ψ|, constrains the form of V . Writtenin terms of V , invariance requires
TrB(
(IN k ⊗ VN→NB)|ψ〉〈ψ|(IN k ⊗ VN→NB)†)
= |ψ〉〈ψ|. (6.43)
Hence, (IN k ⊗ VN→NB)|ψ〉 is some pure state, which, upon tracing out B, leaves thepure state |ψ〉. Therefore, (IN k ⊗ VN→NB)|ψ〉 = |0〉B ⊗ |ψ〉, where |0〉B is some purestate on B. This equation ensures that VN→NB acts trivially on ΣN (|ψ〉), outputting|0〉 on B, when doing so. The action of VN→NB on ΣN (|ψ〉)⊥, which we denote V ⊥N→NBis unconstrained as of yet. In summary, invariance ensures that VN→NB acts triviallyon ΣN (|ψ〉), giving the decomposition
VN→NB = ΠN ⊗ |0〉B + V ⊥N→NB, (6.44)
166
6.8 Proofs
where ΠN is the projector onto ΣN (|ψ〉) and V ⊥N→NB satisfies V ⊥N→NBΠN = 0.
Trace-preservation of Ek constrains V ⊥N→NB. In terms of V , trace-preservationrequires I = V †V (= E†k(I)). Evaluating this in terms of the decomposition of Eq.(6.44), we have
INN = [ΠN + (ΠN ⊗ 〈0|B)V ⊥N→NB + h.c.
+ (V ⊥N→NB)†V ⊥N→NB]⊗ IN . (6.45)
The non-trivial part of this equation is on system N , where the equation may beblock decomposed as[
I 00 I
]=
[ΠN (ΠN ⊗ 〈0|B)V ⊥
((ΠN ⊗ 〈0|B)V ⊥)† (V ⊥)†V ⊥
], (6.46)
showing that (ΠN ⊗ 〈0|B)V ⊥N→NB = 0. With these conditions on V we return to Eq.(6.42),
|λ〉AB|ψ〉NN = IA ⊗ V |φ〉= (IA ⊗ IN ⊗ ΠN )|φ〉 ⊗ |0〉B (6.47)+ (IA ⊗ IN ⊗ V ⊥N→NB)|φ〉. (6.48)
Decompose this equation into three parts according to:
H ' supp(IAN ⊗ ΠN ⊗ |0〉〈0|B)
⊕ supp(IAN ⊗ ΠN ⊗ (I− |0〉〈0|B))
⊕ supp(IAN ⊗ (I− ΠN )⊗ IB). (6.49)
The vector |λ〉AB|ψ〉NN lies entirely in the first two blocks. Later, we will need to usethe fact that IA ⊗ (I− |0〉〈0|)B|λ〉AB 6= 0. This fact follows from |λ〉AB|ψ〉NN havinga non-trivial part in the second block, which we now to follows from the assumptionρ /∈ span(|ψ〉〈ψ|).
Towards reductio ad absurdum, assume that the norm-1 vector |λ〉AB|ψ〉NN liescompletely in the first block. Define |λ0〉A ≡ (IA ⊗ 〈0|B)|λ〉AB. Then, |λ〉AB|ψ〉NN =(IA⊗|0〉〈0|B⊗ IN ⊗ΠN )|λ〉AB|ψ〉NN = |λ0〉A|0〉B|ψ〉NN , where ‖|λ0〉‖ = 1. Projectingthe right-hand side of Eq. (6.47) into the first block,
The last term is zero, as (ΠN ⊗〈0|B)V ⊥N→NB = 0. Then, removing the common factor
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
of |0〉B from the remaining terms, we have
|λ0〉A|ψ〉NN = (IAN ⊗ ΠN )|φ〉. (6.51)
The vector |λ0〉A|ψ〉NN is assumed to be norm-1, and hence (IAN ⊗ ΠN )|φ〉 is aswell. Since ΠN is a projector, ‖(IAN ⊗ ΠN )|φ〉‖ = ‖|φ〉‖ only if (IAN ⊗ ΠN )|φ〉 =|φ〉 = |λ0〉A|ψ〉NN . Tracing out system A, this last equation becomes |ψ〉〈ψ| =TrA (|φ〉〈φ|) = ρ. As this violates the assumption that ρ 6= span(|ψ〉〈ψ|), it must bethat |λ〉AB|ψ〉NN lies at least partly in the second block. From this, 0 6= (IANN ⊗(I − |0〉〈0|)B)|λ〉AB|ψ〉NN ≡ |λ⊥〉AB|ψ〉NN . Defining the matrix V ≡ ΠN ⊗ (IB −|0〉〈0|B)V ⊥N→NB, the second block equation reads
|λ⊥〉AB|ψ〉NN = (IA ⊗ V ⊗ IN )|φ〉. (6.52)
Towards bounding the Schmidt rank of |ψ〉, it is useful to transform this vec-tor equation into a matrix equation by applying partial-transpose to the compositeHilbert space HA ⊗ HN . Hence, the vectors |λ⊥〉AB, |ψ〉NN , and |φ〉ANN are trans-formed into matrices:
λ⊥ : HA → HB
ψ : HN → HNφ : HAN → HN (6.53)
Note that rank(ψ) = dim(ΣN (|ψ〉)). Eq. (6.52) is tranformed into the matrix equa-tion
λ⊥ ⊗ ψ = V φ. (6.54)
It follows that rank(λ⊥ ⊗ ψ) = rank(V φ). On the left hand side,
dim(ΣN (|ψ〉)) = rank(ψ) ≤ rank(λ⊥ ⊗ ψ), (6.55)
using the fact that |λ⊥〉AB 6= 0, as shown earlier. On the right hand side,
rank(V φ) ≤ rank(V ) ≤ dim(ΣN (|ψ〉)⊥), (6.56)
where the last inequality follows from ker(V ) ≥ ker(V ⊥N→NB) ≥ ΣN (|ψ〉). Theabove two inequalities together imply dim(ΣN (|ψ〉)) ≤ dim(ΣN (|ψ〉)⊥). With H 'ΣN (|ψ〉)⊕ ΣN (|ψ〉)⊥, we obtain
dim(HN )
dim(ΣN (|ψ〉))≥ 2. (6.57)
168
6.8 Proofs
• Unitary generation property.–We develop a few results which build up to a proof of Prop. 6.3.4. The following
is a repurposing of Prop. 1.2.2 in [200], specified to our own setting.
Corollary 6.8.1. Consider a quantum system H '⊗
iHi of dimension D =∏
i di,a neighborhood structure N , and a quantum state |ψ〉 ∈ H. Let UNk,|ψ〉 be the neigh-borhood stabilizer groups of |ψ〉. Then, for any element U ∈ 〈UN ,|ψ〉〉k, there exists asequence of (D−1)2 elements Uj, drawn from the UNk,|ψ〉, such that U = U1 . . . U(D−1)2.Also, the group 〈UN ,|ψ〉〉k is connected.
Lemma 6.8.2. Let a Lie subgroup H of a Lie group G be generated by connected Liesubgroups Hα, α ∈ A for set A. Then the Lie algebra h of H is generated by the Liealgebras hα of Hαs.
Proof. Let h = 〈hα〉α ⊆ h. Let H ⊆ H be the corresponding connected Lie subgroup.To show that two Lie algebras are equal, h = h, it suffices to show that their Liegroups are equal, H = H. Thus, it remains to show that H ⊇ H. Since each Hα
is connected, each is the exponential of its Lie algebra hα. Hence, h ⊇ hα for all αimplies H ⊇ Hα for all α. H is the smallest Lie subgroup of G containing all Hα.Thus, H being a group and H ⊇ Hα for all α implies H ⊇ H. Finally, since H = H,we have h = h.
Proposition 6.8.3 (Equality for stabilizer groups/algebras). 〈uNk,|ψ〉〉k = u|ψ〉if and only if 〈UNk,|ψ〉〉k = U|ψ〉.
Proof. (⇐) Equal Lie groups have equal Lie algebras.(⇒) Assume 〈uNk,|ψ〉〉k = u|ψ〉. Let u be the Lie algebra of the Lie group 〈UNk,|ψ〉〉k.
U|ψ〉 and the UNk,|ψ〉 are each connected Lie subgroups of U(H). Therefore, they areeach equal to the exponential of their respective Lie algebras. Furthermore, Coro.6.8.1 ensures that 〈UNk,|ψ〉〉k is connected due to the connectedness of the UNk,|ψ〉.Hence, 〈UNk,|ψ〉〉k and U|ψ〉 being connected implies that if 〈UNk,|ψ〉〉k = U|ψ〉, thenu = u|ψ〉. The UNk,|ψ〉 being connected ensures that, by Lemma 6.8.2, the Lie algebrau is generated by the Lie algebras uNk,|ψ〉. Thus, we have 〈uNk,|ψ〉〉k = u|ψ〉.
Proposition 6.3.4 (Unitary generation property) Given a state |ψ〉 and a neigh-borhood structure N , any element in U|ψ〉 can be decomposed into a finite product ofelements in UNk,|ψ〉 if and only if
〈uNk,|ψ〉〉k = u|ψ〉, (6.58)
where 〈·〉k denotes the smallest Lie algebra which contains all Lie algebras from theset indexed by k.
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
Proof. (⇐) First, we show that 〈uNk,|ψ〉〉k = u|ψ〉 implies the decomposition of elementsin U|ψ〉. By Prop. 6.8.3, the Lie algebra generation implies the Lie group generation.Corollary 6.8.1, then, implies that the Lie group generation ensures that elements ofU|ψ〉 decompose into finite products of elements of the UNk,|ψ〉.
(⇒) 〈uNk,|ψ〉〉k ≤ u|ψ〉 is true by construction. We show that 〈uNk,|ψ〉〉k ≥ u|ψ〉 underthe decomposition assumption. Consider an arbitrary element X ∈ u|ψ〉. Then, byassumption, exp(X) = U ∈ U|ψ〉 admits a decomposition, U = UT . . . U1, with each Uiin a UNk,|ψ〉. As the UNk,|ψ〉 are connected, each element in UNk,|ψ〉 is the exponentiationof an element in uNk,|ψ〉: Ui = exp(Xi). Hence, U = exp(XT ) . . . exp(X1). Iteratingthe Baker-Campbell-Hausdorff formula, we can write U = exp(Y ) for some elementY ∈ 〈uNk,|ψ〉〉k. Then, since U = exp(X) = exp(Y ), X and Y are proportional to oneanother, ensuring λY = X ∈ 〈uNk,|ψ〉〉k, for λ ∈ R.
• Sufficient conditions for FTS.–Theorem 6.3.6. A state |ψ〉 is FTS relative to a connected neighborhood structureN if there exists at least one neighborhood for which
dim(HNk) ≥ 2 · dim(ΣNk(|ψ〉)), (6.59)
and〈uNk,|ψ〉〉k = u|ψ〉. (6.60)
Proof. We construct a finite sequence of CPTP maps which is ensured to stabilizethe target state under the conditions given in the theorem.
The Schmidt span dimension condition ensures that there exists some neighbor-hood for which dim(ΣNk(|ψ〉)) ≤ dim(ΣNk(|ψ〉)⊥). Then, we can choose any subspaceΣ1Nk ≤ dim(ΣNk(|ψ〉)⊥), such that dim(Σ1
Nk) = dim(ΣNk(|ψ〉)). We think of Σ1Nk as
being a “copy” of ΣNk(|ψ〉) lying inside ΣNk(|ψ〉)⊥. For convenience, we drop theindex k, and define Σ0
N ≡ ΣNk(|ψ〉). Then, HN = Σ0N ⊕ Σ1
N ⊕ R, where R is theremaining subspace of HN . Choosing an identification between Σ0
N and Σ1N , we can
writeΣ0N ⊕ Σ1
N ' C2 ⊗ ΣN , (6.61)
such that
Σ0N ' |0〉 ⊗ ΣN , (6.62)
Σ1N ' |1〉 ⊗ ΣN . (6.63)
The first map that we define to be used in our FTS sequence is
EN ≡ (|0〉〈0|Tr ⊗ I)⊕ IR, (6.64)
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6.8 Proofs
with respect to the decomposition HN = (C2 ⊗ ΣN ) ⊕ R. The corresponding globalmap is defined as W ≡ EN ⊗ IN . The global Hilbert space decomposes as
H ' HN ⊗HN = (C2 ⊗ ΣN ⊗HN )⊕ (R⊗HN ), (6.65)
whereby the target state can be written as (|0〉 ⊗ |ψ〉)⊕ 0. From this decompositionand the definition of W , we can see that W(|ψ〉〈ψ|) = |ψ〉〈ψ|, and hence W satisfiesthe invariance condition. Furthermore, the only state orthogonal to |ψ〉 whose densityoperator is mapped to |ψ〉〈ψ| is |ψ′〉 ≡ (|1〉 ⊗ |ψ〉) ⊕ 0. Hence, we can interpret Was correcting an arbitrary error U acting on the C2 qubit of |ψ〉. In brief, the mainstrategy we employ towards stabilizing |ψ〉 is to iterate the following procedure: 1)apply a sequence of invariance-satisfying, neighborhood unitaries to map a state |α〉to |ψ′〉, 2) apply W to map |ψ′〉 to |ψ〉.
As mentioned, the remaining CPTP maps that we define for the FTS sequenceare unitaries. For each vector in |α〉 ∈ ker(〈ψ|), we define
Uα ≡ |ψ〉〈ψ| ⊕ (|ψ′〉〈α|+ |α〉〈ψ′|)⊕ I. (6.66)
This unitary has non-trivial action only on span|α〉, |ψ′〉, acting as Uα|α〉 = |ψ′〉.Thus, we can see that the composition of Uα and W gives a map which takes |α〉 tothe target state |ψ〉. We label the corresponding CPTP map with Uα(·) ≡ Uα · U †α.
In Prop. 6.3.4, we show that the assumed property 〈uNk,|ψ〉〉k = u|ψ〉 ensuresthat any U ∈ U|ψ〉 can be decomposed into a finite product of invariance-satisfying,neighborhood unitaries. Since any Uα, with 〈ψ|α〉 = 0 is in U|ψ〉, such Uα can becomposed from a finite sequence of |ψ〉-preserving neighborhood maps.
Finally, we construct the sequence of CPTP maps which renders |ψ〉 FTS. Let|α〉 label an orthonormal basis set for ker(〈ψ|) with the following convention forthe ordering:
The individual neighborhood maps manifestly satisfy invariance. Thus, it remains toshow that E = |ψ〉〈ψ|Tr. It suffices to show that E(I) = d|ψ〉〈ψ|, where d is the global
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
Hilbert space dimension. In the first step,
WU0′(I) =W(I)
=W([(|0〉〈0|+ |1〉〈1|)⊗ I]⊕ 0 + (0⊗ 0)⊕ I)
= 2(|0〉〈0| ⊗ I)⊕ 0 + (0⊗ 0)⊕ I
= 2ΠN + ΠR, (6.69)
where the decomposition in the second and third lines is H ' [C2 ⊗ (ΣN ⊗ HN )] ⊕[R⊗HN ] and ΠR is the projector onto R⊗HN . In the next step,
WU1′WU0′(I) =W(2U1′(ΠN )) + U1′(ΠR))
=W(2[U1′(ΠN − |1′〉〈1′|)+ U1′(|1′〉〈1′|)] + ΠR)
=W(2[(ΠN − |1′〉〈1′|) + |ψ′〉〈ψ′|+ ΠR)
= 2W(ΠN − |1′〉〈1′|)+ 2W(|ψ′〉〈ψ′|) +W(ΠR)
= 2(ΠN − |1′〉〈1′|) + 2|ψ〉〈ψ|+ ΠR. (6.70)
The key property used above is the fact that all operators above have support in(|0〉 ⊗ ΣN (|ψ〉) ⊕ R) ⊗ HN , where W acts trivially. Similarly, in the next step wehave,
Continuing in this way, the sequence terminates at
WUT ′WU(dk+1)′(2dk|ψ〉〈ψ|+ ΠR) = (dk + T )|ψ〉〈ψ|= d|ψ〉〈ψ|. (6.74)
172
6.8 Proofs
Thus, we have shown that the following sequence renders |ψ〉 FTS,
T∏i=0
(W Ui′) = |ψ〉〈ψ|Tr. (6.75)
Proposition 6.8.4. If |ψ〉 satisfies 〈uNk,|ψ〉〉k = u|ψ〉 with respect to neighborhoodstructure N , then |ψ〉 satisfies Eq. (6.5) with respect to N .
Proof. Assume that |ψ〉 does not satisfy Eq. (6.5). Then⋂k ΣNk(|ψ〉) = S >
span(|ψ〉). We show that 〈uNk,|ψ〉〉k ≤ uS , where uS is the Lie algebra associatedto the Lie group that stabilizes S, US . The definining property of US is that for allU ∈ US , U |s〉〈s′|U † = |s〉〈s′| for all |s〉, |s′〉 ∈ S. Then, the defining property of thecorresponding Lie algebra is that for all X ∈ uS , [X, |s〉〈s′|] = 0 for all |s〉, |s′〉 ∈ S.Consider an arbitrary neighborhood Nk and an element Y ∈ uNk,|ψ〉. By definition,Y satisfies [Y, |r〉〈r′|] = 0 for all |r〉, |r′〉 ∈ ΣNk(|ψ〉). Since S ≤ ΣNk(|ψ〉), we have[Y, |s〉〈s′|] = 0 for all |s〉, |s′〉 ∈ S. Thus, Y ∈ uS . As this inclusion holds for allelements in uNk,|ψ〉 for any k, and uS is closed with respect to linear combination andLie product, then any element in 〈uNk,|ψ〉〉k is contained in uS . Since S > span(|ψ〉),we have uS < u|ψ〉. Finally, since 〈uNk,|ψ〉〉k ≤ uS , it follows that 〈uNk,|ψ〉〉k < u|ψ〉, andin particular, |ψ〉 does not satisfy
〈uNk,|ψ〉〉k 6= u|ψ〉.
• Necessary conditions for RFTS.–Proposition 6.4.3 (Finite correlation length) A quantum state ρ is RFTS withrespect to N only if the following is satisfied: for any two subsystems A and B withdisjoint neighborhood expansions (i.e. AN ∩ BN = ∅), arbitrary observables XA andYB are uncorrelated, that is, Tr (XAYBρ) = Tr (XAρ)Tr (YBρ).
Proof. Assuming ρ is RFTS with respect to N , there exists a sequence of neigh-borhood maps such that ρTr (·) = ET . . . E1(·). Let EAN be the composition ofall such maps which act non-trivially on A, and similarly for EBN with B. Byassumption, EAN and EBN act disjointly. Let Erest be the composition of the re-maining maps. By the robustness assumption, we may reorder the maps to writeρ = ErestEAN EBN (σAN ⊗ τBN ⊗ ωANBN ), with arbitrary input density operators. LetXA and YB be arbitrary observables acting on A and B. We have Tr (XAYBρ) =Tr(XAYBErestEAN EBN (σAN ⊗ τBN ⊗ ωANBN )
). Since, E†rest is unital and both XA
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
and YB are trivial where the map acts, the expression simplifies to Tr (XAYBρ) =Tr(XAEAN (σAN )⊗ YBEBN (τBN )
), where EAN , EBN are defined to act on their re-
spective systems and we have traced out ωANBN . The trace can be separated as
Tr (XAYBρ) = Tr(XAEAN (σAN )
)Tr(YBEBN (τBN )
). For the remaining steps, we
first note that
Tr(XAEAN (σAN )
)= Tr
(XAEAN (σAN )⊗ EBN (τBN )⊗ ω
ANBN
), (6.76)
and
Tr(YBEBN (τBN )
)= Tr
(EAN (σAN )⊗ YBEBN (τBN )⊗ ω
ANBN
). (6.77)
Finally, with Erest being trace-preserving, we may re-insert it into the trace to obtain
Tr (XAYBρ)
= Tr(XAEAN (σAN )
)Tr(YBEBN (τBN )
)= Tr
(Erest[XAEAN (σAN )⊗ EBN (τBN )⊗ ω
ANBN ])
× Tr(Erest[EAN (σAN )⊗ YBEBN (τBN )⊗ ω
ANBN ])
= Tr(XAErestEAN EBN (σAN ⊗ τBN ⊗ ωANBN )
)× Tr
(YBErestEAN EBN (σAN ⊗ τBN ⊗ ωANBN )
)= Tr (XAρ)Tr (YBρ) . (6.78)
In the second to last step we have used the fact that Erest acts trivially on XA andYB.
Lemma 6.4.4 (Recoverability property) Let target pure state ρ = |ψ〉〈ψ| beRFTS with respect to N . If a mapM acts on a subsystem A,M≡ MA ⊗ IA, thenthere exists a sequence of CPTP neighborhood maps EAN each acting only on AN ,such that ρ = El . . . E1 M(ρ).
Proof. Let E ′k be the sequence of neighborhood maps which renders ρ RFTS. Definethe subsequence of maps EAN ≡
∏Nk∩A 6=∅ E
′k. Let ER be the product of the remaining
E ′k. We have that ER EAN (σ) = ρ for any density operator σ. We show that EAN
174
6.8 Proofs
acting on the transformed target state,M(ρ), recovers ρ:
EAN M(ρ) = EAN M ER EAN (σ)
= EAN ER M EAN (σ)
= EAN ER(ρ′)
= ρ,
where σ is an arbitrary density operator and ρ′ =M EAN (σ).
Proposition 6.4.5 (Zero CMI) A quantum state ρ is RFTS with respect to Nonly if the following is satisfied: for any two regions A and B, with AN ∩ B = ∅,I(A : B|C)ρ = 0, where C ≡ AN\A.
Proof. To prove this result, we specify Lemma 6.4.4 to the case whereM = (τATrA)⊗IA, with τA the completely mixed state on A. Lemma 6.4.4 gives EANM(ρ) =(τATrA) ⊗ IA[ρ] = EAN (τA ⊗ ρA). Then, using the fact that AN ∩ B = ∅, we traceout all but AN and B (i.e. all but systems ABC) to obtain ρABC = TrABC (ρ) =
TrABC(EAN (τA ⊗ ρA)
)= EAN (τA ⊗ ρBC). Since ρ is written as a short quantum
Markov chain, we have I(A : B|C)ρ = 0.
We prove a necessary condition for RFTS regarding the commutativity of thecanonical FF Hamiltonian projectors. Towards this, we prove a lemma which con-strains the form of stabilizing CPTP maps.
Lemma 6.8.5. If Ek acting on neighborhood Nk preserves |ψ〉, then, for arbitrary ρ,Ek satisfies
ΠkEk(ρ)Πk = ΠkρΠk + ΠkσΠk, (6.79)
where Πk is the orthogonal projector onto ΣNk(|ψ〉) and σ = Ek(Π⊥k ρΠ⊥k ) ≥ 0.
Proof. If Ek is to preserve |ψ〉 then the Kraus operators of Ek must act trivially onsupp(TrNk [|ψ〉〈ψ|]). This requires the form,
Ki = λiΠk +RiΠ⊥k . (6.80)
Trace preservation of Ek requires that
I =∑i
|λi|2Πk + λ∗iΠkRiΠ⊥k + λiΠ
⊥k R†iΠk + Π⊥k R
†iRiΠ
⊥k . (6.81)
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
From this, it follows that∑
i |λi|2 = 1, Π⊥k∑
iR†iRiΠ
⊥k = Π⊥k , and, most importantly,
Πk(∑i
λ∗iRi)Π⊥k = 0. (6.82)
Finally, applying these trace-preserving conditions to ΠkEk(ρ)Πk, we find
ΠkEk(ρ)Πk = Πk(∑i
|λi|2ΠkρΠk + λiΠkρΠ⊥k R†i
+ h.c.+RiΠ⊥k ρΠ⊥k R
†i )Πk
= ΠkρΠk + Πkρ(Π⊥k∑i
λiR†iΠk)
+ h.c.+∑i
ΠkRiΠ⊥k ρΠ⊥k R
†iΠk
= ΠkρΠk + ΠkσΠk.
We will also make use of the following trace inequality:
Proposition 6.8.6 (Non-commuting penalty). Let Π1 and Π2 be projectors, withΠ1∩2 the projector onto their intersection. Then
Tr (Π1Π2) ≥ Tr (Π1∩2) +1
2Tr(|[Π1,Π2]|2
). (6.83)
Proof. First, note that
1
2|[Π1,Π2]|2 =
1
2(Π1Π2Π1 + Π2Π1Π2 − (Π1Π2)2 − (Π2Π1)2). (6.84)
Taking the trace of both sides and rearranging terms, we have
Tr (Π1Π2) = Tr((Π1Π2)2
)+
1
2Tr(|[Π1,Π2]|2
). (6.85)
Note that, under conjugation, Π1∩2(Π1Π2)2Π1∩2 = Π1∩2. Using that trace is non-increasing under conjugation with respect to a projector, we obtain Tr ((Π1Π2)2) ≥Tr (Π1∩2(Π1Π2)2Π1∩2) = Tr (Π1∩2). Making this replacement, we obtain the desiredresult.
Combining the two results above, we obtain a necessary condition for robust sta-bilization.Proposition 6.4.6 (Commuting FF Hamiltonian) If |ψ〉 is robust quasi-locally
176
6.8 Proofs
stabilizable with respect to neighborhood structure N , then [Πk,Πk] = 0 for all neigh-borhoods Nk, where Πk and Πk are the projectors onto ΣNk(|ψ〉) and ∩j 6=kΣNj(|ψ〉),respectively.
Proof. Assume |ψ〉 is robust stabilizable with respect to the sequence of neighborhoodmaps ET . . . E1. Let Ek be the neighborhood map on Nk and El be the compositionof the remaining neighborhood maps. Robust stabilizability, implies Ek Ek(·) =|ψ〉〈ψ|Tr (·). The invariance condition requires Ej(X) = X for any X ∈ ΣNj(|ψ〉〈ψ|).Since Πk ∈ ΣNj(|ψ〉〈ψ|) for all j 6= k, each Ej with j 6= k must fix Πk. Hence, we have
Ek(Πk) = (∏j 6=k
Ej)(Πk) = Πk. (6.86)
Thus, applying the full sequence of CPTP maps to Πk, we have
|ψ〉〈ψ|Tr(Πk) = Ek Ek(Πk) (6.87)= Ek(Πk). (6.88)
Conjugating both sides of the equation with respect to Πk, we can apply Lemma 6.8.5to obtain
|ψ〉〈ψ|Tr(Πk) = ΠkΠkΠk + ΠkσΠk, (6.89)
where σ is some positive-semidefinite operator. Next, conjugating both sides of theequation with respect to the projector Πk ≡ Πk−|ψ〉〈ψ| kills the left hand side, whileleaving the sum of two positive semidefinite operators on the right hand side,
0 = ΠkΠkΠkΠkΠk + ΠkΠkσΠkΠk. (6.90)
The sum of two positive-semidefinite matrices is zero only if both matrices are zero.Taking the trace of the first zero matrix gives
This only holds if [Πk,Πk] = 0. As the above arguments are made for an arbitraryneighborhood Nk, they must hold for all neighborhoods. Thus, we arrive at the statedconsequence.
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
• Non-operational sufficient conditions for RFTS.–Theorem 6.5.5 (Neighborhood factorization on local restriction) A state|ψ〉 described on the coarse-grained subsystems H '
⊗Ni=1Hi with respect to the
neighborhood structure N is robustly finite time stabilizable (RFTS) if there existsa locally restricted space H '
⊗Ni=1 Hi, with a complement H0 ' H⊥, whereby a
factorization H =⊗M
j=1 Hj gives
|ψ〉 = 0⊕M⊗j=1
|ψj〉 ∈ H0 ⊕M⊗j=1
Hj (6.95)
and for each virtual subsystem Hj there exists a neighborhood Nk such that
I0 ⊕ B(Hj)⊗ Ij ≤ B(HNk)⊗ INk . (6.96)
Proof. Assume the conditions above hold. We construct a finite sequence of com-muting QL CPTP maps which robustly stabilize ρ. First, we construct the mapswhich prepare the locally restricted space. Define the map E0
i : B(Hi) → B(Hi) tobe E0
i (·) = Pi · P †i + PiTr(Pi)
Tr ((I− Pi)·), where Pi is the projector onto Hi. For eachneighborhood Nk we construct a map E0
k ≡⊗
i∈Nk E0i , which prepares support on the
locally restricted space of all coarse-grained subsystems contained in that neighbor-hood.
On the virtual systems, define the maps Ej : B(H0⊕Hj⊗Hj)→ B(H0⊕Hj⊗Hj)as
Ej(·) = I0 ⊕ (ρjTr)j ⊗ Ij. (6.97)
Each virtual subsystem labeled j is associated to a neighborhood Nk on which itsoperators act non-trivially. Correspondingly, each neighborhood-acting map Ej isconstructed from Ej by pre-composing it with E0
k ,
Ej ≡ Ej E0k (·). (6.98)
The Kraus operators of Ej are contained in I0⊕B(Hj)⊗Ij. Hence, by I0⊕B(Hj)⊗Ij ≤B(HNk) ⊗ INk , we have that the Kraus operators of Ej act non-trivially only onneighborhood Nk. Thus, each map Ej is a valid neighborhood map. Finally, we mustshow that an arbitrary sequence of these maps prepares ρ while leaving it invariant.For invariance, we have Ej(ρ) = EjE0
k (ρ) = Ej(0 ⊕⊗M
j=1 ρj) = 0 ⊕⊗M
j=1 ρj = ρ.
To demsonstrate preparation of ρ, we use the fact that E0i Ej = EjE0
i . Consider an
178
6.8 Proofs
arbitrary complete sequence of the neighborhood maps,
EM . . . E1 = (EM E0M) EM−1 . . . E2 E1
= EM EM−1 . . . E2 E1 E0M
We continue in this way, using the commutativity of the support projections withthe Ej to move all of the support projections to act first. Since every coarse-grainedparticle will have been accounted for, we may combine the action of all of theseprojections E0
k into a single projection E ` which has the effect of projecting onto H,
EM . . . E1 = (EM . . . E1) (E0M . . . E0
1 )
= EM . . . E1 E `. (6.99)
Finally, we see that the composition of these maps constitutes a preparation of thetarget state,
EM . . . E1 = I0 ⊕M⊗j=1
(ρjTr) E `
= (0⊕M⊗j=1
ρj)Tr
= ρTr. (6.100)
• Algebraic sufficiency for RFTS.–Theorem 6.5.7 (Algebraically induced factorization) If a set of algebras Aj,Ai ∈ B(H), is complete and commuting, then each Aj has a trivial center and thereexists a decomposition of the Hilbert space H '
⊗Tj=1 Hj for which Aj ' B(Hj)⊗ Ij
for each j.
Proof. First, assume that a neighborhood algebra Aj were reducible, so that thereexists some X ∈ Aj where X ∈ A′j, but X 6= c · I. As X ∈ Aj, it commutes with allelements of Ak for k 6= j, and hence the algebra generated by all the neighborhoodalgebras has a non-trivial commutant, which violates completeness. We obtain theHilbert space factorization as follows. First, for any algebra Aj with trivial centeracting on H, there exists a decomposition H ' Hj ⊗Hj for which Aj = B(Hj)⊗ Ij.Starting with A1, we have H ' H1 ⊗ H1. From this, A′1 = I1 ⊗ B(H1). As thealgebras are all commuting, A2 ≤ A′1 = I1 ⊗ B(H1). Hence, A2 carries a naturalaction on H1, and A2 having a trivial center implies that there is a decomposition
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
H1 = H2⊗H1,2 for which A2 = I1⊗B(H2)⊗ I1,2. So far we have H ' H1⊗H2⊗H1,2.With the introduction of each additional algebra, we obtain another factor in theHilbert space. Continuing in this way, completeness of the set of Aj ensures thatonce all Aj have been included, the global Hilbert space will have been decomposedas H '
⊗j Hj.
The following two lemmas will be used to formulate the decomposition of Prop.6.8.9. This proposition will then be used for proving Prop. 6.5.10.
Lemma 6.8.7. Consider a Hilbert space H '⊗
iHi, a neighborhood Nk containingHp, and a state |ψ〉 ∈ H. Then, the subsystem kernel of |ψ〉〈ψ| on p coincides withthe subsystem kernel of the neighborhood projector Πk on p.
Proof. We show this by direct calculation. With p ∈ Nk,
ker(Trp (|ψ〉〈ψ|)) = ker(Trp (ρNk)). (6.101)
Using the spectral decomposition ρNk =∑
j λj|j〉〈j| along with properties of thekernel function,
ker(Trp (|ψ〉〈ψ|)) = ker(∑j
λjTrp (|j〉〈j|))
= ker(∑j
Trp (|j〉〈j|))
= ker(Trp(TrN k
(Πk ⊗ IN k
)))
= ker(Trp (Πk)). (6.102)
Lemma 6.8.8. Given a positive-semidefinite operator P acting on HA ⊗ HB, letPA = TrB (P ). Then, ker(PA) = ker(ΣA(P )).
Proof. The direction ker(PA) ⊇ ker(ΣA(P )) is trivial since PA ∈ ΣA(P ). For ker(PA) ⊆ker(ΣA(P )), assume PA|v〉 = 0. Since PA ≥ 0, this is equivalent to Tr (|v〉〈v|PA) = 0.In terms of P then, we have Tr (|v〉〈v| ⊗ IP ) = 0. Let Ei
iTr (|v〉〈v| ⊗ EiP ) =0. Since each term must be non-negative, we have Tr (|v〉〈v| ⊗ EiP ) = 0 for alli. We may rewrite this as 〈v|TrB
((I⊗√Ei)P (I⊗
√Ei))|v〉 = 0, which implies
TrB((I⊗√Ei)P (I⊗
√Ei))|v〉 = 0 for all i. Since the POVM is informationally
complete,spanTrB ((I⊗ Ei)P ) |i = 1, . . . , d2
B = ΣA(P ).
Thus, |v〉 ∈ ker(ΣA(P )).
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6.8 Proofs
Proposition 6.8.9. Let⊗N
i=1Hi be a Hilbert space on which we define a neighborhoodstructure N . Let |ψ〉 be any state of this system. For any neighborhood Nk containinga system Hp, consider the reduced state ρp = Tr (|ψ〉〈ψ|) and the decomposition Hp 'supp(ρp)⊕ ker(ρp). There exists a decomposition Hp ' (
⊕lHl ⊗H′l)⊕ ker(ρp) such
that
algΣp(Πk) =
(⊕l
B(Hpl )⊗ IH′pl
)⊕ spanI. (6.103)
Proof. The above decomposition is ensured as long as algΣp(Πk) commutes withall of Isupp(ρp) ⊕ B(ker(ρp)). We show that an arbitrary basis element in Isupp(ρp) ⊕B(ker(ρp)) commutes with all elements in Σp(Πk). Consider the non-orthonormalbasis I, |α〉〈β|, where |α〉, |β〉 are basis elements of ker(ρp). We need only verifythat elements |α〉〈β| commute with Σp(Πk), as I does trivially. Since p ∈ Nk, wemay apply Lemma 6.8.7 to obtain that ker(ρp) = ker(Trp (Πk)). From Lemma 6.8.8we have ker(Trp (Πk)) = ker(Σp(Πk)). Thus, |α〉, |β〉 ∈ ker(Σp(Πk)), ensuring that|α〉〈β| ∈ Σp(Πk)
′.
Note that, by Lem. 6.8.7, since supp(ρp) = supp(Trp (Πk)), the positive semidefiniteoperator Trp (Πk) ∈ alg(Σp(Πk)) has maximal rank in the space B(supp(ρp))⊕ 0.Theorem 6.5.10A state |ψ〉 described on the coarse-grained subsystemsH '
⊗Ni=1Hi
with respect to the neighborhood structure N admits a decomposition |ψ〉 = 0⊕⊗
j |ψj〉induced by the neighborhood algebra-induced factorization H ' H0 ⊕
⊗j Hj, and
hence, is robustly finite time stabilizable (RFTS) with respect to N if1) |ψ〉 satisfies Eq. (6.5) with respect to Nand2) the neighborhood algebras Aj are commuting and complete on the local support
space H.
Proof. Completeness and commutativity of the Aj induce the decomposition H '⊗j Hj. The decomposition ensures that each Aj is of the form I0⊕B(Hj)⊗ Ij. Each
Πk commutes with all elements in Aj for j 6= k. This can only be the case if Πk actsas identity on each factor Hj with j 6= k,
Πk = 0⊕ Πk ⊗⊗j 6=k
I, (6.104)
where we have used the fact that the Πj do not have support on the local kernel spaceH0. Thus, the Πk are mutually commuting with one another. This commutativityalong with satisfaction of Eq. (6.5) ensures that
Π1Π2 . . .ΠT = 0⊕⊗j
Πj = |ψ〉〈ψ|. (6.105)
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
The trace of the left hand side is the product of ranks of projectors Πj and is equal tothe trace of |ψ〉〈ψ|, which is 1. Hence, each projector satisfies Πj = |ψj〉〈ψj|. Thus,
|ψ〉 = 0⊕⊗j
|ψj〉. (6.106)
With this factorization of |ψ〉, as well as the fact that I0⊕B(Hj)⊗ Ij ≤ B(HNj)⊗ IN jfor each j (by construction), Thm. 6.5.5 ensures the |ψ〉 is RFTS with respect toN .
• Matching overlap condition for RFTS.–Proposition 6.5.12 (Matching overlap RFTS) Let
⊗Np=1Hp be a Hilbert space
of coarse-grained particles with respect to a neighborhood structure N that satisfiesthe matching overlap condition. If |ψ〉 satisfies Eq. (6.5) with respect to N and[Πj,Πk] = 0 for all pairs of neighborhood projectors, then |ψ〉 is RFTS.
Proof. We obtain a decomposition of each Hp that constitutes a global change ofbasis leading to a neighborhood factorization as in Prop. 6.5.5 which implies RFTS.Consider an arbitrary coarse-grained particle p with Hilbert space Hp. The decom-position of Hp is induced by the algebra algΣp(Πk)Nk3p. By Prop. 6.8.9, eachalgΣp(Πk) is contained in B(supp(ρp))⊕ spanIker(ρp). We show that, furthermore,algΣp(Πk)Nk3p = B(supp(ρp)) ⊕ spanIker(ρp), by establishing that its center isequal to spanI, Isupp(ρp) ⊕ 0.
Assuming otherwise, there exists an X = X ⊕ 0 /∈ spanI, Isupp(ρp) ⊕ 0 suchthat X ∈ Σp(Πk)
′ for each Nk 3 p. Then, [Ip ⊗Xp,Πk] = 0 for all neighborhoods Nk(including Nk /∈p). Since X acts non-trivially on supp(ρp), we have Ip⊗Xp|ψ〉 = |τ〉 /∈span(|ψ〉). Since |ψ〉 satisfies Eq. (6.5), it is the only vector for which Πk|ψ〉 = |ψ〉for all neighborhoods Nk. However, for |τ〉,
which is a contradiction. Hence, no such X can exist, implying that the center ofalgΣp(Πk)Nk3p is equal to spanI, Isupp(ρp) ⊕ 0. This fact, together with the factthat algΣp(Πk)Nk3p is contained in B(Hsupp(ρp))⊕ spanIker(ρp), ensures that
As described, the matching overlap condition ensures that the intersection of any
182
6.8 Proofs
non-disjoint neighborhoods Nj and Nk is some coarse-grained particle p. Thus, from[Πj,Πk] = 0, we have [Σp(Πj),Σp(Πk)] = 0, abusing notation. Hence, for any twoneighborhoodsNj andNk containing p, we have algΣp(Πj) ≤ algΣp(Πk)′. The al-gebra algΣp(Πk)Nk3p, then, is seen to be generated by a finite number of mutuallycommuting algebras. Given the form of this algebra in Eq. 6.107, these generat-ing subalgebras algΣp(Πk) can only mutually commute if Hsupp(ρp) =
⊗k|Nk3p H
kp,
wherebyalgΣp(Πk) = (B(Hk
p)⊗ IHkp)⊕ spanIker(ρp),
for each neighborhood Nk 3 p.We have obtained a decomposition for each coarse-grained particle Hilbert space
Hp ' (⊗
k|Nk3p Hkp)⊕Hker(ρp). Hence, the global Hilbert space decomposes as
H '⊗p
Hp '⊗p
((⊗k|Nk3p
Hkp)⊕Hker(ρp)
)
'(⊗
p
⊗k|Nk3p
Hkp
)⊕H0 '
(⊗k
⊗p∈Nk
Hkp
)⊕H0
≡(⊗
k
Hk
)⊕H0. (6.108)
By the way this decomposition was formed, the Πk act trivially on all but one of thevirtual factors, Πk = 0⊕ Πk
⊗j 6=k Ij. Hence, |ψ〉 satisfying Eq. (6.5) implies that
Π1Π2 . . .ΠT = 0⊕⊗j
Πj = |ψ〉〈ψ|. (6.109)
Similar to the proof of Thm. 6.5.10, the trace of the left hand side is the product ofranks of projectors Πj and is equal to the trace of |ψ〉〈ψ|, which is 1. Hence, eachprojector satisfies Πj = |ψj〉〈ψj|. Thus,
|ψ〉 = 0⊕⊗j
|ψj〉. (6.110)
With this factorization of |ψ〉, as well as the fact that I0⊕B(Hj)⊗ Ij ≤ B(HNj)⊗ IN jfor each j, Thm. 6.5.5 ensures the |ψ〉 is RFTS with respect to N .
• Efficiency of RFTS.–Proposition 6.6.2 (Lattice circuit size scaling) Consider a scalable lattice neigh-borhood structure NN (see discussion above Prop. 6.6.2 in Sec. 6.6.1). If |ψ〉 is RFTS
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Finite-time stabilization of quantum states with discrete-time quasi-localdynamics
with respect to this neighborhood structure, then |ψ〉 can be stabilized by a dissipativecircuit of size |N 0|(N/c) and depth D = |N 0|diam(N 0)d.
Proof. For any robustly finite-time stabilizable state, the circuit size is equal to thenumber of neighborhoods. From the unit cell definition, the number of neighborhoodsis |N 0|(N/c), the number of neighborhoods per unit cell times the number of unit cells.To bound the depth of the circuit, we devise a scheme which parallelizes the circuitto one with constant depth. In particular, we show that there exists a partitioningof the neighborhoods of N , and hence NN , into |N 0|diam(N 0)d parts such that eachpart consists of a set of mutually disjoint neighborhoods. If the union of the unitcell neighborhoods N 0 is translated in any direction a distance D = diam(N 0), theresulting set is disjoint from N 0. In particular, if we select a single neighborhoodNk ∈ N 0 and construct the set of neighborhoods generated by linear combinationsof Dei for each i = 1, . . . , d, the neighborhoods in this set are ensured to be disjointfrom one another. Hence, the sequence of the corresponding neighborhood maps actin parallel and constitute a layer of the circuit. This set of neighborhoods is generatedby a subgroup (DZ)d = DZ × . . . × DZ of the translation group Zd = Z × . . . × Z.Therefore, the translated copies of Nk for which this did not account each correspondto a coset of (DZ)d in Zd with respect to elements ~m = (m1, . . . ,md) ∈ Zd. Thiscoset group is isomorphic to the finite group Zd
D = ZD × . . . × ZD. The size of thisgroup is |Zd
D| = Dd. Using group action notation, we denote the ~m-translated versionof N0 as ~mN0. Each layer of neighborhood maps corresponds to a set of disjointneighborhoods,
Each neighborhood is accounted for and there are |N 0|Dd layers. With this scheme,we define,
E~m,k ≡∏
~v∈(DZ)d
E(~v+~m)Nk , (6.112)
for k = 1, . . . , |N 0|, ~m ∈ ZdD. The sequence of neighoborhood maps which prepares
the target state can be parallelized as ρTr = EN . . . E1 =∏|N 0|
k=1
∏~m∈ZdD
E~m,k, of whichthere are |N 0|Dd parallelized maps.
Proposition 6.6.7 (Commuting Liouvillian contraction bound) Let Lj bea, possibly infinite, set of bounded norm Liouvillians each acting on a subsystem ofuniformly-bound dimension. Assume that the spectral gaps λj are strictly boundedbelow by ν > 0. Then, there exists R > 0 such that for any subset S ⊆ Lj ofmutually commuting Liouvillians, defining L =
∑S Lj, the contraction is bounded by
η(eLt) ≤ |S|Re−νt. (6.113)
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6.8 Proofs
Proof. From Theorem 6.6.6.a, commutativity implies η(eLt) ≤∑S η(eLjt). With
ν < λj, Theorem 6.6.6.b ensures that, for each Lj ∈ Lj, there exists Rj > 0 suchthat η(eLjt) ≤ Rje
−νt. In [182], it is shown that, for fixed ν, Rj is upper bounded by
a function of order dd2jj , where dj is the dimension of the system on which Lj acts. Let
D ≥ dj be the uniform subsystem dimension bound. Then, we can find a constant R
and c such that for all j, we have R > cDD2> cd
d2jj > Rj. With this,
η(eLt) ≤∑S
η(eLjt) ≤∑S
Rje−νt
≤∑S
Re−νt = |S|Re−νt. (6.114)
Proposition 6.6.9 (Rapid mixing for commuting RFTS) Consider a family offinite systems Hα indexed by α such that for each α there is a neighborhood structureN α with dim(HNαk ) < D. Furthermore, assume the number of neighborhoods scalesno more than polynomially in system size, N α
k ∈ N α and |N α| ≤ b log(dim(Hα)),for some constant b > 0. For each α, let ρα be RFTS with respect to N α by a set ofcommuting neighborhood maps Eαk , where there exists some ν > 0 such that for anyeigenvalue λ ∈ eig(Eαk ), λ = 1 or λ < 1 − ν. Then, there exists a family of boundednorm QL Liouvillians satisfying rapid mixing with respect to ρα.
Proof. For each α, define the neighborhood-acting Liouvillian operators Lαk ≡ Eαk −Iα.These Liouvillians have bounded norm and the gap λk of each Lαk is ensured to satisfyλk > ν > 0. Take Lαkk,α as a set of Liouvillians, and define the sequence of subsetsSα = Lαkk, indexed by α. Then, for each α, the global Liouvillian is
Lα =∑k
Lαk . (6.115)
For each α, this Liouvillian is a sum of commuting terms with D and ν satisfying theconditions in Prop. 6.6.7 with some finite prefactor R. Thus,
Setting c = Rb, δ = 1, and λ = ν verifies rapid mixing.
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Chapter 7
Towards finite-time dissipativequasi-local quantum encoders
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Towards finite-time dissipative quasi-local quantum encoders
7.1 Preliminaries
We explore the notion of a finite-time dissipative encoder. In particular, we investigatethe possibility of quasi-local implementations, giving motivations, rigorous definitions,and a few illustrative examples.
The traditional subspace encoding approach embeds a Hilbert space HQ into alarger Hilbert space H via an isometry. For example, to encode any qubit state|ψ〉 = α|0〉+β|1〉, we could append a number of ancillary systems and apply a unitarytransformation taking (α|0〉+β|1〉)|00 . . .〉 into the encoded state α|0〉+β|1〉 ∈ HQ ≤H. As emphasized in [201], this subspace approach is, in general, inadequate becauseit does not account for the possibility that the encoding system H might couple toother degrees of freedom H′ in the larger space H = H⊗H′. This becomes an issuewhen, for example, H interacts with H′.
As anticipated in [202], and established in [196, 21], the more general notion of anencoding is a subsystem encoding. Ticozzi and Viola [203] give a natural operationalinterpretation for the subsystem principle, which clarifies connections among variousnotions of encoding. They posit that a quantum code is a linear map which preservesthe distinguishability of quantum states.
Definition 7.1.1. A linear map on Hermitian operators, Φ : B(HQ) → B(HP ),defines a 1-isometric encoding if for all ρ1, ρ2 ∈ D(HQ) and p ∈ [0, 1],
||pΦ(ρ1)− (1− p)Φ(ρ2)||1 = ||pρ1 − (1− p)ρ2||1.
As shown in [203], this property is sufficient to ensure that such an encoding is asubsystem encoding: any encoding map Φ : B(HQ) → B(HP ) induces a subsystemdecomposition H ' HS ⊗ HF ⊕ HR, where HS ' HQ and Φ(ρ) = ρ ⊗ τ ⊕ 0R forsome fixed τ ∈ B(HF ). A quantum code, then, is defined as CQ ≡ Φ(D(HQ)) andits elements take the form ρ ⊗ τ ⊕ 0R for fixed τ with respect to the subsystemdecomposition.
In the above notion of an encoder, the input system is an abstract logical quantumsystem, while the output system is a multi-qubit system. In a realistic scenario, wemay desire a description of how to implement the quantum information transfer fromlocally-accessible degrees of freedom into the quantum code. It is useful, then, todecompose the encoder Φ above into two steps. In the first step the abstract logicalquantum information is encoded by ΦL into locally-accessible upload qubits, suchas a number of spatially local spin-1/2 particles. In the second step, the quantuminformation of the data qubits is carried into the code by the map ΦP . The localencoding and the physical encoding compose to give Φ = ΦP ΦL.
The most familiar implementation of such an encoding is a unitary encoder ΦP =UP · U †P . The to-be-encoded quantum information |ψ〉 is initialized in the upload
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7.1 Preliminaries
qubits, while the remaining part of the system is initialized in some fixed pure state|φ〉. Then, the encoder ΦP is a global unitary transformation designed to map |ψ〉⊗|φ〉into the encoded state |ψ〉. In general, the success of the encoding requires theremaining part of the system to be sufficiently well-prepared in the fixed state |φ〉.The unitary encoding is not necessarily robust to errors in the initialization of |φ〉.For example, if the remaining system is afflicted by an error X which transforms |φ〉to an orthogonal state, 〈φ|X|φ〉 = 〈φ|φ′〉 = 0, then the subsequent encoded state issure to be orthogonal to the intended encoded state.
With a dissipative encoder, as opposed to a standard unitary encoder, we areafforded some degree of robustness with respect to the initialization of the remainingsystem.
Definition 7.1.2. Consider a Hilbert space H =⊗
iHi, a quantum code CQ ⊆ B(H),and upload qubitsHS =
⊗i∈SHi with D(HS) 1-isometric to CQ. A dissipative encoder
is a CPTP map ΦD : B(H)→ B(H) which acts as
ΦD(ρ⊗ σ) = Uρ⊗ τ ⊕ 0RU†, (7.1)
where σ is from some set W ⊂ D(HS), while U and τ are those associated to thesubsystem decomposition of the code. The set W is a linear space and we refer to itas the basin of attraction for the encoder ΦD.
In [204] such a dissipative encoder was implemented via continuous-time quasi-local Liouvillian dynamics. As we will describe, this encoding is robust with respect toa non-trivial basin of attraction. Furthermore, the dynamics are autonomous in thatthe control procedure is constant in time, requiring no switching by the controller.A drawback is that, even under the assumption of ideal conditions, the encoding issure to possess non-zero error for any finite time. We explore the construction of adissipative encoder in terms of a finite sequence of quasi-local CPTP maps. Althoughthe dynamics are not autonomous, we show some examples where encoding may beachieved exactly in finite time. Comparing to the continuous-time toric code exampleof [204], the finite-time dissipative encoder we construct for the toric code admits thesame non-trivial basin of attraction. In practice, any encoding, discrete or continuous,is only achieved approximately. However, it is valuable to analyze a scenario in whichthe deviations from the exact encoding are due solely to undesirable errors, as opposedto additionally, a finite waiting time (as in the continuous-time case).
Definition 7.1.3. Consider a Hilbert space H =⊗
iHi, a quantum code CQ ⊆ B(H),upload qubits HS =
⊗i∈SHi with B(HS) 1-isometric to CQ, and a neighborhood
structure N . A finite-time quasi-local dissipative encoder with basin of attractionW ⊆ B(HS) is any product of neighborhood maps Φk such that
ΦT . . . Φ1 = ΦP (7.2)
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Towards finite-time dissipative quasi-local quantum encoders
maps any state ρS ⊗ σS ∈ B(HS) ⊗W into UρS ⊗ τ ⊕ 0RU† where τ and U are the
fixed state and unitary transformation associated to the subsystem decomposition ofthe code CQ.
The utility of an encoding depends on its ability to prevent, recover from, or pro-tect against errors subjected to the system. We establish some terminology regardingquantum error correction. The degree of error prevention of a code can be classifiedaccording to the following [205].
Definition 7.1.4. With respect to error model E : B(H) → B(H), a given codeCQ = Φ(B(HQ)) is
• fixed if E(ρ) = ρ acts as identity on CQ.
• noiseless if for any probability distribution pj,∑
j pjE j acts as a 1-isometryon CQ.
• preserved if E acts as a 1-isometry on CQ.
• correctable if there exists a recovery map R : B(H)→ B(H) such that the codeis noiseless for R E.
• protectable if there exists P : B(H)→ B(H) such that the code is noiseless forE P.
These definitions are presented in order of decreasing strictness. In [203] it is shownthat correctable 1-isometric encodings are equivalent to protectable 1-isometric en-codings.
The quantum codes that we consider fall under the stabilizer formalism [20]. Here,the subspace code is specified as the common +1 eigenspace of a set of commutingoperators Sj, which are all elements of the N -qubit Pauli group. The logical oper-ators Xα and Zα commute with all stabilizers and satisfy the commutation relationsXαZβ + (−1)δαβZβXα = 0.
Our strategy for constructing finite-time dissipative encoders employs correctionmaps that correspond to stabilizer generators.
A correction map Φj corresponding to stabilizer operator Sj acts trivially on oper-ators with support in the code, has range in the +1 eigenspace of Sj, and coherentlymaps the −1 eigenspace of Sj into the +1 eigenspace. Often, the coherent mappingfrom the −1 to +1 eigenspace corresponds to undoing an error from the Pauli-group.An error E is detectable by measurement of Sj if E, Sj = 0; that is, if E exchangesthe ± eigenspaces of Sj. Assuming E is from the Pauli-group, reapplying E undoesthe detected error. In general, the correction map Kraus operators involve a Pauli-group correction operator Cj satisfying Cj, Sj = 0. The Kraus operators of the
190
7.2 Repetition code
correction map correspond to correcting an error conditional on the measurementoutcome of Sj:
K(0)j =
1
2(I + Sj), K
(1)j =
1
2Cj(I− Sj), (7.3)
We find that the following equivalent form will be more useful for our purposes,
K(0)j =
1
2(I + Sj), K
(1)j =
1
2(I + Sj)Cj. (7.4)
For a given stabilizer Sj, there are many choices of correction operators Cj. Thechallenge in constructing a working finite-time dissipative encoder, then, lies in prop-erly choosing the correction operators along with the order of the correction maps.Through a number of examples, we will arrive at a list of principles which guide theconstruction of a finite-time dissipative encoder for general stabilizer codes.
The finite-time dissipative encoders that we construct are closely related to thenotion of discrete-time quasi-local stabilization [206, 194]. In particular, any finite-time dissipative encoder ΦD made from correction maps constitutes a conditionalstabilization procedure for any state |ψ〉 in the code. By construction, correctionmaps leave invariant any state in the code. Furthermore, if the system is conditionedon |ψ〉〈ψ| ⊗ σ where σ is any state in the basin of attraction W , then ΦD maps theinput into |ψ〉〈ψ|.
Our correction map approach is not the only means of constructing a finite-timedissipative encoder. As we will demonstrate at the end of Section 7.2, the techniqueof sequential generation can be implemented towards making a finite-time dissipativeencoder for the repetition code with no constraints on the basin of attraction. Itwould be interesting to explore if such approaches could be applied to more generalcodes.
7.2 Repetition code
We demonstrate a finite-time quasi-local dissipative encoder for the quantum 3-qubitrepetition code. This stabilizer code is correctable with respect to single bit fliperrors. The code space is span|000〉, |111〉 ≤ H⊗3. Sufficient stabilizer generatorsare ZZI, IZZ. We choose the logical operators to be X = XXX and Z = ZII.
We show that the repetition code may be encoded from a localized data qubitusing a sequence of two-body CPTP maps. The two correction maps are Φ12 andΦ23, which we consider to act in that order. (One can construct maps acting inthe opposite order which will work as an encoder, but we choose to analyze onecase for simplicity of presentation.) Each stabilizer generator admits the same set of
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Towards finite-time dissipative quasi-local quantum encoders
neighborhood-acting correction operators on its respective neighborhood,
XI, IX,XZ,ZX, Y I, IY, Y Z, ZY .
To arrive at a principle for choosing the correction operators, consider any codewith two stabilizer generators S1 and S2. Let Φ2 Φ1 be the composition of twocorrection maps. The Kraus operators of the composed map are K(i1,i2) = 1
2(I +
S2)Ci22
12(I +S1)Ci1
1 . In order for the range of these composed maps to be in the code,it is necessary and sufficient that C2 commutes with S1. For necessity, consider thatC2 and S1 do not commute. Then, it must be that C2, S1 = 0, since Pauli groupelements either commute or anti-commute. The Kraus operator K(i1,1) would be ofthe form 1
2(I+S2)1
2(I−S1)C2C
i11 , having a range which is orthogonal to the code. That
commutativity is sufficient follows from the fact that the range of the composed map’sKraus operators is the range of 1
4(I + S2)(I + S1), which is equal to the code. Note
that this condition rules out the issue of choosing redundant correction operators; if acorrection operator is used for two different stabilizers, then, in the composed map’sKraus operators, one of the correction operators necessarily anti-commutes with astabilizer to the right of it.
For the repetition code, the set of candidate correction operators for C23 is re-duced to IIX, IZX, IIY, IZY . We can narrow down further by the requirementthat the logical operators of the code (i.e. X = XXX and Z = ZII) be left in-variant. This requires that correction operators commute with the logical operators,leaving C12 ∈ IXI, ZY I and C23 ∈ IIX, IZY . Furthermore, for general cor-rection maps, correction operators are only ever defined up to multiplication by thecorresponding stabilizer. For instance, the Kraus operator of the first map can bewritten equivalently with respect to either correction operator,
1
2(III + ZZI)ZY I =
1
2(ZZI + III)(ZZI)ZY I =
1
2(III + ZZI)IXI. (7.5)
Therefore, our requirements have singled out one possibility for the set of correctionmaps. The key property of these correction maps is that the range of their compositionis in the code and the logical operator values are invariant. Having specified thecorrection maps, we determine the basin of attraction W ⊆ D(H23) which ensuresthat ΦD(ρ⊗ σ) = ρ for σ ∈ W .
For encoding to be achieved, the logical operator coordinates of the output densitymatrix must coincide with those of the data qubit:
Tr(X iZjρ
)= Tr
(XiZjΦD(ρ⊗ σ)
), (7.6)
for all i, j ∈ 0, 1. We have chosen ΦD to leave the values of the logical operators
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7.2 Repetition code
invariant so that the above equation simplifies to
Tr(X iZjρ
)= Tr
(XiZj(ρ⊗ σ)
). (7.7)
Evaluating this for the repetition code, we obtain
Tr(X iZjρ
)= Tr
((X iZjρ)⊗ [(XX)iσ]
), (7.8)
giving Tr ((XX)iσ) = 1. The basin of attraction is therefore the set of densityoperators σ with support contained in the +1 eigenspace of IXX.
The above conclusions can be generalized to the N -qubit repetition code. Thisstabilizer error correcting code is correctable with respect to N−1
2uncorrelated bit flip
errors. The code is defined as the subspace span|0〉⊗N , |1〉⊗N ≤ H⊗N . Sufficientstabilizer generators are ZiZi+1N−1
i=1 . The logical operators can be taken as X =X⊗N and Z = Z ⊗ I⊗N−1. As with the 3-qubit repetition code, we find that thecorrection operator for stabilizer ZiZi+1 is Xi+1. Applying Eq. (7.6) as before, wefind that the basin of attraction W is the set of density operators σ with support inthe +1 eigenspace of X⊗N−1. One state which can be locally prepared in the basin ofattraction is |+〉⊗N−1. Consider that the pairs of qubits in this system were subject toany degree of XX, Y Y , ZZ , Y Z, or ZY interaction. The initialized state |+〉⊗N−1
would be altered, but because it would remain in the +1 eigenspace of X⊗N−1, thesubsequent encoding would still function properly.
Next we present a finite-time dissipative encoder that deviates from the abovescheme, but admits a global basin of attraction. The scheme utilizes the concept ofsequential generation [193]. Consider the two-body Kraus operators defined by
K(j) =1∑i=0
|ii〉〈ij|. (7.9)
The corresponding superoperator is easily verified to be trace-preserving. Intuitively,this map traces out the second system, while classically copying the |0〉 or |1〉 statefrom the first qubit onto the second.
Consider a 1-D chain of N qubits. Let a sequence of N − 1 of these maps act on
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Towards finite-time dissipative quasi-local quantum encoders
subsequent nearest neighbor qubits. The Kraus operators of the composed map are
Applying an arbitrary Kraus operator K(~j) ≡ K(jN )N−1,N . . . K
(j2)1,2 to the product of a
data qubit state and an arbitrary state, we obtain
K(~j)(α|0〉+ β|1〉)⊗ |φ〉2...N =∑i
|i〉〈i|(α|0〉+ β|1〉)|i . . . ii〉〈j2 . . . jN ||φ〉
= (α|0〉⊗N + β|1〉⊗N)〈j2 . . . jN ||φ〉(7.11)
Letting |ψ〉 ≡ α|0〉+β|1〉 and |ψ〉 ≡ α|0〉⊗N+β|1〉⊗N , the corresponding superoperatoraction is therefore∑
~j
K(~j)|ψφ〉〈ψφ|K(~j)†
= |ψ〉〈ψ|∑~j
〈j2 . . . jN ||φ〉〈φ||j2 . . . jN〉
= |ψ〉〈ψ|. (7.12)
Thus, the encoding is achieved independent of the state of the remaining system.In other words, the basin of attraction is global. The main distinction between thisencoding and the previous one based of the stabilizer formalism is that the individualdissipative maps above do not leave the values of the logical operators invariant. Thisshows that, while the invariance condition may be useful for constructing dissipativeencoders for stabilizer codes, it is not requisite for constructing a general finite-timedissipative encoder.
We have demonstrated that a single localized qubit can be dissipatively encodedinto the repetition code using a finite sequence of nearest neighbor acting CPTP maps.We demonstrated the basin of attraction for this dissipative encoding as any state onthe remaining qubits with support contained in the +1 eigenspace of X⊗N , allowingthe encoding to be achieved with some degree of robustness in the initialization. Inthe remaining sections, we build off of this example to construct finite-time dissipativeencoders for more compelling examples of stabilizer codes.
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7.3 Toric code
7.3 Toric code
We now demonstrate a finite-time dissipative encoding of the toric code [207, 208].The 2-D toric code employs a 2-D lattice of 2L2 qubits arranged on the edges of thesquares of a grid. In this way, neighboring quartets of qubits either surround a squareof the grid (plaquette) or surround the intersection of a vertical and a horizontalline of the grid (vertex). The name “toric code” is due to identifying the eastern andwestern border qubits as neighboring one another, while similarly for the northern andsouthern border qubits; hence, the geometry and topology induced by the adjacencyof qubits is that of a flat 2-torus.
The Hamiltonian of the system is constructed using this structure. To each pla-quette p and to each vertex v (see Fig. 7.1a) we assign a four-body Hamiltonianacting on the corresponding qubits defined by
Hp ≡ (Z⊗4)p ⊗ Ip (7.13)
andHv ≡ (X⊗4)v ⊗ Iv. (7.14)
These operators establish the toric code as a stabilizer code as follows. Since eachplaquette overlaps with an even number of systems in any vertex, we have [Hp, Hv] = 0for all p and v.
HT ≡ −(∑p
Hp +∑v
Hv). (7.15)
A crucial consequence of the toroidal geometry is that the plaquette and vertex Hamil-tonians are not algebraically independent,∏
p
Hp = I and∏v
Hv = I. (7.16)
This implies that HT is at least 4-fold degenerate. Since the above identities generateall algebraic dependence of the plaquette and vertex Hamiltonians, the ground spaceof HT is exactly 4-fold degenerate. This ground space constitutes the toric code andis equivalently defined as the space of vectors |ψ〉 satisfying Hp|ψ〉 = Hv|ψ〉 = |ψ〉for all p and v. Later on, we will use the fact that it suffices to define the toric codeas the +1 eigenspace of all but one Hp and all but one Hv. This follows from thealgebraic redundancy expressed in Eq. (7.16).
In [204], the authors construct a Liouvillian constructed as a sum of plaquette-and vertex-acting Liouvillian terms. This Liouvillian generates a continuous-timeencoding from two localized physical qubits into the toric code. The labeling schemeof the qubits is borrowed adapted from [204] and depicted in Fig. 7.1b. The qubits
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Towards finite-time dissipative quasi-local quantum encoders
(a) Plaquette and vertex stabilizer oper-ators
(b) Qubit labeling scheme
Figure 7.1: a) The four-body stabilizer operators of the toric code act on vertices v asX⊗4 or on plaquettes as Z⊗4. b) A1 and A2 are the two upload qubits whose initialstate is mapped into the toric code. B1 and C1 are the systems (in addition to A1)on which the code’s logical operators for the first encoded qubit are defined to act.The same holds for B2 and C2 with respect to the second encoded qubit. D denotesthe remaining qubits. The systems B1, C1, B1, and C2 must be properly initialized inorder to achieve a faithful encoding.
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7.3 Toric code
(a) Logical Xs of toric code (b) Logical Zs of toric code
Figure 7.2: A definition of logical operators for the toric code. Logical operatorsconsist of topologically non-trivial loops of local bit-flip or local phase-flip errors.
A1 and A2 are chosen so as to share a plaquette and vertex which are labeled p∗and v∗. Then, the qubits of the vertical and horizontal strips which pass through p∗(except for qubits A1 and A2) are each prepared in |+〉. These strips are labeled B1
and B2, respectively. Simliarly, the qubits of the bands passing through v∗ are eachprepared in |0〉. These strips are labeled C1 and C2, respectively. The logical operatorsare chosen as
X1 = XA1 ⊗X⊗L−1B1 ⊗ Irest
Z1 = ZA1 ⊗ Z⊗L−1C1 ⊗ Irest
X2 = XA2 ⊗X⊗L−1B2 ⊗ Irest
Z1 = XA2 ⊗ Z⊗L−1C2 ⊗ Irest. (7.17)
The authors of [204] show that, with this initialization, the state of A1⊗A2 is driventowards the corresponding state of the toric code by means of well-chosen Liouvilliandynamics.
We review the correction maps used to construct the Liouvillian in [204]. Thenwe show that a judicious ordering of these maps constitutes a finite-time dissipativeencoder. The stabilizer generators are Hp, Hv, where the set ranges over all pla-quettes and vertices except for p∗ and v∗. Correction maps Φp are Φv are associatedto each of these stabilizer generators.
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Towards finite-time dissipative quasi-local quantum encoders
Figure 7.3: Sequences of correction maps. The ordering of correction maps andlocation of correction operators are chosen so that 1) correction operators commutewith all four logical operators and 2) subsequent correction operators are appliedwhere no correction map has acted previously.
Plaquettes and vertices are labeled according to their lattice coordinates withrespect to p∗ and v∗. The plaquette pα,β and vertex vα,β lie α sites north and β siteseast of p∗ and v∗, respectively. We use a tensor product structure where, for eachplaquette and vertex system, the north, east, south, and west qubits areN⊗E⊗S⊗W .
correction maps, respectively. The scheme devised in [204] for “pushing” errors to-wards A1 and A2 suggests a choice of ordering for the correction maps as depicted inFig. 7.3. From our analysis of the repetition code, we expect that the key feature ofthis choice of correction operators and ordering is that subsequent correction opera-tors commute with all previous stabilizer operators. This property is necessary andsufficient for the range of the composed map to be in the code.
Each plaquette correction map commutes with each vertex correction map sinceexchange of their Kraus operators can, at most, accrue an irrelevant global phase (itis canceled in the superoperator). Without loss of generality, we consider the vertex
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7.3 Toric code
maps to act first. As seen in Fig. 7.3, the ordering among the vertex (resp. plaquette)correction maps is chosen such that each subsequent correction operator acts whereno previous correction map (and hence stabilizer operator) has acted. This verifiesthat subsequent correction operators commute with all previous stabilizer operators.
Let ~i ∈ 0, 12(L2−1) indicate the i = 0, 1 of the plaquette and vertex Krausoperators. Then, moving all correction operators to the right of the stabilizers, eachKraus operator of the encoder is written as follows
K(~i) = (P1Ci11 ) . . . (PtC
itt )
= P+(Ci11 . . . Cit
t )
= P+C~i, (7.19)
where Pj = 12(I + Hj), P+ is the projector into the toric code, and t = 2(L2 − 1) is
the total number of correction maps. The encoder can be written as
ΦD(·) = P+∑~i
C(~i) · C(~i)†P+.
This verifies that the range of the encoder is contained in the code itself.Finally, we determine the initial conditions of the input state that ensure ρ on
A1 ⊗ A2 is encoded into ρ of the code. Let the initial state be ρA1A2 ⊗ σBCD. Thebasin of attraction for σ is determined by the analogue of Eq. (7.6),
Tr(X iA1ZjA1XkA2Z lA2ρ)
= Tr(Xi
1Zj
1Xk
2Zl
2ΦD(ρA1A2 ⊗ σBCD)), (7.20)
for all i, j, k, l = 0, 1. Since the correction operators and stabilizers commute withthe logical operators, we have Φ†D(X
i
1Zj
1Xk
2Zl
2) = Xi
1Zj
1Xk
2Zl
2. With this, the aboveequation simplifies to
Tr(X iA1ZjA1XkA2Z lA2ρ)
= Tr((X iA1ZjA1XkA2Z lA2ρ)⊗ ((X iZjXkZ l)⊗L−1
BC σBCD))
= Tr(X iA1ZjA1XkA2Z lA2ρ)Tr((X iZjXkZ l)⊗L−1
BC σBC), (7.21)
where in the last step, we have traced out D to obtain σBC = TrD (σBCD). Hence, thestate of systemD does not affect the encoding. This determines the basin of attractionto be states σBCD with reduced state on BC satisfying Tr
((X iZjXkZ l)⊗L−1
BC σBC)
= 1for all i, j, k, l. A sufficient choice of initial state, as given in [204], is the pure state|φ〉BC = |+〉⊗L−1
B1 |0〉⊗L−1C1 |+〉⊗L−1
B2 |0〉⊗L−1C2 . This demonstrates that an initialization can
be implemented locally. The full basin of attraction can also be described as anydensity operator with support in the +1 eigenspace of each of the four commutingoperators X⊗L−1
B1 , Z⊗L−1C1 , X⊗L−1
B2 , and Z⊗L−1C2 .
199
Towards finite-time dissipative quasi-local quantum encoders
7.4 General stabilizer codesWe investigate finite-time dissipative encoders for several other quantum error cor-recting codes in the stabilizer formalism [20]. We use the following principles in con-structing a finite-time dissipative encoder for a given stabilizer code with stabilizersSj, logical operators X i, Zi, and data qubits A1 ⊗ . . .⊗Ak:
(a) Choose correction operators which commute with the logical operators X i, Zi.This ensures that the basin of attraction is easily determined.
(b) Choose an order for the stabilizers and a corresponding set of correction opera-tors so that subsequent correction operators commute with all previous stabilizeroperators. This ensures that the range of the correction maps is in the code.
(c) Basin of attraction is calculated by requiring
Tr(X i1A1Zj1A1. . . X ik
AkZjkAkρ)
= Tr(Xi11 Z
j11 . . . X
ikk Z
jkk (ρA ⊗ σA)
)(7.22)
for all il, jl ∈ 0, 1, where A and A denote A1 ⊗ . . .⊗Ak and its complement,respectively.
Note that Eq. (7.22) is the generalization of Eq. (7.6), where we have simplified usingthe invariance of the logical operators with respect to ΦD. We use these principles todesign finite-time dissipative encoders for a few well-known stabilizer codes.
9-Qubit Shor code: The Shor code defined on 9-qubits has logical operators
X = XXX XXX XXX
Z = ZZZ ZZZ ZZZ. (7.23)
For each stabilizer we choose a corresponding correction operator using the first prin-ciple above.
S1 = ZZI III III C1 = IXX III III
S2 = ZIZ III III C2 = XXI III III
S3 = III ZZI III C3 = III IXX III
S4 = III ZIZ III C4 = III XXI III
S5 = III III ZZI C5 = III III IXX
S6 = III III ZIZ C6 = III III XXI
S7 = XXX XXX III C7 = ZII III ZII
S8 = XXX III XXX C8 = ZII ZII III. (7.24)
200
7.4 General stabilizer codes
To each stabilizer we associate a correction map Φj with Kraus operators K(i)j =
12(I + Sj)C
ij, where i = 0, 1, giving C0
j = I and C1j = Cj. Note that the quasi-locality
of each Kraus operator is slightly increased from that of the corresponding stabilizeroperator. This is necessarily the case if ever the form of a stabilizer operator anda logical operator are identical on qubits which the stabilizer acts non-trivially on.Otherwise, the correction operator could not anti-commute with the stabilizer whilecommuting with the logical operator as is needed. This increase in neighborhood sizewill occur whenever the logical operators act as collective unitaries and the stabilizerscontain either only X or only Z operators.
The finite-time dissipative encoding map is defined as ΦD = Φ1 . . .Φ7Φ8. Sincethe correction operators commute with all of the preceding stabilizers, the Krausoperators of ΦD can be written as
K(i1)1 . . . K
(i7)7 K
(i8)8 = P+Ci1
1 . . . Ci77 C
i88 , (7.25)
where P+ is the product of the stabilizer projectors and hence the projector into thecode. This shows that the output of ΦD is necessarily in the code. We choose thefirst qubit to be the data qubit whose state is to be mapped into the code. Since thecorrection operators and stabilizers commute with the logical operators, the logicaloperators are preserved by ΦD. The initialized state is of the form ρ1 ⊗ σ2...9. Thebasin of attraction for σ is determined by
Tr(X iZjρ
)= Tr
(XiZjΦD(ρ⊗ σ)
). (7.26)
Using the fact that Φ†D(XiZj) = X
iZj, we obtain
Tr(X iZjρ
)= Tr
(X iZjρ⊗ (X iZj)⊗8σ
)= Tr
(X iZjρ
)Tr((X iZj)⊗8σ
),
determining the basin of attraction to be states satisfying Tr ((X iZj)⊗8σ) = 1. Thereare no product states which satisfy this requirement. One pure state satisfying thiscondition is |φ〉 = (|00〉 + |11〉)⊗4. Although this state is not a product state, itsentanglement is restricted to pairs of neighboring qubits and can be prepared from aproduct state by a circuit of depth one.
7-Qubit Steane code: The Steane code defined on 7-qubits has logical operators
X = XXX XXX X
Z = ZZZ ZZZ Z. (7.27)
201
Towards finite-time dissipative quasi-local quantum encoders
The stabilizers and sufficient correction operators are
S1 = III XXX X C1 = ZII ZIZ Z
S2 = IXX IIX X C2 = ZIZ III I
S3 = XIX IXI X C3 = ZZI III III
S4 = III ZZZ Z C4 = XII IXI I
S5 = IZZ IIZ Z C5 = XIX III I
S6 = ZIZ IZI Z C6 = XXI III I. (7.28)
As before, to each stabilizer we associate a correction map Φj with Kraus opera-tors K(i)
j = 12(I + Sj)C
ij. The finite-time dissipative encoding map is defined as
ΦD = Φ1 . . .Φ5Φ6. Since the correction operators commute with all of the precedingstabilizers, the Kraus operators of ΦD can be written as
K(i1)1 . . . K
(i5)5 K
(i6)6 = P+Ci1
1 . . . Ci55 C
i66 , (7.29)
where P+ is the projector into the code, ensuring that the output of ΦD is necessarilyin the code. We choose the first qubit to be the data qubit. Since the correctionoperators and stabilizers commute with the logical operators, the logical operatorsare preserved by ΦD. The initialized state is of the form ρ1 ⊗ σ2...7. The basin ofattraction for σ is determined by
Tr(X iZjρ
)= Tr
(XiZjΦD(ρ⊗ σ)
). (7.30)
Using the fact that Φ†D(XiZj) = X
iZj, we obtain
Tr(X iZjρ
)= Tr
(X iZjρ⊗ (X iZj)⊗6σ
)= Tr
(X iZjρ
)Tr((X iZj)⊗6σ
),
determining the basin of attraction to be states satisfying Tr ((X iZj)⊗6σ) = 1. Thereare no product states which satisfy this requirement, but, as before, a product ofbipartite maximally entangled states serves the purpose: |φ〉 = (|00〉+ |11〉)⊗3.
5-Qubit code: The logical operators for the 5-qubit code are
X = XXXXX
Z = ZZZZZ. (7.31)
202
7.5 Further questions
The stabilizer operators are
S1 = XZZXI
S2 = IXZZX
S3 = XIXZZ
S4 = ZXIXZ. (7.32)
The 5-qubit stabilizer operators are 4-body. We are unable to find corresponding4-body correction operators which satisfy the sufficient properties for constructing afinite-time dissipative encoder. It is possible that the 5-qubit code is not amenable toconstructing a finite-time dissipative encoder using the approach we have outlined.
7.5 Further questions(a) Both the repetition code and toric code are examples of generalized surface
codes. How can finite-time dissipative encoders be constructed for (generalized)surface codes? What about Bacon-Shor, Haah, or Bombìn’s codes?
(b) Given a finite-time dissipative encoder constructed from the stabilizer formalismas above, does the “Lindbladized” version constitute a continuous-time dissipa-tive encoder?
(c) Can the sequential generation approach be extended to the toric code using itstensor network representation?
203
Chapter 8
Summary and outlook
205
Summary and outlook
In this thesis we have described results on two fronts: quantum joinability andquasi-local stabilization. The part-whole relationship features as the unifying themeof the work. On the first front, we have analyzed the question of whether or not adescription of the whole is consistent with constraints placed on the parts. On thesecond front, we have investigated preparing features of the whole using dynamicswhich are constrained to address specified parts. In both, we have confronted thephenomenon of multipartite quantum entanglement and its nuances which distinguishthe quantum from the classical cases.
Respectively, these results contribute to our understanding of the nature of quan-tum relationships among the parts of a quantum system and to our understandingof the limits to and opportunities for controlling multipartite entanglement usingengineered dissipation. We have drawn from a diverse mathematical tool set encom-passing linear algebra, group theory, representation theory, Lie theory, operator alge-bra, among others. Additionally, we have developed several mathematical conceptsthat we have found to be useful, including homocorrelation map, degree of agreement,Schmidt span, and neighborhood algebra.
In our exploration of quantum joinability, we began by analyzing the joinabil-ity and sharability of bipartite quantum states. By restricting to the special classesof Werner and isotropic states, we determined simple analytical expressions for 1-nsharability, namely, that the sum of the bipartite concurrences cannot exceed (d− 1)for Werner states and cannot exceed Cmax,d for isotropic states. Then, we determinedanalytical conditions for characterizing the joinability of these bipartite classes ofstates in the three-party setting. In other words, we determined what trios of Werneror isotropic states the pairs among Alice-Bob-Charlie are able to simultaneously pos-sess. Surprisingly, we found that the entanglement content of the joined bipartitestates does not suffice to determine the resulting joinability properties. In this three-party setting, we investigated the role that the classical joining limitations play inrestricting quantum joining.
The comparison of the quantum and classical joinability settings illuminated an in-triguing distinction. In particular, the quantum joinability limitations do not obey thetetrahedral symmetry of the classical joinability limitations in the three-bit setting.We discovered that the tetrahedral symmetry could be regained in the quantum caseby incorporating additional joinability settings. In particular, we were prompted todevelop the notion of channel joinability, whereby one asks whether a set of subsystemquantum channels (causal relationships) are consistent with some quantum channelon the whole. This led us to develop the unifying framework of generalized joinability,which encompasses the quantum state joinability case, the quantum channel joinabil-ity case, as well as variants. Many problems regarding the part-whole relationshipin multiparty quantum settings, such as the quantum marginal problem, the asym-metric cloning problem, and various quantum extension problems, are encapsulated
206
Summary and outlook
by this framework. An important step was to introduce the homocorrelation map asa natural way to represent quantum channels with bipartite operators, making themgeometrically comparable to quantum states. Using this tool, it is possible to directlycontrast the joinability properties of quantum states with those of quantum channels.As shown in Fig. 3.5, by directly comparing the joinability limitations for Wernerstates and depolarizing channels, the union of their joinability bounds regains thetetrahedral symmetry of the classical joinability scenario. Taking quantum channelsto express causal relationships and bipartite states to express acausal relationships,we find it intruiging that quantum mechanics distinguishes between the two in a waythat the classical probability theory does not. From these joinability findings, it istempting to view quantum causal and quantum acausal relationships as two puzzlepieces, neither of which completes the picture alone.
This distinction between the joinability properties of quantum channels and quan-tum states led us to further investigate the quantum causal-acausal asymmetry. Weidentified a difference between quantum states and quantum channels in the corre-lations that are obtainable from each. Namely, bipartite quantum states are lim-ited in their allowed degree of agreement, whereas quantum channels are limited intheir allowed degree of disagreement. This difference is made explicit by representingquantum channels with the homocorrelation map. We showed how these differences,expressed in terms of agreement bounds, in turn inform the joinability properties ofchannels vs states.
We view the quantum joinability results that we have presented as a startingpoint and a framework for further exploration. Throughout our analysis of quan-tum joinability, we have only considered scenarios with a pre-defined tensor prod-uct structure, and consequently all operator reductions are obtained via the usualpartial-trace construction. However, it is important to appreciate that this was not anecessary restriction. Following [170], one may also consider a more general notion ofa reduced state, which results from appropriately restricting the global state to a dis-tinguished operator subspace. Such a notion of reduction is operationally motivatedin situations where a tensor product structure is not uniquely or naturally affordedon physical grounds (notably, systems of indistinguishable particles or operationalquantum theory, see e.g. [132]). This points to a further extension of the presentjoinability framework “beyond subsystems”. This perspective also highlights anotherdifference between quantum causal and acausal relationships. In the acausal case, weare free to consider various factorizations of the multipartite Hilbert space, or evenforgo mention of the factorization, viewing all states on equal footing. It seems thatthe same does not hold for quantum causal relationships expressed by quantum chan-nels. Here, it seems that the input-output distinction is critical. Yet, do we need tomake such a sharp distinction between these subsystems? What operational groundslead to the need for a fixed subsystem distinction in the causal case, while allowing
207
Summary and outlook
for some freedom in the acausal case? We expect that it would be fruitful to connectthe insights from our joinability work to the recent, intriguing finding regarding theindefiniteness of causal order in quantum theory [209].
In the second half of the thesis, we investigated the potential of using quasi-localcontrolled dissipation for achieving the tasks of quantum state stabilization and en-coding quantum information into a quantum error-correcting code. We began ourexploration of quasi-local stabilization in the setting of continuous-time control dy-namics. We discovered conditions determining whether a general mixed state of afinite-dimensional multi-partite quantum system may be the unique fixed point for anatural class of quasi-local frustration-free Markovian dynamics – for given localityconstraints. In any case that a target state had been diagnosed as stabilizable, wehave provided a constructive procedure to synthesize stabilizing dynamics.
We have presented a number of quantum information processing and physicallymotivated examples demonstrating how our tools can naturally complement and gen-uinely extend available techniques for fixed-point convergence and stability analysis– including quasi-local stabilization of Gibbs states of non-commuting Hamiltonians(albeit for finite system size). Altogether, beside filling a major gap in the existingpure-state quasi-local stabilizability analysis, we believe that our results will havedirect relevance to dissipative quantum information processing and quantum engi-neering, notably, open-system quantum simulators.
We then moved on to explore quasi-local stabilization in the setting of discrete-time control dynamics. Specifically, we investigated the task of stabilizing a targetpure state in with a finite sequence of quasi-local-acting dissipative maps. In parallel,ongoing work [194] (not presented in this thesis), we have been developing a methodof alternating completely-positive trace-preserving map projectors to drive the sys-tem asymptotically towards a target state. With this scheme, many quasi-locallystabilizable states are only stabilized asymptotically, requiring infinite-time for exactstabilization. For a subset of states, however, we found that the sequence of stabi-lizing maps converges in a finite number of steps, exhibiting finite-time stabilization.Exact stabilization in finite-time is desirable from a control-theoretic, as well as prac-tical, perspective. Thus, the aim of our work was to develop control strategies andunderstand the features of target states which enable finite-time stabilization. Weshow that the set of finite-time stabilizable states is strict subset of the asymptoti-cally stabilizable states. For this subset of states, finite-time stabilization is achievedby designing the completely-positive trace-preserving maps to act collaboratively, by-passing the frustrated interactions of the alternating projection approach that ledto infinite-time stabilization. With the scheme that we developed, we showed thatcertain asymptotically stabilizable states, such as certain AKLT states [185], can, inprinciple, be quasi-locally stabilized exactly in finite time.
Beyond finite-time stabilization, we have investigated the notion of robust finite-
208
Summary and outlook
time stabilization, whereby the stabilization is achieved independently of the orderof the dissipative maps. While exact robust finite-time stabilization is very desir-able from an implementation perspective, we found that the set states amenable tothis task is strictly contained in the set of finite-time stabilizable states, as expected.Nevertheless, we showed that relevant multipartite entangled states, such as the re-sourceful graph states of measurement-based quantum computing, are still robustlyfinite-time stabilizable. Our general approach to establishing robust finite-time sta-bilization of a target state was to discover a virtual subsystem factorization of thetarget state, whereby the role of each neighborhood-acting map simplifies to preparinga pure state factor of the virtual subsystems.
In developing the use of a virtual subsystem decomposition for robust finite-timestabilization, we have clarified the role played by “commuting structures” towardsefficiently stabilizing a target state. We believe that the techniques used for “discov-ering” such commuting structures (e.g. the neighborhood algebra in Sec. 6.5.9) maybe of general interest to researchers working with virtual subsystem decompositions.
As robust finite-time stabilization constitutes an efficient means of preparing atarget state, we have sought to connect it to notions of efficient preparation in thecontinuous-time setting, such as rapid mixing. We showed that, for scalable cases ofrobust finite-time stabilization where the dissipative maps mutually commute, there,indeed, exist rapid mixing continuous-time quasi-local dynamics which stabilize thetarget state. Lastly, we have highlighted the stabilization results which may be ex-tended to the case of a target mixed state. We leave a number of directions here tofuture work.
Beyond quasi-local stabilization of a target state, we have also started to inves-tigate the task of encoding quantum information into a quantum error-correctingcode using a finite sequence of quasi-local dissipative maps. By focusing on quantumerror-correcting codes within the stabilizer formalism, we develop several principlesthat aid the design of finite-time “dissipative encoders” and give a number of relevantexamples. Of particular interest is the dissipative encoder we construct for the toriccode.
A number of research questions are prompted by the present analysis and call forfurther investigation. Determining whether our necessary condition for frustration-free quasi-local stabilization is, as we conjecture, always sufficient on its own even fornon-full-rank states is a first obvious issue to address. From a physical standpoint, inorder to both understand the role of Hamiltonan control and to make contact withnaturally occurring dissipative dynamics, it is important to scrutinize the extent towhich the stabilizing dynamics obtained in our framework may be compatible withrigorous derivations of the QDS – in particular in the weak-coupling-limit, where theinterplay between Hamiltonian and dissipative components is crucial and demands tobe carefully accounted for [136, 210].
209
Summary and outlook
Still within the present quasi-local stabilization setting, an interesting mathemat-ical question is to obtain a global characterization of the geometrical and topologi-cal properties of the frustration-free quasi-local stabilizable set, beginning from purestates. As alluded to in the last remark above, answering these questions may alsohave practical implications, in terms of approximate quasi-local stabilization, whichwe briefly discussed in Section 5.5.5. Related to that, while our main focus has indeedbeen on exact asymptotic stabilization, it may also be beneficial (possibly necessary)to tailor approximate methods for analysis and/or synthesis to specific classes ofstates. A natural starting point could be provided here by graph states, which haverecently been shown to arise as arbitrarily accurate approximations of ground statesof two-body frustration-free Hamiltonians [211].
From a more general perspective, the present analysis fits within our broaderprogram of understanding controlled open-quantum system dynamics subject to a re-source constraint. In that respect, an important next step will be to tackle differentkinds of constraints – including, for instance, less restrictive notions of quasi-locality,which allow for exponentially decaying interactions in space, as in [156]; or possiblyreformulating the quasi-local constraint away from “real” space and the associatedtensor-product-decomposition, but rather relative to a preferred operator subspace,in the spirit of “generalized entanglement” [170]. Lastly, and perhaps somewhatcounter-intuitively, as our results on separable non-FFQLS evidence, it is interest-ing to acknowledge that the quasi-local stabilization problem need not be trivial forclassical probability distributions. In work not presented in this thesis, we have be-gun to investigate further the connections between quasi-local stabilization and thequantum marginal problem. A practical motivation for making this connection is to-wards simplifying our characterizations of “naturally-occuring” many-body quantumstates. Recent work has investigated the use of ideas from the quantum marginalproblem towards efficient quantum state tomography [212]. They achieve quantumtomography using just quasi-local expectation values, for example, depending only onjust the two-body reduced density matrices. The extent to which naturally occurringquantum states can be characterized by their marginals is not well understood. But,if naturally occurring states are taken to be steady states of quasi-local dynamics,and we can relate steady states of quasi-local dynamics to states that are uniquelydetermined by their marginals, then there is hope for “efficient” characterization ofsuch states.
Several open questions also remain in our pursuit of understanding finite-timestabilization. One clear, outstanding problem is to obtain constructive proceduresfor synthesizing the sequences of stabilizing neighborhood unitary maps for finite-time stabilization. Furthermore, we would like to understand if, as conjectured, the“unitary generation property” is truly a necessary condition for finite-time stabiliza-tion. Towards robust finite-time stabilization, we have made significant progress in
210
Summary and outlook
understanding the role of “commuting structures”. While we have shown that thecommuting structure inherent in the virtual subsystem factorization ensures robustfinite-time stabilization, we would like to understand the extent to which it is neces-sary. In thinking towards approximate robust finite-time stabilization, analogous toapproximate frustration-free quasi-local stabilization of Sec. 5.5.5, we expect that itwould be fruitful to explore the notion of “approximate factorization” with respect tovirtual subsystems. More generally, we would like to explore the use of virtual sub-system factorization for analyzing the efficiency of approximate preparation statesusing either continuous-time or discrete-time dynamics. A last, important directionto pursue would be to assess the robustness of these finite-time stabilization schemesagainst various implementation errors. Intuitively, we expect the robust finite-timestabilization schemes to exhibit some degree of resilience to errors.
LOCC: local operations and classical communication, pg. 26
MPS: matrix product state, pg. 183
NN: nearest-neighbor, pg. 149 and Fig. 6.1
PEPS: projected entangled pair state, pg. 183
POVM: positive-operator valued measure, pg. 137
QDS: quantum dynamical semigroup, pg. 114
QIP: quantum information processing, pg. 13
QL: quasi-local, pg. 124
QLS: quasi-locally stabilizable, pg. 126
RFTS: robust finite-time stabilizable, pg. 189
213
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