Physik Department Matrix Product Formalism

Physik Department

Matrix Product Formalism

Diplomarbeitvon

Marıa Gracia Eckholt Perotti

Angefertigt an der

Technische Universitat Munchenund am

Max-Planck-Institut fur Quantenoptik

Garching, September 2005

First Adviser: Prof. Dr. Ignacio Cirac

Second Adviser: Dr. Juan Jose Garcıa–RipollDr. Michael Wolf

Each path is only one of a million paths. Therefore, you must always keep in mind that apath is only a path. If you feel that you must not follow it, you need not stay with it underany circumstances. Any path is only a path. There is no affront to yourself or others indropping it if that is what your heart tells you to do. But your decision to keep on the pathor to leave it must be free of fear and ambition. I warn you: look at every path closely anddeliberately. Try it as many times as you think necessary. Then ask yourself and yourselfalone one question. It is this:

Does this path have a heart?

All paths are the same. They lead nowhere. They are paths going through the brush orinto the brush or under the brush. Does this path have a heart is the only question. If itdoes, then the path is good. If it doesn’t, then it is of no use.

Carlos Castaneda

i

Contents

1 Introduction and Motivation 1

2 Entanglement 52.1 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Entanglement Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Entanglement Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Positive Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.2 Entanglement Witnesses . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Matrix Product States 153.1 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 MPS in the AKLT model . . . . . . . . . . . . . . . . . . . . . . . 163.1.2 MPS from slightly entangled states . . . . . . . . . . . . . . . . . . 183.1.3 MPS in DMRG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1.4 Simple examples of MPS . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Formal Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.1 Expectation values . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.2 Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.3 Transfer Matrix and Normal Form . . . . . . . . . . . . . . . . . . 23

3.3 Some physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1 Decay of correlation functions . . . . . . . . . . . . . . . . . . . . . 253.3.2 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 An application: Calculation of ground-states . . . . . . . . . . . . . . . . . 283.4.1 DMRG in brief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4.3 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4.4 Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4.5 Writing the code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4.6 Some results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Matrix Product Density Operators 394.1 MPDO from MPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Formal Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.1 Expectation values . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.2 Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

iii

iv

4.3 Entanglement in MPDO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Conclusions and Outlook 47

A Notation and review of selected topics in Linear Algebra 49A.1 Basic Mathematical Notation . . . . . . . . . . . . . . . . . . . . . . . . . 49A.2 Selected topics in Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . 49

B Normal Form for MPS from the Schmidt decomposition 51

C Normal Form implies normalized states for OBC 61

Chapter 1

Introduction and Motivation

Entanglement is one of the most striking features of quantum theory. After playing asignificant role in the foundations of quantum mechanics, it has been recently rediscov-ered as a new physical resource with potential commercial applications such as quantumcryptography, better frequency standards or quantum-enhanced positioning, and clocksynchronization. The ability to generate entangled states is one of the basic requirementsfor building quantum computers. Hence, efficient experimental methods for detection,verification, and estimation of quantum entanglement are of great practical importance.For this task we need a complete theoretical framework which provides us with the toolsfor characterizing entanglement.

It is well known that it is not difficult to say when a quantum pure state is entangled.On the other hand, it is usually very hard to identify whether a given mixed state isentangled. Indeed, it is more relevant for experimental applications to investigate thestructure of mixed state entanglement, because in real settings we often have either in-complete information about the system or interactions with the environment, known asdecoherence, both of which give rise to mixed states.

The ultimate aim of this work is the characterization and detection of entangle-ment in mixed states. We will focus on the matrix-product-density-operator (MPDO)–representation for mixed states and deal with the characterization of entanglement bymeans of the partial-transpose (PT): one of the standard criterion for detecting entangle-ment. The reason of using these specifications lies on the fact that this criterion can beimplemented efficiently to mixed-quantum-states written in the MPDO–representation,i.e. the PT can be easily handled with only a linear growth of computational effort onthe size of the system. The key to this lies in the fact that having ρ written as an MPDO,it is only determined by a set of matrices {M}. Indeed, there is not only one set but ainfinite number of them (Fig. 1.1) leading to the same state ρ.

In view of this, in this thesis we are interested in answering these two questions:

• Does there exist a standard set {M} that completely represents the classof sets that define ρ? In other words, we are looking for a set {M} in one-to-one

1

2 Chapter 1. Introduction and Motivation

Figure 1.1: All these sets define the same ρ.

correspondence with the state ρ. In order to do this, we will review the normal formin the matrix-product formalism for pure states and extend this to a normal formfor mixed states.

• Is it possible to study entanglement properties of ρ by means of studyingproperties of the set {M}? In particular, we would like to see if the positivityof a density operator ρ is a necessary and sufficient condition for the positivity ofthis set of standard matrices describing the state in the MPDO formalism. If thisis the case, it would be sufficient to check the positivity of these matrices after PTfor detecting entanglement; simpler than working directly with ρ.

The structure of this thesis is as follows:

In chapter 2 we will consider the PT as a mathematical tool for entanglement–detection, together with an overview of the theory of entanglement.

Although its roots date back from the end of the 80s and considerable advances havebeen achieved during the last couple of years, there exists no comprehensive publicationsummarizing the basic features of the matrix-product formalism. The following two chap-ters give a complete, self-contained description.

In chapter 3 we start introducing the matrix product states (MPS) from differentperspectives:

• The class of MPS first saw light as the ground state of the AKLT model, an exactlysolvable model in condensed matter physics. In this section we briefly explain themodel and how it leads to the matrix-product representation.

• MPS appear as a natural and efficient way to represent states of slightly entangledsystems.

• We also explain briefly the roll that MPS have played in the Density Matrix Renor-malization Group (DMRG) method.

The chapter follows with the explicit derivation of some formal aspects, namely the cal-culation of expectation values using transfer matrices and the normal form for MPS. For

3

completeness, the normal form is derived from a set of conditions imposed on a transfermatrix. Later, to give a flavor of what this states represent, some physical propertiesare derived. To close the chapter we apply MPS to the ground state calculation withinthe framework of the DMRG method. We summarize the most relevant features of thistechnique and give a detailed account of the algorithm using MPS. Finally, we give someresults for the specific problem of the Ising model.

In chapter 4 we follow more or less the same structure. Here matrix product densityoperators are introduced as an extension of matrix product states to the mixed–state sce-nario. We derive the calculation of expectation values using transfer matrices and workon the normal form for MPDO. To close the chapter, we study the implementation of PTon MPDO.

Finally in chapter 5 we briefly summarize the results and give an outlook on futurework.

There are also three appendices at the end of this work.

In appendix A we explain the basic mathematical notation that we use. We also in-clude some selected topics in linear algebra to make this work more self-contained, namelythe definition of rank, unitary matrices, a brief comment on the determination of eigen-vectors and eigenvalues and the singular value decomposition.

In appendix B we treat the derivation of the normal form for MPS from the Schmidtdecomposition used in chapter 3.

Finally, in appendix C, we show that the normal form for MPS, in the case of OBC,implies normalized states. This fact simplifies in a very convenient way the formulationof the algorithm for the ground-state calculation, described at the end of chapter 3.

4 Chapter 1. Introduction and Motivation

Chapter 2

Entanglement

If two systems interacted in the past it is, in general, not possible to assign a singlestate vector to either of the two subsystems [14]. This is also known as the principle ofnon-separability and expresses much of what entanglement is about.

First recognized by Einstein, Podolsky and Rosen [12] and Schrodinger [31], it is oneof the most astonishing features of the quantum formalism. The main problem in Entan-glement Theory is that we do not fully understand what entanglement is. More precisely,we only know is its mathematical definition and its manifestations [5, 7, 4].

Entanglement appears as the consequence of the combination of two of the quantumpostulates:

the state of a quantum

system is described by

a vector in a complex

Hilbert space

+

the Hilbert space of a

composite system is the

tensor product of the

two local spaces

=

∃ superposition of pure

states that cannot be

written as the tensor

product of pure states

in each local space

Antipodean to entangled states are the separable states, i.e., a state is entangled ifand only if it is not separable.

Whether a given state is entangled or just classically correlated is easy to determinefor pure states. However, for arbitrary mixed states it is a hard problem [16]. We will seethis later.

2.1 Separability

Deciding whether several systems are entangled or whether they are just classicallycorrelated is known as the separability problem. In this section we present the separabilitycondition for pure and mixed states, i.e., the definition of entangled states. We will bereferring to bipartite systems in a Hilbert space H = HA ⊗HB.

5

6 Chapter 2. Entanglement

Pure States A pure state |ψ〉 is entangled if and only if it is not separable, i.e., it cannotbe written as a product vector

|ψ〉 = |ψA〉 ⊗ |ψB〉.

In this case the criterion, for deciding if the state is entangled or not, is very simple.First we introduce an useful tool [24, 13].

Theorem 1 (Schmidt decomposition) Suppose |ψ〉 is a pure state of a com-posite system, AB. Then there are orthonormal states {|iA〉} for system A, andorthonormal states {|iB〉} of system B such that

|ψ〉 =∑

i

λi|iA〉|iB〉,

where λi are non-negative real numbers satisfying∑

i λ2i = 1 known as Schmidt

coefficients. If there is no degeneracy, this decomposition is unique up to arbitraryopposite phases in |iA〉 and |iB〉.

The Schmidt rank is defined as the number of non-vanishing Schmidt coefficients.

Then, the criterion for pure states is

|ψ〉 is pure ⇔ |ψ〉 has Schmidt rank one.

Mixed States A mixed state ρ is entangled if and only if it is not separable, i.e., itcannot be written as [39]

ρ =N∑

i=1

pi

[

|ψiA〉〈ψiA| ⊗ |ψiB〉〈ψiB|]

where N ∈ N+ is arbitrary; |ψiA〉 ∈ HA, |ψiB〉 ∈ HB are arbitrary but normalized

and pi ≥ 0 with∑N

i=1 pi = 1.

That is, a separable state can be prepared by two distant observers who receiveinstructions from a common classical source and prepare the different pure states|ψiA〉 and |ψiB〉 with probability pi (Fig. 2.1). So, entangled states are those thatcannot be created using local operations and classical communication.

The criteria for entanglement of mixed state are many and diverse. Here we startintroducing two of them [26, 20]. The symbol >i indicates the transposition ofsubsystem i, i.e., partial transposition of the entire system with respect to i (seesection 2.3)

Theorem 2 (Peres) If ρ is separable then

ρ>A ≥ 0 and ρ>B = (ρ>A)> ≥ 0.

Theorem 3 (Horodecki) A state ρ of a C2 ⊗ C

2 or C2 ⊗ C

3 system is separableif and only if its partial transposition is a positive operator.

2.2. Entanglement Measures 7

Figure 2.1: Separable-states factory. A classical source gives withprobability pi the output i, indicating far away partners which state to prepare.

2.2 Entanglement Measures

Quantifying quantum entanglement is one of the central topics in quantum informa-tion theory. How can entanglement be “measured” or quantified, how can entanglementbe classified, i.e., what physically different types of entanglement exist, and finally howdoes entanglement behave as a physical resource for quantum communication, quantumcomputation, etc.?

First of all, we need to know what an entanglement measure is [28]. We answer thisimportant question by stating the conditions that every measure of entanglement E hasto satisfy:

� Entanglement is non-negative. It is zero if and only if the stateis separable

E(ψ) ≥ 0 ∀ψ, E(ψ) = 0 ⇔ ψ is separable� Entanglement of independent systems is additive

E(ψ⊗n) = nE(ψ)� Entanglement is conserved under local unitary operations

ψ → Uψ, U = UA ⊗ UB : E(ψ) = E(Uψ)↪→ a local change of basis has no effect on E

� Its expectation value cannot be increased by local nonunitaryoperations

ψ −local nonunitary→ {pj, ψj} :∑

j

pjE(ψj) ≤ E(ψ)

↪→ monotonicity under local operationsand classical communication (LOCC)

For more on this see the pioneering paper on entanglement measures [8].

A pure state’s entanglement is measured by its entropy of entanglement E(ψ)

|ψ〉 =∑

i

pi|ψiA〉 ⊗ |ψiB〉 : E(ψ) = S(ρA) = S(ρB) (2.1)


i.e., the apparent entropy of any of the systems considered alone, where

S(ρ) = −Tr(ρ logρ) (2.2)

is the von Neumann entropy, ρA = TrB|ψ〉〈ψ| is the reduced density matrix of A,obtained after tracing over B’s degrees of freedom, and the logarithm is to base two(the information is stored in qubits). The entropy measures how much uncertainty thereis in the state of the physical system. For example, if ρA and ρB describe pure states(there is no uncertainty in the individual systems), then E(ψ) = 0 (there are no quantumcorrelations between them).

We define an ebit as the amount of entanglement in a maximally entangled state oftwo qubits, for which E = 1.

Another possibility is to use the rank of the Schmidt decomposition (SD) as ameasure. If A is a subset of n qubits and B the rest of them, the SD of |ψ〉 with respectto the partition A : B reads

|ψ〉 =

χA∑

α=1

λα|ψ[A]α〉 ⊗ |ψ[B]α〉

The rank χA of ρA (the reduced density matrix for block A) is a natural measure [37] ofthe entanglement between the qubits in A and those in B. Therefore, a good measure toquantify the entanglement of state |ψ〉 would be the maximal value of χA over all possiblebipartite splits A : B of the n qubits, namely

χ := maxA

χA

or the related entanglement measure Eχ

Eχ := log2(χ)

In the bipartite setting, Eχ upper bounds the more standard measure entropy of entan-glement.

For mixed states we have a whole zoo of measures, there is not a unique measure ofentanglement. The choice of one measure or another depends on what you need. We willsee some examples in what follows.

In principle, there are two approaches to quantify entanglement [8, 19]:

Abstract approach A state function can be used to quantify entanglement if it satisfiesthe natural properties stated before as definition of a measure.

1. Von Neumann entropy S: already introduced in (2.2).

2. Relative Entropy of Entanglement ER: it is based on the idea of distance; thecloser the state is to the set of separable states, the less entangled it is.

2.3. Entanglement Detection 9

3. Other measures: Squashed Entanglement Esq, Renyi Entropy Eα, Logarithmof the Negativity EN , Concurrence C, etc.

Operational approach The system is more entangled if it allows for better performanceof some task impossible without entanglement.

1. Entanglement of Formation EoF : having a large number n of Bell states, wewant to produce as many (high-fidelity) copies |ψ〉 using LOCC, getting finallym copies, therefore ψ’s E of formation is the limiting ratio n/m.

2. Distillable entanglement ED: performing the reverse process, it is the limitingratio m/n, when having a large number m of copies of |ψ〉 and we want todistill as many Bell states using LOCC, getting finally n EPR pairs.

3. Other measures: Entanglement Cost EC , Entanglement of Assistance EoA,etc.

All these measures are equivalent in certain limits, e.g. [17]. We have so many defin-itions not only due to the diverse interpretations, but because calculating some of themare of the Big Open Problems of QIT.

2.3 Entanglement Detection

Entangled states of many qubits are needed for quantum information tasks such asmeasurement based quantum computation [29], error correction [15] or quantum cryptog-raphy [10], to mention only few. Thus, it is important to study, both theoretically andexperimentally, multipartite entanglement and to provide efficient methods to verify if ina given experiment entanglement is really present.

Although, to detect entanglement is not an easy job. Here we introduce some ideas oftwo formalisms that deal with entanglement detection: positive maps and entanglementwitnesses.

2.3.1 Positive Maps

Any admissible physical transformation of a density matrix can be specified throughsome operators {Ki} such that,

ρ→ ρi =KiρK

†i

pi: pi = Tr(KiρK

†i ) (2.3)

{Ki} are known as Kraus operators [21]. These transformations define what is called acompletely positive map (CPM),

κ(ρ) =∑

i

KiρK†i

which fulfils the following properties:


1. Sends positive operators into positive operators,

∀ρ ≥ 0, κ(ρ) ≥ 0

i.e., a positive map.

2. It’s also positive for composite systems,

∀ρAB ≥ 0, (IA ⊗ κ)(ρAB) ≥ 0

because any physical transformation should still remain meaningful when it is justperformed on a subset of the parties.

Maps that are positive, but not completely positive, define unphysical operations. Thisproperty makes them useful for the detection of entanglement:

Any positive map acting on a product state gives a positive operator. Therefore,the same is valid for separable states. But if acting on some ρAB this mapproduces to a non-positive operator, then one can conclude that the state isentangled.

We can see this in more detail, consider we have a separable state

ρAB =N∑

i=1

pi(ρiA ⊗ ρiB

)

and we apply the positive, but non-completely positive, map κ on subsystem B

ρAB → ρ′AB =N∑

i=1

pi(ρiA ⊗ κ(ρiB)

)

given that κ is a positive map, κ(ρiB) will also be legitimate density matrices. So, itfollows that none of the eigenvalues of ρ′AB is negative. This is a necessary condition forρAB being separable.

For every entangled state there is a positive map detecting it. This is an straightfor-ward consequence of the following theorem [20].

Theorem 4 (Horodecki) A state ρ ∈ HA⊗HB is separable if and only if for all positivemaps

ε : HB → HC

we have(IA ⊗ ε)ρ ≥ 0

This translates the problem of detecting entanglement to the characterization of allthe positive maps.

We now move to the study of a concrete positive map.


Partial Transposition

The most known positive map which is not completely positive is the matrix transpo-

sition. The transposition θ is the map

θ(C) → C>

where C> is the matrix obtained by exchanging C’s rows and columns and it satisfies theidentity

(C>)−1 = (C−1)>

writing the matrix elements(c)>i,j = cj,i.

From this we define the map(θ ⊗ I) (C) → C>A

which is called partial transposition. If we work with the matrix elements we wouldwrite

(c)>A

iAiB ,jAjB= cjAiB ,iAjB .

As we saw before, right after the definition of a completely positive map, if we ap-ply (θ ⊗ I) (ρ) = ρ and ρ is a non-positive operator, then ρ is entangled. Our criterionreduces to transpose part of ρ and diagonalize the resultant matrix. This is an easilycomputable criterion for entanglement in mixed states [26], seen in theorem 2. A draw-back of PT is that it is not a sufficient condition; it has only been proved to be a sufficientcondition for pure states and for composite systems having dimensions 2×2 and 2×3 [20].

Regardless of the fact that PT is a non-physical operation (consequently, it cannotbe use to detect entanglement experimentally), it can be understood as antiunitary timeinversion operation in one subsystem; it means that e.g. Alice inverses time while Bobdoes not. We can understand more this and the effectiveness of PT in the following.

According to Wigner’s theorem [41], every symmetry transformation should always beimplemented by a unitary (U) or antiunitary (A) matrix. If we are working with a binarycomposite system, i.e., H = Ha ⊗Hb, the direct product of unitary matrices Ua ⊗ Ub (orantiunitary matrices Aa ⊗ Ab) is a unitary (or antiunitary) matrix in H. Nevertheless,the combination of a unitary and an antiunitary transformation Ua ⊗ Ab (or Aa ⊗ Ub)results in a transformation which is neither unitary nor antiunitary in H, whose action ona general ket of the composite system |ψ〉 ∈ H, furthermore, cannot be properly defined[30]. However, its action on a product state is, but for a phase ambiguity, well defined.As a separable state ρs ∈ H can always be rewritten as a statistical mixture of productstates,

ρs =∑

i

pi(|ai〉〈ai| ⊗ |bi〉〈bi|), 1 ≥ pi ≥ 0,∑

i

pi = 1

the action of such operations on ρs leads to a ρ′s

ρs → ρ′s =∑

i

pi(|a′i〉〈a′i| ⊗ |b′i〉〈b′i|)


where |a′i〉 := Ua|ai〉 ∈ Ha, |b′i〉 := Ab|bi〉 ∈ Hb, which is also physical (a positive definedhermitian matrix with normalized trace). Separable states are characterized by this:any local1 symmetry transformation, which obviously transforms local physical states intolocal physical states, also transforms the global physical state into another physical state.

There is only one independent antiunitary symmetry which physical meaning is time

reversal. Any other antiunitary transformation can be expressed as the product of a uni-tary matrix times time reversal. Therefore, quantum separability of composite systemsimplies the lack of correlation between the time arrows of their subsystems, as if separablesystems do not have memory of a unique time direction in the sense entangled states haveand they are thus compatible with a time evolution which factorizes into the product oftwo opposed time evolutions still leading to a physical state.

2.3.2 Entanglement Witnesses

Entanglement detection in an experiment is a hard problem, since reconstructing thewhole density matrix is usually not possible and the quantum state is only partially known.One can typically measure a few observables and still one would like to detect some of theentangled states. In this direction appears another approach for detecting entanglement,the so-called entanglement witnesses (EW).

An EW is a hermitian operator (an observable) W such that if Tr[Wρ] < 0, then ρ isentangled.

For every entangled state there is an EW W detecting it. This is a consequence of aspecial formulation of the Hahn-Banach theorem:

Theorem 5 (Hahn-Banach) Let S be a convex set in a finite dimensional Banachspace. Let ρ be a point in the space with ρ /∈ S. Then there exists a hyperplane2 thatseparates ρ from S.

Therefore, for every entangled ρ /∈ S3 there exists a hyperplane, described by a Her-mitian operator W 4, which separates ρ from S, such that Tr[ρW ] < 0, whereas ∀σ ∈S : Tr[σW ] ≥ 0. See Fig. (2.2)

As theorem 4 for positive maps, this theorem is quite powerful from a theoretical pointof view. However, we have the same handicap: it is not useful for constructing witnessesthat detect entanglement in a given state ρ.

An entanglement witness only gives one condition at detecting entanglement, while fora map has to be positively definite (i.e., there are many that have to be fulfilled). Thus

1Local means that it refers to the subsystem2A linear subspace with dimension one less than the dimension of the space itself.3S, set of separable ρ’s, is a convex and closed set.4In operator space, Hermitian operators define planes: {ρ : Tr[ρW ] = const.}.


Figure 2.2: Geometric Hahn-Banach theorem.W is a valid entanglement witness for all entangled ρ’s in the red-lined zone.

a map is much stronger. However, EW are able to provide a more detailed classificationof entangled states [32].


Chapter 3

Matrix Product States

A position-dependent unnormalized matrix product state for a one-dimensional systemof size N is defined as,

|ψmps〉 =d∑

s1,...sN=1

Tr (A[1]s1A[2]s2 ...A[N ]sN ) |s1, ...sN〉 (3.1)

A[i]si :

Matrix associated to site i

and its state si, whosedimension is bounded bysome fixed number Di×Di+1.They parametrize the state.

d :Dimension of the Hilbertspace corresponding tothe physical system.

They are a class of states that yields local descriptions of multipartite quantum states,giving a very good approximation [33] with only a polynomial number of parameters insome 1D problems. In the special case of open boundary conditions (OBC) we haveD1 = DN+1 = 1.

In this chapter we start introducing matrix product states from three different per-spectives. First, from their roots as the ground state of the AKLT model, where theywere originally introduced as Valence-Bond Solid (VBS) states. Second, in a more math-ematical scenario through the Schmidt decomposition. Third, we give a short descriptionof their roll in the Density Matrix Renormalization Group (DMRG). We end this firstintroductory section with some examples. Later, we treat formal aspects of the formalismconcerning calculations of expectation values and the definition of the matrices A for agiven state. Then, we study some physical properties of these states. To close this chap-ter, we deal with an application of all the machinery we have built: the calculation ofground states and correlation functions.

3.1 The Basics

We acquaint the reader three different pictures of matrix product states. We startwith a physical approach, follow later in a more mathematical direction and end with

15

16 Chapter 3. Matrix Product States

their appearance within a numerical method that during the last years has become verysuccessful simulating condensed-matter systems. The purpose of this section is merely togive an idea of where do the MPS come from.

3.1.1 MPS in the AKLT model

The AKLT [2, 3] is an exactly solvable model of an antiferromagnetic spin-1 chainexhibiting strong quantum fluctuations. Proposed by Affleck, Kennedy, Lieb and Tasaki,they were able to establish the existence of a Haldane gap (and related phenomena) rig-orously. This was the first rigorous demonstration that exotic behavior in integer spinchains predicted by Haldane is indeed possible. We will briefly summarize it here.

The goal was to create an instance for a Hamiltonian, for a one-dimensional isotropicspin chain, with a continuous symmetry, exponentially decaying correlation functions, agap and a unique (in the thermodynamic limit) ground state.

Figure 3.1: A Valence Bond.

The key to the model is the idea of a valence bond. Given two spin-1/2’s, a valencebond is formed by putting them in the singlet state ↑↓ − ↓↑, as represented in Fig. 3.1.

As depicted in Fig. 3.2, consider a spin-1 chain. Each spin can be regarded as thesymmetric part of the product of two virtual spin-1/2’s

1

2⊗ 1

2= 0 ⊕ 1

↪→Antisym. ↪→Symmetric

We construct a state with a valence bond between each pair of adjacent sites i and i+ 1by forming a singlet out of one of the spin-1/2’s at site i and one at site i + 1 (this alsopreserves the translational invariance in a simple way) [1]. After doing this we must sym-metrize the two spin-1/2’s at each site to restore a spin-1 at each site.

If we label the virtual spins at each site α and β, the state of the spin at a given siteis

3.1. The Basics 17

Figure 3.2: Valence Bond Solid state (VBS).

|sαβ〉 :=1√2(|α〉|β〉 + |β〉|α〉)

for α, β = ↑, ↓, while |sαβ〉 = |sβα〉 denote the same state. This gives us an orthogonalbasis in the symmetric part of the tensor product (LHS refers to the real space, RHSrefers to the virtual space):

|+1〉 =|s↑↑〉√

2|0〉 = |s↑↓〉 = |s↓↑〉

|−1〉 =|s↓↓〉√

2

or equivalently, we can apply the following projector over each site to map the state tothe real space

P = |+1〉〈s↑↑| + |0〉〈s↑↓| + |0〉〈s↓↑| + |−1〉〈s↓↓|Generalizing to different projectors P [i] for different sites i:

P [i] =∑

si,α,β

A[i]si

α,β|si〉〈αβ| (3.2)

where the elements of the matrix A denote the coefficients in the projector P .

For a system of two particles we can make the following construction


|AKLT [N=2]〉α1β2=∑

β1,α2

εβ1α2 |sα1β1〉|sα2β2

〉 (3.3)

where ε is the Levy-Civita antisymmetric tensor, it preserves the rotational invarianceand contracts two adjacent virtual spins from different sites to form a singlet. This kindof state has only spin 0 or 1.

We construct the Hamiltonian H as the sum of link Hamiltonians Hi for each site. Ifwe want to have (3.3) as ground state, we choose H as the sum of projection operatorsonto spin-2 for each neighboring pair

H =∑

i

Hi =∑

i

[1

2~Si · ~Si+1 +

1

6(~Si · ~Si+1)

2 +1

3

]

(3.4)

where ~Si are spin-1 operators. Obviously H ≥ 0, since Hi are projectors. Therefore, ifwe find an state such that Hi|s〉 = 0 for all i, it will be the ground state. Thus, (3.3) iseffectively the exact ground state for (3.4) with ground-state energy zero.

Considering a chain of arbitrary size N , this ground state can be written as the sum oftensor products of states |s1〉⊗· · ·⊗|sn〉 with the coefficients cs1...sN

expressed as productsof matrices A[1]s1 . . . A[N ]sN , i.e., a matrix-product state. This can be easily seen whenwe map explicitly (3.3) to the real space using (3.2)’s at each site for generic coefficients A.

3.1.2 MPS from slightly entangled states

Another approach to MPS is [37] by G. Vidal, developed independently to previousworks. There, a particular decomposition for the coefficients of an arbitrary state isintroduced |ψ〉 ∈ H⊗n

2 in the computational basis {|0〉, |1〉}

|ψ〉 =1∑

i1=1

· · ·1∑

in=0

ci1···in |i1〉 ⊗ · · · ⊗ |in〉

ci1i2···in =∑

α1··· ,αn−1

Γ[1]i1α1λ[1]α1

Γ[2]i2α1α2λ[2]α2

Γ[3]i3α2α3· · ·Γ[n]inαn−1

(3.5)

It is obvious that by contracting Λ ↔ λ ’s or λ↔ Λ ’s together we obtain (3.1).

The motivation of [37] is to show that any quantum computation with pure states canbe efficiently simulated with a classical computer provided the amount of entanglementinvolved is sufficiently restricted. We will only describe the state decomposition and finishwith a brief justification of the work.

Decomposition (3.5) employs n tensors {Γ[1], . . . ,Γ[n]} and n−1 vectors {λ[1], . . . , λ[n−1]}, whose indices il and αl take values in {0, 1} and {1, . . . , χ}, respectively.

3.1. The Basics 19

This decomposition can be found for all possible states. It consists of a concatenationof n − 1 Schmidt decompositions and depends on the particular way qubits have beenordered from 1 to n. We start with the first SD of |ψ〉 at the partition 1 : 2 . . . n

|ψ〉 =∑

α1

λ[1]α1|φ[1]α1

〉|φ[2 . . . n]α1〉

=∑

i1,α1

Γ[1]i1α1λ[1]α1

|i1〉|φ[2 . . . n]α1〉

and we have expanded each Schmidt vector for qubit 1 in the terms of the computationalbasis. Then we proceed as follows:

1. expand each Schmidt vector for qubit 2, i.e., |φ[2 . . . n]α1〉, in a local basis

|φ[2 . . . n]α1〉 =

∑

i2

|i2〉|τ [3 . . . n]α1i2〉

2. write |τ [3 . . . n]α1i2〉 in terms of the at most χ Schmidt vectors corresponding to thepartition 1 2 : 3 . . . n for the second half1

|τ [3 . . . n]α1i2〉 =

χ∑

α2=1

Γ[2]i2α1α2λ[2]α2

|φ[3 . . . n]α2〉

3. substitute this in the previous decomposition of |ψ〉 to obtain

|ψ〉 =∑

i1,i2α1,α2

Γ[1]i1α1λ[1]α1

Γ[2]i2α1α2λ[2]α2

|i1i2〉|φ[3 . . . n]α2〉

Iterating these steps for the Schmidt vectors |φ[3 . . . n]α2〉, |φ[4 . . . n]α3

〉, . . . , |φ[n]αn−1〉

one can finally express |ψ〉 as

|ψ〉 =∑

i1,···inα1,··· ,αn−1

Γ[1]i1α1λ[1]α1

Γ[2]i2α1α2λ[2]α2

Γ[3]i3α2α3· · ·Γ[n]inαn−1

|i1 · · · in〉

Contracting Λ ↔ λ ’s or λ ↔ Λ ’s together it can be readily identified with an matrix-product state.

Hence, (3.5) reexpresses the 2n coefficients ci1···in of |ψ〉 in terms of about (2χ2 + χ)nparameters, i.e.,

n qubit state ↔ n exp(Eχ) parameters

where Eχ is an entanglement measure introduced in section 2.2. Therefore, this leadsto an efficient description of |ψ〉 if Eχ scales as O(log(n)), because in that case onlypoly(n) parameters are required.

1We can do this because {|τ [3 . . . n]α1i2〉} ∈ span{|φ[3 . . . n]α2

〉}


3.1.3 MPS in DMRG

As we will see in the last section of this chapter, matrix-product states also appearin a very well known method that enables us to treat quantum many-body systems, to an-alyze and understand the physical properties of certain of these condensed matter systemswith unprecedented precision [11]. This method is the Density Matrix RenormalizationGroup [40] (DMRG), originally envisioned for 1D systems with short-range interactionsat zero temperatures, during the last years it has been successfully extended to other situ-ations [11]. This fact was already noticed by Ostlund et al. [25] who established DMRG’smathematical foundations in terms of the MPS for White’s infinite algorithm [40]. Thestandard DMRG method was originally introduced in a partially ad hoc manner withoutfully understanding the reasons of its success, but with MPS there exists now a coherenttheoretical picture of it which underlies Quantum Information Theory concepts [36].

3.1.4 Simple examples of MPS

The basic idea of MPS is to associate a set of d matrices to each site of the system(Fig. 3.3). The bounds between adjacent sites are represented by the product of thecorresponding matrices. Through all this work, the physical picture we will have on mindis an array of spins either in a chain for OBC or in a ring for PBC.

Figure 3.3: MPS are based on a local description.

For a clearer understanding of MPS and their notation, we will give three simple ex-amples of these states showing their corresponding matrices. We particularly consideringPBC and equal dimension D for all the matrices. Dealing with qubits, we have that thephysical dimension of our system is d = 2, using the computational notation for the statess = 0, 1.

State |000 . . . 0〉 Using dimension D = 1, we have the one-by-one matrices:

A[i]0 = 1 A[i]1 = 0

GHZ state |000 . . . 0〉 + |111 . . . 1〉 Using dimension D = 2 we have:

A[i]0 =

(1 00 0

)

A[i]1 =

(0 00 1

)

3.2. Formal Aspects 21

W state |001〉 + |010〉 + |100〉 With D = 3 the matrices are:

A[1]0 =

1 0 00 1 00 0 0

A[2]0 =

1 0 00 0 00 0 1

A[3]0 =

0 0 00 1 00 0 1

A[1]1 =

0 0 00 0 00 0 1

A[2]1 =

0 0 00 1 00 0 0

A[3]1 =

1 0 00 0 00 0 0

3.2 Formal Aspects

In this section we study two powerful instruments for working with MPS: the cal-culation of expectation values and the definition of the matrices A for a given state bymeans of a normal form. We follow with a different derivation of the normal form thatlinks several ideas and it is included for completeness of the picture. We start with thecalculation of correlation functions to launch an important tool: the transfer matrix.

3.2.1 Expectation values

Given the state (3.1), we would like to calculate the expectation value of some operatorO which is the tensor product of local operators Oi for each site i

O = O1 ⊗ · · · ⊗ON

Gathering what we have and using (A+B) = A+ B

〈ψmps|O|ψmps〉 =d∑

s1...,sN

s′1...,s′

N=1

Tr

(N∏

i=1

A[i]si

)

Tr

(N∏

i=1

A[i]s′

i

)N∏

i=1

〈s′i|Oi|si〉

we can arrange this using Tr(A⊗B) = (TrA)(TrB)

=d∑

s1...,sN

s′1...,s′

N=1

Tr

[N∏

i=1

A[i]si ⊗N∏

i=1

A[i]s′

i

]N∏

i=1

〈s′i|Oi|si〉

and also considering that (A1 ⊗B1)(A2 ⊗B2) = A1A2 ⊗B1B2

=d∑

s1...,sN

s′1...,s′

N=1

Tr

[N∏

i=1

(

A[i]si ⊗ A[i]s′

i

)]

N∏

i=1

〈s′i|Oi|si〉

the trace commutes with the sum symbol since both are sums and there is no harm inputting together both product symbols

= Tr

d∑

s1...,sN

s′1...,s′

N=1

N∏

i=1

〈s′i|Oi|si〉(

A[i]si ⊗ A[i]s′

i

)


finally the sum over states s commutes with the product over sites i

= Tr

N∏

i=1

d∑

si,s′i=1

〈s′i|Oi|si〉(

A[i]si ⊗ A[i]s′

i

)

and we make the following definition:

EOi:=

d∑

si,s′i=1

〈s′i|Oi|si〉(

A[i]si ⊗ A[i]s′

i

)

(3.6)

where we baptize EOiunder the name of transfer matrix, we end up having,

〈ψmps|O|ψmps〉 = Tr [EO1. . . EON

] (3.7)

Therefore, the calculation of expectation values of products of local observables reducesto the task of multiplying a set of transfer matrices and tracing the result.

3.2.2 Normal Form

To simplify several algorithms it is useful to choose a particular set of conditions onthe MPS. Given a state |ψ〉, the choice of the matrices A[i]si is not unique. For example,the change over all the A’s

A[i]si −→ X[i]A[i]siX[i+ 1]−1

does not alter the state they describe, for any set of non-singular X’s. Therefore, we canchoose a gauge condition(s) at each site to fix any mathematical freedom(s) we have. Thisconditions constitute the normal form.

Starting at the k-th site, the conditions are slightly different if we move to the rightor to the left of it. The conditions to the left of the k-th site in our 1D system are

∑

si

(A[i]si)†A[i]si = I (3.8)

∑

si

A[i]siΛ[i] (A[i]si)† = Λ[i− 1] (3.9)

where Λ[i] is a diagonal matrix with the corresponding eigenvalues sorted in decreasingorder. To the right the requirements are

∑

si

A[i]si (A[i]si)† = I (3.10)

∑

si

(A[i]si)† Λ[i− 1]A[i]si = Λ[i] (3.11)

Introducing MPS as it was done by Vidal in [37], these conditions appear naturally.For more on this see appendix B.


3.2.3 Transfer Matrix and Normal Form

This part is included for completeness of the derivation of the normal form. The idea isto obtain the same conditions than before imposing some constraints on EI: an eigenvalueequation for the right eigenvector and one for the left eigenvector, both corresponding tothe highest eigenvalue of this transfer matrix. This will shed some light on how to obtainthe matrices A that obey the normal form.

For each site i we defined the transfer matrix EOiin (3.6). When the local operator

Oi is the identity I we will use the following notation

EI =d∑

si=1

A[i]si ⊗ A[i]si =: E[i] (3.12)

We will consider the case of a translational invariant system, i.e., E[i] = E ∀ i; it is thesimplest but results can be easily generalized. We observe that E is symmetric under theexchange of tensor factors together with complex conjugation. It can be shown that atleast its eigenvector with the highest eigenvalue ξmax has the same symmetry:

• we set ξmax = 1 renormalizing the matrix E and we assume that it is non-degenerate

E|χ〉 = ξmax|χ〉

• complex conjugating and remembering that ξmax ∈ R

E ¯|χ〉 = ξmax ¯|χ〉

• applying the flip operator F, which exchanges tensor factors

FE ¯|χ〉 = ξmaxF ¯|χ〉

• considering the symmetry E has, E = FE

E F ¯|χ〉︸︷︷︸

|χ′〉

= ξmax F ¯|χ〉︸︷︷︸

|χ′〉

• we have assumed that ξmax is non-degenerate, therefore

|χ′〉 = |χ〉

and it has the same symmetry as E.

The right eigenvector corresponding to the highest eigenvalue (set to one) is |φ〉 ,

E|φ〉 = 1|φ〉 (3.13)


According to the Schmidt decomposition, for any state |φ〉 there are orthonormal states|uk〉 and |vk〉 such that:

|φ〉 =∑

m,l

φml|ml〉 =∑

k

λk|ukvk〉, λk ≥ 0 ∀ k

We assume the decomposition has full Schmidt rank, i.e., its rank coincides with thedimensionality of the space the vector belongs to. There are unitaries U and V such that

|φ〉 =∑

k

λk(U ⊗ V )|kk〉

where the {|k〉} are still orthonormal. Due to the symmetry of the problem mentionedbefore, we find that U = V (if the decomposition does not have full rank we have somefreedom in this matrices and can choose them like that). Consequently,

|φ〉 = (U ⊗ U)(√

Λ ⊗√

Λ)∑

k

|kk〉

where Λ is diagonal and positive and its eigenvalues are the Schmidt coefficients of thedecomposition. We redefine E as folows

(1√ΛU † ⊗ 1√

ΛU †

)

E(

U√

Λ ⊗ U√

Λ)

︸︷︷︸

↪→E

∑

k

|kk〉 = 1∑

k

|kk〉

therefore

As →(

1√ΛU †

)

As(

U√

Λ)

=: XAsX−1

which implies a change in the definition of the A’s. We observe that this modificationdoes not vary the state |ψ〉.

Having X = DU , where D is a diagonal matrix and U a unitary, gives us a hint forfinding the new A’s: we should do a singular value decomposition (SVD) of the old ones.We keep one of the unitaries as the new A. Then, we need a second condition to uniquelydetermine these matrices. This is because in the SVD of A (A = V DU †) D is uniquebut U and V are not. This procedure of the SVD will be explained in more detail insubsection 3.4.3.

Now consider the corresponding left eigenvector |ϕ〉,

〈ϕ|E = 1〈ϕ| (3.14)

we can write it, analogously to previous steps, like

|ϕ〉 = (U ⊗ U)∑

k

λk|kk〉

3.3. Some physical properties 25

and doing the following redefinition of E

∑

k

λk〈kk| (U † ⊗ U †)E(U ⊗ U)︸︷︷︸

↪→E

=∑

k

λk〈kk|

therefore

As → U †AsU

and this reassignment does not change the state |ψ〉 either.

We summarize the general case (proceeding from right to left) in the following box, thefreedoms we have to choose the matrices A[i]si giving no observable effect on the statethey describe:

Freedom Gauge Condition

A[i]si →∑

siA[i]si (A[i]si)† = I,

X[i]A[i]siX[i+ 1]−1 E[i]|φ〉 = |φ〉 : |φ〉 =∑

k |kk〉X[i] → U [i]X[i]

∑

si(A[i]si)† Λ[i− 1]A[i]si = Λ[i],

: U [i] unitary 〈ϕi−1|E[i] = 〈ϕi| : |ϕi〉 =∑

k λ[i]k|kk〉

3.3 Some physical properties

3.3.1 Decay of correlation functions

We want to calculate the correlation function between sites i and i + ∆ of a generalMPS of a system with N sites |ψN〉: 〈ψN |σixσi+∆

x |ψN〉 =: 〈σixσi+∆x 〉ψN

. We are interestedin its behavior in the limit of a long chain as a function of the distance ∆ between the twosites. We will use what we have already learned for the calculation of expectation values.

We will observe the following:

In an infinite and translational invariant chain, the correlation between twosites decays exponentially with the distance ∆ between them

|ψN〉 =d∑

i1...,iN=1

Tr(Ai1Ai2 . . . AiN )|i1i2 . . . iN〉

limN→∞

〈σixσi+∆x 〉ψN

∆ � 1−−−→ e−c∆, c = const.

Proof:


Using our machinery for the calculation of expectation values and considering that, ingeneral, our state is not normalized, the correlation function that we want to calculate is


=Tr[EI . . . EIEσi

xEI . . . EIEσi+∆

xEI . . . EI

]

Tr[EI . . . EI

]

grouping equal factors we have a simplified version of this expression

=Tr[Ei−1

IEσi

xE∆−1

IEσi+∆

xEN−i−∆

I

]

Tr[EN

I

]

using the cyclicly property of the trace operation Tr(ABC) = Tr(BCA) = Tr(CAB) itreduces to

=Tr[Eσi

xE∆−1

IEσi+∆

xEN−∆+1

I

]

Tr[EN

I

]

Choosing the basis where EI is diagonal we have

EI =m∑

i=0

λi|λRi 〉〈λLi | : λ0 ≥ λ1 ≥ λ2 ≥ · · ·λm

in the limit of an infinite chain ENI

is dominated by the term of the highest eigenvalue

limN→∞

ENI

−→ λN0 |λR0 〉〈λL0 |

Therefore, we obtain the expression

limN→∞


=λN−∆+1

0 Tr〈λL0 |EσixE∆−1

IEσi+∆

x|λR0 〉

λN0 〈λL0 |λR0 〉that simplifying it

= λ−(∆−1)0

〈λL0 |EσixE∆−1

IEσi+∆

x|λR0 〉

〈λL0 |λR0 〉and expressing E∆−1

Ialso in the diagonal basis

=m∑

i=0

(λiλ0

)∆−1 〈λL0 |Eσix|λRi 〉〈λLi |Eσi+∆

x|λR0 〉

〈λL0 |λR0 〉We can finally make the following approximation for ∆ � 1 and big enough:

limN→∞∆�1


'〈λL0 |Eσi

x|λR0 〉〈λL0 |Eσi+∆

x|λR0 〉

〈λL0 |λR0 〉

+

(λ1

λ0

)∆ 〈λL0 |Eσix|λR1 〉〈λL1 |Eσi+∆

x|λR0 〉

〈λL0 |λR0 〉We are now able to recognize two special cases:

• If 〈λL0 |Eσx|λR0 〉 6= 0 we only keep the first term and the correlations have an infinite

range.

• If 〈λL0 |Eσx|λR0 〉 = 0 we only keep the second term and, as λ1/λ0 ≤ 1, the correlations

decay exponentially with ∆.

which gives us the behavior that we wanted to show.

3.3. Some physical properties 27

3.3.2 Entanglement

We want to measure the entanglement of a piece of chain with the rest of it. We aregoing to use as entanglement measure the von Neumann entropy (2.2).

We enunciate the following result:

If we have a 1D system of size N described by an MPS, such that A[k]i ∈MD×D, and take a piece with longitude L < N , then the entropy of this smallerblock has 2 logdD as an upper-bound independently of the size L.

∀ |ψMPS〉 : A[k]i ∈MD×D ∀k, if ρL = Trs/∈L|ψMPS〉〈ψMPS|⇒ S(ρL) ≤ 2 logdD ∀L

Proof:

In general, our state is given by

|ψ〉 =

d,D∑

s1...,sN=1α1...,αN−1

A[1]s1α1A[2]s2α1,α2

. . . A[N ]sN−1

αN−1|s1〉|s2〉 . . . |sN〉

Using the following notation for the different blocks in which for convenience we havedivided our system

|αm〉 :=

d,D∑

s1...,sm−1

α1...,αm−1=1

A[1]s1α1. . . A[m− 1]sm−1

αm−1,αm|s1〉 . . . |sm−1〉

|αm, αm+L〉 :=

d,D∑

sm...,sm+L−1

αm+1...,αm+L−1=1

A[m]sm

αm,αm+1. . . A[m+ L− 1]sN−1

αm+L−1,αm+L|sm〉 . . . |sm+L−1〉

|αm+L〉 :=

d,D∑

sm+L...,sN

αm+L+1...,αN−1=1

A[m+ L]sm+Lαm+L,αm+L+1

. . . A[N ]sN−1

αN−1|sm+L〉 . . . |sN〉

we can easily rewrite our state |ψ〉 like

|ψ〉 =D∑

αm,αm+L=1

|αm〉|αm, αm+L〉|αm+L〉

or its density matrix ρ = |ψ〉〈ψ| as

ρ =D∑

αm,αm+L

α′

m,α′

m+L=1

|αm〉|αm, αm+L〉|αm+L〉〈α′m|〈α′

m, α′m+L|〈α′

m+L|


Tracing out all sites not in our subsystem of length L, the matrix we obtain is

ρL =D∑

αm,αm+L=1

|αm, αm+L〉〈αm, αm+L|

which has as much rank D2, i.e., the largest number of columns of ρL that constitutes alinearly independent set is as much of this size.

For any state σ, we saw that the von Neumann entropy, measured in q-dits, is definedas

S(σ) = −Tr(σ logdσ)

if we know σ has eigenvalues {λ1, λ2, . . . , λR} we have a simplified expression

S(σ) = −R∑

i=1

λi logdλi

An upper-bound for the entropy is the case of a maximally mixed state, i.e., λ1 = λ2 =. . . = 1/R, where R is the rank of the matrix σ

S(σ) ≤ −R∑

i=1

1

Rlogd

1

R= logdR

Therefore, for our ρL we have

S(ρL) ≤ logdRL ≤ logdD2 = 2 logdD

independently of the size L, that is what we wanted to show.

In practice S(ρL) will grow with L and saturate at a value S(ρ∞) ≤ 2logdD.

3.4 An application: Calculation of ground-states

For concreteness, we will consider a one-dimensional array of N spins, with openboundary conditions (OBC), that interact via an Ising-type next-neighbour interactionplus an external magnetic field.

Figure 3.4: 1D spin chain

3.4. An application: Calculation of ground-states 29

Thus, the interaction is described by a Hamiltonian H with the following structure:

H =N−1∑

i=1

σijσi+1j + λ

N∑

i=1

σik (3.15)

where λ gives the ratio of the strengths of the magnetic field and the interaction betweensites.

We would like to find the ground state of this system. But, if we want to describethis quantum many-body system, we know that the number of parameters describing itsphysical state grows exponentially with the number of particles, N . However, during thelast decade several numerical methods to describe certain many-body systems have beenput forward. One of such methods is the so-called density matrix renormalization group(DMRG), which will be introduced in the next subsection 3.4.1. There is a deep connec-tion between DMRG and MPS.

3.4.1 DMRG in brief

Density Matrix Renormalization Group [40] (DMRG) is a numerical technique forfinding accurate approximations of the ground state and the low-lying excited states ofstrongly interacting quantum lattice systems. It traces its roots to Wilson’s numericalrenormalization group (RG) treatment of the impurity problem [42] and it is weakly re-lated to real space renormalization groups [34]. The accuracy of this method, with amodest amount of computational effort, is remarkable for 1D systems and it is limited bythe dimensionality or range of the interaction.

To give a flavor, here are few ideas on some simpler methods to which DMRG is closelyrelated and the ideas that leaded to the following step:

Exact Diagonalization Gives an exact solution, but the maximum system size thatcan be treated is severely limited by the exponential growth of the Hilbert space.However, diagonalization algorithms work remarkably well for finding the lowesteigenvalues of the large, sparse matrices found in quantum lattice problems.

⇒ to formulate a variational diagonalization scheme that also truncates

the Hilbert space used to represent the Hamiltonian in a controlled way.

Wilson’s Numerical Renormalization Group Integrates out unimportant degrees offreedom progressively using a succession of renormalization group transformations(only the low-energy eigenstates obtained for a system of size L will be importantin making up the low-energy states of a system of size L + 1). The general stepsare:

a) We start with a block of length L and its m (some prefixed number) lowestenergy eigenstates from previous steps as an approximate basis


{|ψj〉}j=1...,m

In the initial step, L is small enough to exactly diagonalize the Hamiltonian.

b) We add a new site to the block

Our basis is {|ψj〉|i〉}.c) We project our enlarged basis onto a subspace of dimension m, keeping the

low-energy states of the system and recovering a smaller basis

{|ψj〉}j=1...,m

The truncation scheme is iterated.

How to treat the boundaries of the isolated block after its enlargement is crucial informulating an accurate RG procedure, it has been the main problem for Wilson’smethod. Some ideas to do this are:

• Combination of boundary conditions method: the new basis is formed from apply-ing several different boundary conditions to the edge of the block and keepingthe low-lying eigenstates.

• Superblock method: the general behavior at the boundaries is provided au-tomatically by embedding the block of interest in a larger superblock. Thismethod is more promising for applications to interacting systems, but onestate of the superblock can, in general, project onto many states of the systemblock.

⇒ DMRG chooses an optimal way to do this projection.

DMRG scheme It proceeds after the enlargement of the block (step b in Wilson’smethod) introducing an environment block, which is the enlarged block’s mirrorimage as it is assumed to have reflexive symmetry

The total system is what is called the superblock. The reduced density matrix of theenlarged system (not the whole superblock) is built and its most probable eigenstatesare kept as the new basis. This last step is the so-called density matrix projection.The whole procedure is iterated until the quantities of interest (computed at everystep) have converged to the desired accuracy.


3.4.2 The Algorithm

DMRG can be viewed as an iterative method that for a fixed D determines the matri-ces whose state |ψmps〉 minimizes the energy in a variational sense. We explain in detailhow to do this.

We start setting all A[i] of the initial state with random entries. In general, theseA[i]’s are tensors with three indices, i.e., A[i] → A[i]α,si,β; where α and β run from 1 toD and si from 1 to d. Remember that making the following change in the A[i]’s does notchange the state |ψmps〉:

A[i] → A[i] = XiA[i]X−1i+1 : Xi ∈MD×D

|ψmps〉 → |ψmps〉

Beginning at location i, if we are running the algorithm to the right we are lookingfor a matrix A[i] such that the following condition is fulfilled:

∑

α,si

A[i](α,si),β¯A[i](α,si),β′ = δβ,β′ (3.16)

which is (3.8) written component-wise. If we are going to the left (3.9) reads

∑

si,β

A[i]α,(si,β)¯A[i]α′,(si,β) = δα,α′ (3.17)

Conditions (3.16) and (3.17) can be thought as A[i] being a unitary matrix (they arerectangular matrices so they are not proper unitaries) or that it is normalized (they willlead to a normalized state). In general, A[i] will not be normalized. Therefore, we applyto A[i] the normalization routine explained in subsection 3.4.3, which allows us to finda unitary A[i]. We repeat this sequentially to each matrix until we reach the end of thechain and then back until we have normalized all the matrices and being back to theinitial one.

We construct the effective Hamiltonian at this site, which is the Hamiltonian as afunction of the matrix A[i] of the site we are at, and proceed to the minimization of theenergy in terms of this A[i]. How to do this is elaborated in detail in subsection 3.4.4.From this procedure we obtain an optimal A[i] that minimizes the energy of the state atthis step. We normalize the state again and move to the next site to proceed in the sameway, until the energy E converges. That is, the procedure is continued until a fixed pointis reached, something which always occurs since the energy is a monotonically decreas-ing function of the step number [36]. This is a variational method which always converges.

At the end we have all the A[i]’s that describe the ground state of our Hamiltonianand we can evaluate with them all expectation values as explained in 3.2.1.

There is a generalization to periodic boundary conditions (PBC). The main idea isto assume that the spins are in a ring configuration, so all of then are treated on the


same footing. The matrices A are determined in clockwise order, then improve followinga counterclockwise ordering, then clockwise again, until a fixed point is reached.

For the sake of readability we restrict our description to OBC. As a reminder, in theMPS (3.1) the A[i]’s are matrices whose dimension is bounded by some fixed number D(the number of states kept by the DMRG method) and d is the dimension of the Hilbertspace corresponding to the physical systems. Also, For OBC we have that DN = D1 = 1,so A[1]s1 = ( ~A[1]s1)T and A[N ]sN = ~A[N ]sN .

3.4.3 Normalization

In general, given that A[i] is the “only one” not normalized, using (3.16) and C.22 wehave that:

〈ψmps|ψmps〉 =d∑

si=1

D∑

κ,η=1

Tr(A[i]κ,si,ηA[i]κ,si,η

)

=d∑

si=1

Tr(A[i]siA[i]†si

)(3.18)

If a given A[k] is not unitary, then we proceed to its singular value decomposition(SVD) (usually the normalization is run from k = 1 to k = N , left to right, or fromk = N to k = 1, right to left):

A[k](α,sk),β = U [k](α,sk),(α,sk)′D[k](α,sk)′,β′V [k]β′,β (3.19)

A[k]α,(sk,β) = U [k]α,α′D[k]α′,(sk,β)′V [k](sk,β)′,(sk,β) (3.20)

Depending on which condition we ask to the A[k]’s, (3.16) or (3.17), we choose a de-terminate partition for the indices in the SVD, (3.19) or (3.20) respectively (the firstcorresponds to left to right and the second to right to left normalization). As they aresimilar, we will just work with one of them, lets say (3.20). For (3.19) everything can bedone in a similar way.

In (3.20), U [k]α,α′ and V [k](sk,β)′,(sk,β) are unitary matrices; D[k]α′,(sk,β)′ is diagonal,but rectangular:

(Diag 0

0 Om×n

)

α′,(sk,β)′

As α′ < (sk, β)′, we can take away some zero columns so that we have a square matrixD[k]α′,α′′ . Now, V [k](sk,β)′,(sk,β) is in fact of the same dimensions as A[k]α,(sk,β)

V [k](sk,β)′,(sk,β) → V [k]α′′,(sk,β)

2In Appendix C: Normal Form implies normalized states for OBC


and we can make the following identification

A[k]α,(sk,β) = U [k]α,α′D[k]α′,α′′

︸︷︷︸V [k]α′′,(sk,β)︸︷︷︸

X[k]−1 A[k]α,(sk,β)

ThereforeA[k] := X[k]A[k]

A[k − 1] := A[k − 1]X[k]−1

and the state hasn’t changed, as stated before.

3.4.4 Effective Hamiltonian

Fixing our attention on site i, in general, the Hamiltonian H of the system can bedecomposed in the following way, according to the partition depicted in Fig. 3.5:

H = HL +HLi +Hi +HiR +HR (3.21)

Figure 3.5: Partition of the chain for site i

where:

• HL refers to all the interactions within block A;

• HLi refers to all the interactions between block A and site i;

• Hi refers to all the interactions of site i alone;

• HiR refers to all the interactions between site i and block B;

• HR refers to all the interactions within block B.

In our case H has the form (3.15). Therefore:

〈ψmps|H|ψmps〉 = (Eσj1Eσj

2EI · · · ) + (EIEσj

2Eσj

3· · · ) + · · · (3.22)

In reference to the site i we are interested on, we can decompose it in the following way:

〈ψmps|H|ψmps〉 = AiEI[i]Bi + FiEσj [i]Bi + CiEI[i]Di + CiEσj [i]Gi

+λ[

A′iEI[i]Bi + CiEσk [i]Bi + CiEI[i]D

′i

]

(3.23)


where we have defined the following matrices which refer only to the rest of the sites:

Ai := Eσj [1]Eσj [2]EI[3] · · ·EI[i− 1] +

+EI[1]Eσj [2]Eσj [3] · · ·EI[i− 1] + · · ·· · · + EI[1] · · ·EI[i− 3]Eσj [i − 2]Eσj [i − 1]

Bi := EI[i+ 1] · · ·EI[N ]

Ci := EI[1] · · ·EI[i− 1]

Di := Eσj [i + 1]Eσj [i + 2]EI[i+ 3] · · ·EI[N ] +

+EI[i+ 1]Eσj [i + 2]Eσj [i + 3] · · ·EI[N ] + · · ·· · · + EI[i+ 1] · · ·EI[N − 2]Eσj [N − 1]Eσj [N]

Fi := EI[1] · · ·EI[i− 2]Eσj [i − 1]

Gi := Eσj [i + 1]EI[i+ 2] · · ·EI[N ]

A′i := Eσk [1]EI[2] · · ·EI[i− 1] +

+EI[1]Eσk [2] · · ·EI[i− 1] + · · ·· · · + EI[1] · · ·EI[i− 2]Eσk [i − 1]

D′i := Eσk [i + 1]EI[i+ 2] · · ·EI[N ] +

+EI[i+ 1]Eσk [i + 2] · · ·EI[N ] + · · ·· · · + EI[i+ 1] · · ·EI[N − 1]Eσk [N]

It is very important to realise that:

• The energy is a quadratic form of our unknown variable at site i:

〈ψmps|H|ψmps〉 = ~A[i]†H ′ ~A[i]

where ~A[i] = A[i](α,si,β).

• Recursive forms can be found for the matrices Ai, Bi, Ci . . . between the differentsites i.

If all the matrices A[j] are given, except for the one j = i, then:

E =〈ψmps|H|ψmps〉〈ψmps|ψmps〉

=Tr [AiEI[i]Bi]

Tr [A[i]A[i]†]+Tr [FiEσj [i]Bi]

Tr [A[i]A[i]†]

+Tr [CiEI[i]Di]

Tr [A[i]A[i]†]+Tr [CiEσj [i]Gi]

Tr [A[i]A[i]†]

+ λTr [A′

iEI[i]Bi + CiEσk [i]Bi + CiEI[i]D′i]

Tr [A[i]A[i]†]

The dependence of E on A[i] is contained in the bold letters. Playing a little bit around

with the indices it is possible to put all the numerators in the form ~A[i]†M [i] ~A[i]. Adding


up all these contributions (M [i], . . .), we get an effective Hamiltonian for site i which wecall H[i].

For the minimization of E in function of A[i] we have the following expression:

E =~A[i]†H[i] ~A[i]

~A[i]† ~A[i](3.24)

Minimizing E with respect to A[i] is equivalent to solving the eigenvalues equation:

H[i] ~A[i] = E ~A[i] (3.25)

and we keep the ~A[i] corresponding to the minimal E of the spectrum.

3.4.5 Writing the code

The program has been written in C++ using MPSC++.

Why C++ and not some other programming language? The most remarkable reasonsfor us to choose this programming language are:

• It is object-oriented : it allows to design applications from a point of view more likea communication between objects that in a structured sequence of code and it alsoallows the reusability of code in a more logical and productive way

• Its brevity

• Allows modular structure: it can be made up of several source code files that arecompiled separately and then linked together

• Its speed : the resulting code from a C++ compilation is very efficient

MPSC++ is a C++ library created by J. J. Garcıa–Ripoll for working with MatrixProduct States. These objects can be used to accurately simulate large 1D quantum me-chanical systems, such as boson, fermions or spin chains. In general, they can be used tocompute ground states [36], thermal states [35], or do time evolution [38, 35].

The program we have designed is capable of finding ground states of Hamiltonianswith nearest-neighbor interactions, that is, Hamiltonians of the form

H =N−1∑

i=1

Q1[i] ⊗Q1[i+ 1] +N∑

i=1

Q2[i]

with OBC, where Q1[i] and Q2[i] are some local operators acting on site i. The programimplements the algorithm described in section 3.4.2, iterating over the lattice a fixed


number of sweeps. A more elaborate criterion for the number of steps could be also im-plemented, by computing the change of energy between different sweeps and conditioningthe quitting of the program to some preassigned maximal energy difference or tolerance.

We have also created a subroutine for calculating expectation values, specially withthe aim of evaluating correlation functions that can be compared with the literature.

3.4.6 Some results

We started with a very simple case, looking for the ground state of the Hamiltonian:

H =9∑

i=1

σizσi+1z + λ

10∑

i=1

σix

of a short spin chain with N=10 sites. The matrices A[i]’s are stored in a text document.We observed, as seen in Fig. 3.6, that the energy converges very fast with this algorithmand we obtain very good results for a small value of Dmax (the dimension D that ourmatrices have) in comparison to the value 2N which is of the order O(2N) ∼ 103.

Also for longer chains we have observed a fast convergence with a small dimensionDmax for the matrices. Until the present we have studied problems of up to hundreds ofsites without problems.

For the same Hamiltonian, considering a chain of 100 spins and different values of theparameter λ, we have studied the behavior of the correlation function

Cxx(∆) = 〈σ1xσ

1+∆x 〉 − 〈σ1

x〉〈σ1+∆x 〉

between site 1 and site 1 + ∆, as a function of the distance ∆ between the sites.

We have already seen theoretically what happens for the case of a translational in-variant and infinite long chain when the distance ∆ is very big: correlations decay expo-nentially. This is exactly what can be observed in Fig. 3.7 where we have plotted thecorrelation function Cxx(∆) as a function of the distance ∆ between pairs of sites. Weobserved that, at least, for parameter λ no longer than 8/9 and distances longer than 20sites correlations go down to zero.

It is remarkable that the degree of confidence (chi square) for each fitting was seento decrease as the parameter λ of our Hamiltonian increases. The explanation for this isinside the model itself: our Hamiltonian is of the Ising-type [22, 27], which has a criticalpoint at λ = 1. Very near to that regime the correlations no longer decay exponentiallybut logarithmically and DMRG’s accuracy decreases (we start needing longer Dmax).

However, it can be seen how, as we approach the critical point, correlations have alonger range.


Figure 3.6: Running the programme


10 20 30

-0.10

-0.05

0.00

0.05

Cxx(D

)=

<s

x1s

x1+D>

-<s

x1><s

x1+D>

l0

1/2

2/3

3/4

4/5

5/6

6/7

7/8

8/9

D

15 20 25 30

-0.004

-0.003

-0.002

-0.001

0.000

0.001

D

Cxx(D

)=

<s

x1s

x1+D>

-<s

x1><s

x1+D>

Exp. decay e-a

a ~ 0.00

a ~ 0.71

a ~ 1.19

a ~ 1.65

a ~ 2.09

a ~ 2.52

a ~ 2.93

a ~ 2.92

a ~ 3.14

Figure 3.7: Above: Correlation function Cxx(∆), for different valuesof parameter λ, between sites at distance ∆. Bellow: Zoom for exponential fit.

Chapter 4

Matrix Product Density Operators

Just like for pure states, to represent a mixed state we need an exponentially largenumber of coefficients

ρ =d∑

i1...,iNj1...,jN=1

ρi1...iN j1...jN |i1 . . . iN〉〈j1 . . . jN |

The success of the MPS representation encourages us to find a similar representationfor mixed states. This problem has already been considered [35], bringing forth the classof matrix product density operators (MPDO),

ρ =d∑

s1...,sN

s′1...,s′

N=1

Tr(

M [1]s1,s′

1M [2]s2,s′

2 . . .M [N ]sN ,s′

N

)

|s1 . . . sN〉〈s′1 . . . s′N | (4.1)

M [i]si,s′

i :

Matrix associated to site i

and its states si and s′

i, whose

dimension is bounded bysome fixed number D2

i×D2

i+1.They parametrize the state.

d :

Dimension ofthe Hilbert spacecorrespondingto the physicalsystem.

Here the M ’s play the role of the A’s in (3.1).

4.1 MPDO from MPS

A simple way to introduce MPDO is through the class of MPS. This can be done usingthe concept of purification [24], a purely mathematical procedure, which states:

If we are given a state ρA of a quantum system, it is always possible to intro-duce another system, a reference system R, and define a pure state |AR〉 forthe joint system AR such that ρA = TrR(|AR〉〈AR|).

39

40 Chapter 4. Matrix Product Density Operators

Then, if we have the state ρ, it can be purified into an MPS:

To each physical system |si〉 we associate an ancillary state |ai〉 of dimensiondi at most dDiDi+1 [35], such that the whole system is in the pure state

|ψ〉 =

d,di∑

s1...,sN

a1...,aN=1

Tr (A[1]s1,a1A[2]s2,a2 . . . A[N ]sN ,aN ) |s1a1, . . . sNaN〉 (4.2)

These ancillary systems form the so-called reference system. The MPDO ρ isobtained after tracing over the ancillas,

ρ = Tra|ψ〉〈ψ|

obtaining through this operation the following expression for the matrices Mof the MPDO

M [i]si,s′

i =

di∑

a=1

A[i]si,a ⊗ A[i]s′

i,a

Conversely, the ancillary matrices A[i]si,a can be recovered from the matricesM [i]si,si by means of an eigenvalue decomposition [35].

For a given state ρ, the number of different purifications that we can realize is infinite.This is an inconvenient if we try to characterize a state through purifications. Although,the minimal purification (which has the minimal dimension for the ancillary system) isunique up to unitary actions on the ancillary system.

4.2 Formal Aspects

4.2.1 Expectation values

We suppose the case that we are given the state (4.1) and we would like to calculatethe expectation value for some operator O which is the tensor product of local operatorsOi for each site i

O = O1 ⊗ · · · ⊗ON

According to the density matrix formalism, we want to calculate

〈O〉ρ = Tr[Oρ]

taking what we have until now

=d∑

s1...,sN

s′1...,s′

N=1

Tr[

O1 ⊗ · · · ⊗ON Tr(

M [1]s1,s′

1 . . .M [N ]sN ,s′

N

)

|s1 . . . sN〉〈s′1 . . . s′N |]


using the fact that the trace of a number is the number itself and that Tr|a〉〈b| = 〈b|a〉

=d∑

s1...,sN

s′1...,s′

N=1

Tr(

M [1]s1,s′

1 . . .M [N ]sN ,s′

N

)

〈s′1 . . . s′N | O1 ⊗ · · · ⊗ON |s1 . . . sN〉

or what is the same

=d∑

s1...,sN

s′1...,s′

N=1

Tr(

M [1]s1,s′

1 . . .M [N ]sN ,s′

N

)

〈s′1|O1|s1〉 ⊗ · · · ⊗ 〈s′N |ON |sN〉

considering that 〈s′i|Oi|si〉 are just numbers, we can reexpress this as

=d∑

s1...,sN

s′1...,s′

N=1

Tr(

〈s′1|O1|s1〉M [1]s1,s′

1 . . . 〈s′N |ON |sN〉M [N ]sN ,s′

N

)

the trace commutes with the sum symbol since both are sums

= Tr

d∑

s1,s′1=1

〈s′1|O1|s1〉M [1]s1,s′

1 . . .d∑

sN ,s′

N=1

〈s′N |ON |sN〉M [N ]sN ,s′

N

and we make the following definition:

EOi:=

d∑

si,s′i=1

〈s′i|Oi|si〉M [i]si,s′

i (4.3)

We baptize this matrices EOiwith the name of transfer matrix for MPDO. We use the

same symbol than for MPS, in case there should be made a difference it can be understoodthrough the context. Using this notation

〈O〉ρ = Tr [EO1. . . EON

] (4.4)

Therefore, the calculation of expectation values of operators that are products of localobservables reduces to the task of multiplying a set of transfer matrices and tracing theresult, as it happened before for MPS.

4.2.2 Normal Form

The difficulty to define entanglement for mixed states gives rise to the many differentmeasures to quantify it. The reason for this theoretical hardship lies on the fact that wecannot easily define an analogue to the Schmidt decomposition for a general mixed stateof a composite system. We also have that mixed states can be expanded in terms of purestates in infinitely many different ways and it is not clear which decomposition, if any,


should be favored.

In the case of MPDO, due to the similar matrix-product construction, we have thesame problem than for MPS: given a state ρ, the choice of the matrices M [i]si,s

′

i is notunique. In this part, we will try to construct a normal form for MPDO.

In an attempt to find a standard form to define any MPDO, we start generalizing thepreceding gauge conditions for MPS to the following (written component-wise)

∑

si,β

M [i]si,si

αα′,ββ = δαα′ (4.5)

∑

si,α,α′

M [i]si,si

αα′,ββ′Λ[i− 1]αα′ = Λ[i]ββ′ (4.6)

where Λ[i] is a diagonal matrix. For the sake of readability we will just refer to theconditions to the right. These two conditions are a must for MPDO, because the condi-tions for mixed states do have to converge to the pure-state conditions in the case thatwe deal with such a state: when M [i]si,s

′

i = A[i]si ⊗ A[i]s′

i ∀ i, the expressions (4.5) and(4.6) reduce to the ones for MPS, (3.10) and (3.11) respectively. It also happens for thematrices in (4.2) when dealing with purifications.

As (3.10) for MPS gives a normalized state, the first condition (4.5) also ensures thatthe trace of ρ equals one.

However, these two conditions might not be enough for our purposes. At least, inM [i]si,s

′

i we have an extra liberty that we did not have in A[i]si , that is a second physicalindex s′i. Also the dimensionality of the matrices is bigger.

The first approach we followed was to extend (4.5) and (4.6), including the cases whensi 6= s′i, to the following conditions:

∑

si,s′i,β

M [i]si,s

′

i

αα′,ββ = δαα′ (4.7)

∑

si,s′i,α,α′

M [i]si,s

′

i

αα′,ββ′Λ[i− 1]αα′ = Λ[i]ββ′ (4.8)

implying that

∑

si,s′i 6=si,β

M [i]si,s

′

i

αα′,ββ = O (4.9)

∑

si,s′i 6=si,α,α′

M [i]si,s

′

i

αα′,ββ′Λ[i− 1]αα′ = O (4.10)

Therefore, (4.9) and (4.10) would give us additional constrains on the A[i]si ’s in the case


of a pure state:

∑

si,s′i 6=si

A[i]si

(

A[i]s′

i

)†

= O

∑

si,s′i 6=si

(A[i]si)† Λ[i− 1]A[i]s′

i = O

But in general, an MPS does not have to fulfil these conditions. What we are doing isrestricting ourselves to states that do satisfy this and not considering all the possiblestates as we would like to.

We change direction and start asking ourselves what do we know about these matricesM [i]. The first we think of are the conditions a density matrix ρ has to obey [24]:

1. It is a positive matrix, ρ ≥ 0

2. Its trace equals one, Trρ = 1

The second condition we already know how to attain it. For the first one we startwith a less restrictive condition: ρ has to be Hermitian. Considering the expression (4.1),its adjoint matrix would be

ρ† =d∑

s1...,sN

s′1...,s′

N=1

Tr(

M [1]s′

1,s1M [2]s′

2,s2 . . . M [N ]s′

N ,sN

)

|s1 . . . sN〉〈s′1 . . . s′N |

If ρ = ρ† and we are looking for a unique representation, our normal form should takeinto account that

M [i]si,s′

i = M [i]s′

i,si (4.11)

whose physical meaning is that ρ is Hermitian, i.e. its eigenvalues are real.

Going back to expression (4.1)

ρ =d∑

s1...,sN

s′1...,s′

N=1

Tr(

M [1]s1,s′

1M [2]s2,s′

2 . . .M [N ]sN ,s′

N

)

|s1 . . . sN〉〈s′1 . . . s′N |

the numbers that we use to label our sites have no physical relevance further than orderingour system. Consequently, from which end do we start numbering should not matter.Therefore, our state above must be the same than

ρ =d∑

s1...,sN

s′1...,s′

N=1

Tr(

M [N ]sN ,s′

N . . .M [2]s2,s′

2M [1]s1,s′

1

)

|s1 . . . sN〉〈s′1 . . . s′N |


considering that the trace of square matrices fulfils Tr[A] = Tr[A>]

=d∑

s1...,sN

s′1...,s′

N=1

Tr(

M [N ]sN ,s′

N . . .M [2]s2,s′

2M [1]s1,s′

1

)>

|s1 . . . sN〉〈s′1 . . . s′N |

and that the transposition of product of matrices is such that (AB)> = B>A>

=d∑

s1...,sN

s′1...,s′

N=1

Tr(

M [1]s1,s′

1>M [2]s2,s′

2> . . .M [N ]sN ,s′

N>)

|s1 . . . sN〉〈s′1 . . . s′N |

this together with (4.1) leads us to conclude that matrices M [i]si,s′

i , in the representationwe are looking for, are symmetric

M [i]si,s′

i = M [i]si,s′

i> (4.12)

Now, lets pay attention both (4.11) and (4.12) together, written component-wise:{

M [i]αsiβ,α′s′iβ′ = M [i]αs′iβ,α′siβ′

M [i]αsiβ,α′s′iβ′ = M [i]α′siβ′,αs′iβ

combining them we have that M [i] is Hermitian

M [i] = M [i]†

i.e. M [i]αsiβ,α′s′iβ′ = M [i]α′s′iβ

′,αsiβ, and as a consequence a normal matrix M [i]M [i]† =

M [i]†M [i]. This fact allows us for its unitary diagonalization, that written component-wise reads

M [i]αsiβ,α′s′iβ′ =

∑

µ

U [i]αsiβ,µD[i]µ,µU [i]µ,α′s′iβ′

assuming that M [i]si,s′

i are positive maps

=∑

µ

U [i]αsiβ,µ

√

D[i]µ,µ

√

D[i]µ,µU [i]µ,α′s′iβ′

where µ runs from one to as much the rank of M [i], which is no greater than d×D×D.Defining A[i]si,µ

α,β := U [i]αsiβ,µ

√

D[i]µ,µ we have

M [i]si,s′

i =∑

µ

A[i]si,µ ⊗ A[i]s′

i,µ

which is equivalent to use the process of purification described in section 4.1.

We have reduced our problem of characterizing the matrices M [i] to the use of purifi-cations. The advantage of this is that we can always find a purification for any state, the

4.3. Entanglement in MPDO 45

disadvantage is that there is an infinite number of them possible.

We are interested now in fixing the purification we will use. We can always do this in astraightforward manner, as we will show now. Given the state ρ, we look for its diagonalform:

ρ =∑

j

pj|ψj〉〈ψj|

given that there is no degeneracy this decomposition is unique, we will assume this con-dition. Then, we construct our purification like:

|ψ〉 =∑

j

√pj (|ψj〉 ⊗ |ψj〉ancilla)

With this procedure we have overcome the problem of fixing the ancillary system. Nowwe look for |ψ〉’s matrix-product representation

|ψ〉 =d∑

s1,...sN

s′1,...s′

N=1

Tr(

A[1]s1,s′

1A[2]s2,s′

2 . . . A[N ]sN ,s′

N

)

|s1s′1, . . . sNs

′N〉

which is symmetric under the exchange of indices si ↔ s′i over all i’s, and from thereconstruct matrices M [i]

M [i]si,si =d∑

s′i=1

A[i]si,s′

i ⊗ A[i]si,s′

i

which are now unique under this specifications and by construction fulfil all the conditionsthat were established before.

4.3 Entanglement in MPDO

As we saw in section 2.3.1, an easily computable criterion for detecting entanglement inmixed states is the partial transpose (PT). It is considered easy because it consists of onlytwo operations: transposition of one “part” of the system and diagonalization of the re-sultant matrix. But certainly, as the size of our system grows this becomes a difficult task.

On the other hand, this criterion might be applied to MPDO in an efficient way.

Consider a general state ρ of two parties A and B

ρ =∑

sA,s′

A

sB ,s′

B

ρsA,s′

A;sB ,s

′

B|sA〉〈s′A| ⊗ |sB〉〈s′B|


if we transpose system A we obtain

ρ>A =∑

sA,s′

A

sB ,s′

B

ρsA,s′

A;sB ,s

′

B|s′A〉〈sA| ⊗ |sB〉〈s′B|

as sA and s′A are dummy indices this is the same than

ρ>A =∑

s′A,sA

sB ,s′

B

ρsA,s′

A;sB ,s

′

B|sA〉〈s′A| ⊗ |sB〉〈s′B|

Translated to MPDO, we only have to make the following change

M [i]si,s′

i −→M [i]s′

i,si

over all sites i affected by the transposition.

According to our normal form, our matrices M [i]si,s′

i are positive maps. This conditionautomatically implies that ρ is positive, but the opposite does not have to be necessarilytrue. We expect that the matrices we obtain from our normal form1 do fulfil this, i.e.

M [i] ≥ 0 ⇔ ρ ≥ 0

Altogether this would mean that, to apply the PT criterion on an MPDO, we wouldonly have to do the change M [i]si,s

′

i → M [i]s′

i,si and test the positivity of the new M [i].If D is fixed, the cost of the operation for every matrix is fixed and for ρ it would onlymean a linear growth with the size of the system.

1Or some definite set of matrices M [i].

Chapter 5

Conclusions and Outlook

The goal of this thesis has been the characterization and detection of entanglementin mixed states, a fundamental problem in QIT. With this purpose, we have investigatedthe very promising formulation for mixed states known as MPDO, which we have triedto put into a more complete framework.

We have reviewed and analyzed the known results for the MPS–representation for purestates. We have tracked its roots through different perspectives, which have given us adeeper understanding of what this representation actually means and what it is good for.It has been shown that there exists a robust formulation for defining the states, detectingentanglement and performing calculations of practical relevance. We have also studiedsome physical properties that naturally arise from this class of states. As an application ofthis formalism, within the frame of a real setting, we have introduced the DMRG method.The formulation of DMRG in terms of MPS has provided us with a more profound under-standing of this numerical technique. We have given a detailed description of the DMRGalgorithm we have implemented and we have also presented some of the results we haveobtained.

Using this pure–state formalism as background, we have worked on the formal basis ofthe mixed–state formalism. We introduced MPDO as a natural extension of MPS to thecase of mixed states. In particular, we have shown how to calculate expectation values forthis states and we have also given a possible way of characterizing the matrices that definethem. We have done this characterization based on our previous knowledge of MPS andthe purification technique. We are aware that, in the general case, this characterizationmay not be the optimal one (in terms of the dimension D).

Finally, we have analyzed the implementation of the MPDO–formulation in the taskof detecting entanglement in mixed states without major success.

The challenging problem for future work consists in finding an optimal characteriza-tion of the matrices in MPDO, independent of any nonphysical freedom that could bemathematically encountered. Instead of using purifications, it would be more desirableand convenient to be able to find these matrices from any other set of matrices defining

47

48 Chapter 5. Conclusions and Outlook

the same state we are interested in. For that we would need a procedure as the SVDin the case of MPS. It remains to be seen if such formulation can lead us to an efficiententanglement–detection scheme.

The detection and characterization of entanglement in mixed states is a very interestingproblem, due to its importance for the development of a complete theory of entanglement.We envisage continuing actively in this track in future work.

In any case, we hope that the present work helps to elucidate the basics of the matrix-product formalism and we wish this techniques find further applications with all thesuccess they have attain until the present.

Appendix A

Notation and review of selectedtopics in Linear Algebra

A.1 Basic Mathematical Notation

N+ the positive natural numbers

C the complex numbersCn complex vector space of complex n-vectors, Mn,1

~A vector ∈Mn,1

Mm,n m-by-n complex matricesMn n-by-n complex matricesI identity matrix in Mn

O zero matrix in Mm,n

A matrix with the complex conjugates entries of A ∈Mm,n

A> transpose of A ∈Mm,n

A† Hermitian adjoint of A ∈Mm,n, A>

A−1 inverse of a nonsingular A ∈Mn

A1/2 unique positive semidefinite square root of a positive semidefinite A ∈Mn

⊗ Kronecker (tensor) product⊕ direct sumspan(S) span of a subset S of a vector spaceTr[A] trace of A ∈Mn

a := b a is defined as b

A.2 Selected topics in Linear Algebra

For a more detailed account see [24, 9, 18].

Rank If A ∈ Mm,n, rank A is the largest number of columns/rows of A that constitutea linearly independent set. This set of columns/rows is not unique but the numberof elements of this set is unique.

49

50 Appendix A. Notation and review of selected topics in Linear Algebra

Unitary matrices A matrix U ∈Mn is said to be unitary if U †U = I. This is equivalentto the following statement: the columns/rows of U form an orthonormal set.

Determination of Eigenvectors and Eigenvalues It is equivalent to the matrix di-agonalization for A ∈ Mn, considering a matrix A that can be diagonalized. Eacheigenvector is paired with a corresponding eigenvalue. Mathematically, two kinds ofeigenvectors need to be distinguished:

• Right Eigenvector, defined as a column vector |λiR〉 satisfying A|λiR〉 = λi|λiR〉• Left Eigenvector, defined as a column vector |λjL〉 satisfying 〈λjL|A = λj〈λjL|

If i = j, then both eigenvectors correspond to the same eigenvalue. The followingresult is known as the principle of biorthogonality :

If |λiR〉 and |λjL〉 are the corresponding right eigenvector and left eigenvec-tor of A with eigenvalues λi and λj respectively, then

〈λjL|A|λiR〉 = 〈λjL|(λi|λiR〉) = λi(〈λjL|λiR〉)= (λj〈λjL|)|λiR〉 = λj(〈λjL|λiR〉)

Since λi 6= λj we end up with

〈λjL|λiR〉 = δij

that is, |λiR〉 and |λjL〉 are orthogonal for i 6= j.

Therefore,〈λjL|A|λiR〉 = λiδij

and we can write A asA =

∑

i

λi|λiR〉〈λiL|

where λi are in general complex numbers. This is the most general way of writingthe diagonal form for matrix A.

If A = A†, then left eigenvectors coincide with right eigenvectors and we are leftwith the more familiar expression

A =∑

i

λi|λi〉〈λi|

where λi are all real numbers.

Singular Value Decomposition Let A be a square matrix. Then there exist unitarymatrices U and V , and a diagonal matrix D with non-negative entries such that

A = UDV

The diagonal elements of D are called the singular values of A, i.e. the eigenvaluesof | A | := (A†A)1/2 counted with multiplicities.

Appendix B

Normal Form for MPS from theSchmidt decomposition

The normal conditions that we have treated before, arise as a natural consequence ofthe decomposition introduced in section 3.1.2, as we will see in the following.

• We start partitioning the system from the left end. This will give us the conditionsfor normalizing the state starting from that point.

For partition 1 : 2 . . . n we have

|ψ〉 =∑

α1

λ[1]α1|φ[1]α1

〉|φ[2 . . . n]α1〉

we can expand each Schmidt vector for qubit 1 in terms of the computational basis

|φ[1]α1〉 =

∑

i1

Γ[1]i1α1|i1〉

due the orthonormality of the Schmidt vectors and of the computational basis, wehave

〈φ[1]α′

1|φ[1]α1

〉 =∑

i1,i′1

Γ[1]i′1α′

1

Γ[1]i1α1〈i′1|i1〉

=∑

i1,i′1

Γ[1]i′1α′

1

Γ[1]i1α1δi′1i1

=∑

i1

Γ[1]i1α′

1

Γ[1]i1α1

= δα′

1α1

making the identification Γ[1]i1α1= ~A[1]i1α1

we find

∑

s1

(

~A[1]s1)†

~A[1]s1 = I

51

52 Appendix B. Normal Form for MPS from the Schmidt decomposition

For partition 1 2 : 3 . . . n we have

|ψ〉 =∑

α2

λ[2]α2|φ[1 2]α2

〉|φ[3 . . . n]α2〉

we can expand each Schmidt vector for qubits 1 and 2 in terms of the computationalbasis

|φ[1 2]α2〉 =

∑

α1,i1,i2

Γ[1]i1α1λ[1]α1

Γ[2]i2α1α2|i1i2〉


〈φ[1 2]α′

2|φ[1 2]α2

〉 =∑

α1,α′

1

i1,i′1i2,i′2

Γ[1]i′1α′

1

Γ[1]i1α1λ[1]α′

1λ[1]α1

Γ[2]i′2α′

1α′

2

Γ[2]i2α1α2·

· 〈i′1|i1〉〈i′2|i2〉=

∑

α1,α′

1

i1,i′1i2,i′2

Γ[1]i′1α′

1

Γ[1]i1α1λ[1]α′

1λ[1]α1

Γ[2]i′2α′

1α′

2

Γ[2]i2α1α2·

· δi′1i1δi′2i2=

∑

α1,α′

1

i1,i2

Γ[1]i1α′

1

Γ[1]i1α1λ[1]α′

1λ[1]α1

Γ[2]i2α′

1α′

2

Γ[2]i2α1α2

= δα′

2α2

making the identifications Γ[1]i1α1= ~A[1]i1α1

and λ[1]α1Γ[2]i2α1α2

= A[2]i1α1α2, and using

the previous result for A[1], we find

∑

s2

(A[2]s2)†A[2]s2 = I

Iteratively we find that for any particle i

∑

si

(A[i]si)†A[i]si = I

just as our first gauge condition (3.8).

Going back to partition 1 : 2 . . . n

|ψ〉 =∑

α1

λ[1]α1|φ[1]α1

〉|φ[2 . . . n]α1〉

Appendix B. Normal Form for MPS from the Schmidt decomposition 53

we can expand each Schmidt vector for qubits 2 . . . n in terms of the computationalbasis for qubit 2 and the Schmidt vectors for the rest 3 . . . n

|φ[2 . . . n]α1〉 =

∑

i2,α2

|i2〉Γ[2]i2α1α2λ[2]α2

|φ[3 . . . n]α2〉

due the orthonormality of the Schmidt vectors and of the vectors in the computa-tional basis, we have

〈φ[2 . . . n]α′

1|φ[2 . . . n]α1

〉 =∑

i2,i′2α2,α′

2

〈i′2|i2〉Γ[2]i′2α′

1α′

2

Γ[2]i2α1α2λ[2]α′

2λ[2]α2

·

· 〈φ[3 . . . n]α′

2|φ[3 . . . n]α2

〉=

∑

i2,i′2α2,α′

2

δi′2i2Γ[2]i′2α′

1α′

2

Γ[2]i2α1α2λ[2]α′

2λ[2]α2

δα′

2α2

=∑

i2,α2

Γ[2]i2α′

1α2Γ[2]i2α1α2

λ[2]α2λ[2]α2

= δα′

1α1

therefore

∑

i2,α2

λ[1]α1Γ[2]i2α1α2

λ[1]α1Γ[2]i2α1α2

λ[2]α2λ[2]α2

= λ[1]α1λ[1]α1

we obtain the last expression multiplying by λ[1]α′

1λ[1]α1

on both sides. Then,making the identification λ[1]α1

Γ[2]i2α1α2= A[2]i2α1α2

and λ[i]αiλ[i]αi

= Λ[i]αiαi, where

Λ[i] is a diagonal matrix that we associate to site i, we find

∑

s2

A[2]s2Λ[2] (A[2]s2)† = Λ[1]

For partition 1 2 : 3 . . . n we had

|ψ〉 =∑

α2

λ[2]α2|φ[1 2]α2

〉|φ[3 . . . n]α2〉

we can expand each Schmidt vector for qubits 3 . . . n in terms of the computationalbasis for qubit 3 and the Schmidt vectors for the rest 4 . . . n

|φ[3 . . . n]α2〉 =

∑

i3,α3

|i3〉Γ[3]i3α2α3λ[3]α3

|φ[4 . . . n]α3〉

due the orthonormality of the Schmidt vectors and of the computational basis, we


have

〈φ[3 . . . n]α′

2|φ[3 . . . n]α2

〉 =∑

i3,i′3α3,α′

3

〈i′3|i3〉Γ[3]i′3α′

2α′

3

Γ[3]i3α2α3λ[3]α′

3λ[3]α3

·

· 〈φ[4 . . . n]α′

3|φ[4 . . . n]α3

〉=

∑

i3,i′3α3,α′

3

δi′3i3Γ[3]i′3α′

2α′

3

Γ[3]i3α2α3λ[3]α′

3λ[3]α3

δα′

3α3

=∑

i3,α3

Γ[3]i3α′

2α3Γ[3]i3α2α3

λ[3]α3λ[3]α3

= δα′

2α2

therefore

∑

i3,α3

λ[2]α2Γ[3]i3α2α3

λ[2]α2Γ[3]i3α2α3

λ[3]α3λ[3]α3

= λ[2]α2λ[2]α2

we obtain the last expression multiplying by λ[2]α′

2λ[2]α2

on both sides. Then,making the identification λ[2]α2

Γ[3]i3α2α3= A[3]i3α2α3

and λ[i]αiλ[i]αi

= Λ[i]αiαi, where

Λ[i] is a diagonal matrix that we associate to site i, we find

∑

s3

A[3]s3Λ[3] (A[3]s3)† = Λ[2]


∑

si

A[i]siΛ[i] (A[i]si)† = Λ[i− 1]

just as our second gauge condition (3.9).

• Next, we follow partitioning the system from the right end. This will give us theconditions for normalizing the state starting from that point.

For partition 1 . . . n− 1 : n we have

|ψ〉 =∑

αn−1

λ[n− 1]αn−1|φ[1 . . . n− 1]αn−1

〉|φ[n]αn−1〉

we can expand each Schmidt vector for qubit n in terms of the computational basis

|φ[n]αn−1〉 =

∑

i1

Γ[n]inαn−1|in〉



〈φ[n]α′

n−1|φ[n]αn−1

〉 =∑

in,i′n

Γ[n]i′nα′

n−1

Γ[n]inαn−1〈i′n|in〉

=∑

in,i′n

Γ[n]i′1α′

n−1

Γ[n]inαn−1δi′nin

=∑

in

Γ[n]inα′

n−1

Γ[n]inαn−1= δα′

n−1αn−1

making the identification Γ[n]inαn−1= A[n]inαn−1

we find

∑

s1

~A[n]sn

(

~A[n]sn

)†

= I

For partition 1 . . . n− 2 : n− 1 n we have

|ψ〉 =∑

αn−2

λ[n− 2]αn−2|φ[1 . . . n− 2]αn−2

〉|φ[n− 1 n]αn−2〉

we can expand each Schmidt vector for qubits n− 1 and n in terms of the compu-tational basis

|φ[n− 1 n]αn−2〉 =

∑

αn−1,in−1,in

Γ[n− 1]in−1

αn−2αn−1λ[n− 1]αn−1

Γ[n]inαn−1|in−1in〉


〈φ[n− 1 n]α′

n−2|φ[n− 1 n]αn−2

〉 =∑

αn−1,α′

n−1

in−1,i′n−1

in,i′n

Γ[n− 1]i′n−1

α′

n−2α′

n−1

·

· Γ[n− 1]in−1

αn−2αn−1λ[n− 1]α′

n−1λ[n− 1]αn−1

Γ[n]i′nα′

n−1

Γ[n]inαn−1·

· 〈i′n−1|in−1〉〈i′n|in〉=

∑

αn−1,α′

n−1

in−1,i′n−1

in,i′n

Γ[n− 1]i′n−1

α′

n−2α′

n−1

·

· Γ[n− 1]in−1


n−1λ[n− 1]αn−1

Γ[n]i′nα′

n−1

Γ[n]inαn−1·

· δi′n−1in−1δi′nin

=∑

αn−1,α′

n−1

in−1,in

Γ[n− 1]in−1

α′

n−2α′

n−1

·

· Γ[n− 1]in−1


n−1λ[n− 1]αn−1

Γ[n]inα′

n−1

Γ[n]inαn−1

= δα′

2α2


therefore

∑

in−1

αn−1

Γ[n− 1]in−1

α′

n−2αn−1Γ[n− 1]in−1


λ[n− 1]αn−1=

= δα′

2α2

where we used the previous result for Γ[n]. Making the identificationΓ[n− 1]in−1


= A[n− 1]in−1αn−2αn−1

we find

∑

sn−1

A[n− 1]sn−1 (A[n− 1]sn−1)† = I


∑

si

A[i]si (A[i]si)† = I

just as our gauge condition (3.10).

Going back to partition 1 . . . n− 1 : n

|ψ〉 =∑

αn−1

λ[n− 1]αn−1|φ[1 . . . n− 1]αn−1

〉|φ[n]αn−1〉

we can expand each Schmidt vector for qubits 1 . . . n − 1 in terms of the compu-tational basis for qubit n − 1 and the Schmidt vectors for the rest of the system1 . . . n− 2

|φ[1 . . . n− 1]αn−1〉 =

∑

in−1

αn−2

|in−1〉λ[n− 2]αn−2Γ[n− 1]in−1

αn−2αn−1|φ[1 . . . n− 2]αn−2

〉

due the orthonormality of the Schmidt vectors and of the vectors in the computa-


tional basis, we have

〈φ[1 . . . n− 1]α′

n−1|φ[1 . . . n− 1]αn−1

〉 =∑

in−1,i′n−1

αn−2,α′

n−2

〈i′n−1|in−1〉λ[n− 2]α′

n−2·

· λ[n− 2]αn−2Γ[n− 1]

i′n−1

α′

n−2α′

n−1

Γ[n− 1]in−1

αn−2αn−1·

· 〈φ[1 . . . n− 2]α′

n−2|φ[1 . . . n− 2]αn−2

〉=

∑

in−1,i′n−1

αn−2,α′

n−2

δi′n−1in−1λ[n− 2]α′

n−2·

· λ[n− 2]αn−2Γ[n− 1]

i′n−1

α′

n−2α′

n−1

Γ[n− 1]in−1

αn−2αn−1δα′

n−2αn−2

=∑

in−1,αn−2

λ[n− 2]αn−2·

· λ[n− 2]αn−2Γ[n− 1]

in−1

αn−2α′

n−1

Γ[n− 1]in−1

αn−2αn−1

= δα′

n−1αn−1

therefore

∑

in−1,αn−2

λ[n− 2]αn−2λ[n− 2]αn−2

Γ[n− 1]in−1

αn−2α′

n−1

λ[n− 1]αn−1·

· Γ[n− 1]in−1


=

= λ[n− 1]αn−1λ[n− 1]αn−1

obtaining the last expression multiplying by λ[n− 1]αn−1λ[n− 1]α′

n−1on both sides.

Then, making the identification Γ[n− 1]in−1αn−2αn−1

λ[n− 1]αn−1= A[n− 1]in−1

αn−2αn−1and

λ[i]αiλ[i]αi

= Λ[i]αiαi, where Λ[i] is a diagonal matrix that we associate to site i, we

find∑

sn−1

(A[n− 1]sn−1)† Λ[n− 2]A[n− 1]sn−1 = Λ[n− 1]

For partition 1 . . . n− 2 : n− 1 n we had

|ψ〉 =∑

αn−2

λ[n− 2]αn−2|φ[1 . . . n− 2]αn−2

〉|φ[n− 1 n]αn−2〉

we can expand each Schmidt vector for qubits 1 . . . n − 2 in terms of the compu-tational basis for qubit n − 2 and the Schmidt vectors for the rest of the particles1 . . . n− 3

|φ[1 . . . n− 2]αn−2〉 =

∑

in−2

αn−3

|in−2〉λ[n− 3]αn−3Γ[n− 2]in−2

αn−3αn−2|φ[1 . . . n− 3]αn−3

〉



〈φ[1 . . . n− 2]α′

n−2|φ[1 . . . n− 2]αn−2

〉 =∑

in−2,i′n−2

αn−3,α′

n−3

〈i′n−2|in−2〉λ[n− 3]α′

n−3·

· λ[n− 3]αn−3Γ[n− 2]

i′n−2

α′

n−3α′

n−2

Γ[n− 2]in−2

αn−3αn−2·

· 〈φ[1 . . . n− 3]α′

n−3|φ[1 . . . n− 3]αn−3

〉=

∑

in−2,i′n−2

αn−3,α′

n−3

δi′n−2in−2λ[n− 3]α′

n−3·

· λ[n− 3]αn−3Γ[n− 2]

i′n−2

α′

n−3α′

n−2

Γ[n− 2]in−2

αn−3αn−2·

· δα′

n−3αn−3

=∑

in−2,αn−3

λ[n− 3]αn−3·

· λ[n− 3]αn−3Γ[n− 2]

in−2

αn−3α′

n−2

Γ[n− 2]in−2

αn−3αn−2

= δα′

n−2αn−2

consequently

∑

in−2

αn−3

λ[n− 3]αn−3λ[n− 3]αn−3

Γ[n− 2]in−2

αn−3α′

n−2

λ[n− 2]αn−2·

· Γ[n− 2]in−2


=

= λ[n− 2]αn−2λ[n− 2]αn−2

obtaining the last expression multiplying by λ[n− 2]α′

n−2λ[n− 2]αn−2

on both sides.

Then, making the identification Γ[n− 2]in−2αn−3αn−2

λ[n− 2]αn−2= A[n− 2]in−2

αn−3αn−2and

λ[i]αiλ[i]αi

= Λ[i]αiαi, where Λ[i] is a diagonal matrix that we associate to site i, we

find ∑

sn−2

(A[n− 2]sn−2)† Λ[n− 3]A[n− 2]sn−2 = Λ[n− 2]

Iteratively we find that for any particle i∑

si

(A[i]si)† Λ[i− 1]A[i]si = Λ[i]

just as our gauge condition (3.11).

Now we can say more about these matrices. Provided that the Schmidt decompositionat each step has no degeneracy, this decomposition:

{Γ[1], . . . ,Γ[n]}; {λ[1], . . . , λ[n− 1]}


is unique, up to arbitrary phases in the basis states. Therefore, the matrices

{A[1], . . . , A[n]}

are unique, up to the same extend, given a fixed |ψ〉.


Appendix C

Normal Form implies normalizedstates for OBC

Condition (3.16) implies that, thinking that all our A[i]’s fulfill it:

〈ψmps|ψmps〉 = 〈ψmps|I|ψmps〉= Tr [EI1

· · ·EIN]

= Tr

[N∏

i=1

d∑

si=1

A[i]si ⊗ A[i]si

]

= Tr

[D∑

α,β...=1

(d∑

s1=1

A[1]α,s1,βA[1]α′,s1,β′

)

· · ·

· · ·(

d∑

sN=1

A[N ]ν,sN ,µA[N ]ν′,sN ,µ′

)]

= 1 (C.1)

The last equality comes from the fact that, starting from the left:

D∑

β,β′=1

(d∑

s1=1

A[1]α,s1,βA[1]α′,s1,β′

)(d∑

s2=1

A[2]β,sN ,γA[N ]β′,s2,γ′

)

As A[1]s1 is a vector transposed, we have α, α′ = 1. So, inserting a∑D

α=1 δα,α′ after∑d

s1=1

makes no harm. Using (3.16) we get:

D∑

β,β′=1

δβ,β′

(d∑

s2=1

A[2]β,sN ,γA[2]β′,s2,γ′

)

And so on, until the end in the right side we finally get δµ,µ′ , where µ, µ′ = 1 as A[N ]sN

is also a vector. Therefore,

〈ψmps|ψmps〉 = Tr [δµ,µ′ ]

= 1

our state |ψmps〉 is normalized.

61

62 Appendix C. Normal Form implies normalized states for OBC

Bibliography

[1] I. Affleck, Phys. Rev. Lett. 54, 966 (1985).

[2] I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki, Phys. Rev. Lett. 59, 799 (1987).

[3] I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki, Commun. Math. Phys. 115, 477(1988).

[4] A. Barenco, D. Deutsch, A. Ekert and R. Jozsa, Phys. Rev. Lett. 74, 4083 (1995).

[5] J. S. Bell, Physics 1, 194 (1964).

[6] C. H. Bennett, H. J. Bernstein, S. Popescu and B. Schumacher, Phys. Rev. A 53,2046 (1996).

[7] C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and W. K. Wootters, Phys.Rev. Lett. 70, 1895 (1993).

[8] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters, Phys. Rev. A54, 3824 (1996).

[9] R. Bhatia, Matrix Analysis, New York: Springer Verlag (1997).

[10] R. Cleve, D. Gottesman and H. K. Lo, Phys. Rev. Lett. 83, 648 (1999)

[11] Density-Matrix Renormalization, Eds. I. Peschel, et al., Belin: Springer Verlag(1999).

[12] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935).

[13] A. Ekert and P. Knight, Am. J. Phys. 63 (5), 415 (1995).

[14] B. d’Espagnat, Conceptual Foundations of Quantum Mechanics, 2nd Ed. Reading,MA: Benjamin (1976).

[15] D. Gottesman, Phys. Rev. A 54, 1862 (1996).

[16] L. Gurvits, quant-ph/0303055 (2003).

[17] P. Hayden, M. Horodecki and B. M. Terhal, J. Phys. A, 34(35), 6891-6898 (2001).

63

64 Bibliography

[18] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge: Cambridge UniversityPress (1985).

[19] M. Horodecki, Quant. Inf. Comp. 1, No 1, 3-26 (2001).

[20] M. Horodecki, P. Horodecki and R. Horodecki, Phys. Lett. A 223, 1 (1996).

[21] K. Kraus, States, Effects and Operations: Fundamental notions of quantum theory,New York: Wiley (1991).

[22] E. Lieb, T. Schultz and D. Mattis, Ann. Phys. 16, 407 (1961).

[23] C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, 447 (1988).

[24] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cam-bridge University Press (2000).

[25] Y. S. Ostlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995).

[26] A. Peres, Phys. Rev. Lett. 76, 1413 (1997).

[27] P. Pfeuty, Ann. Phys. 57, 79 (1970).

[28] M. B. Plenio and S. Virmani, quant-ph/0504163 (2005).

[29] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001).

[30] A. Sanpera, R. Tarrach and G. Vidal, quant-ph/9707041 (1997).

[31] E. Schrodinger, Naturwissenschaften 23, 807 (1935).

[32] B. M. Terhal and P. Horodecki, Phys. Rev. A 61, 040301 (2000)

[33] F. Verstraete and J. I. Cirac, cond-mat/0505140 (2005).

[34] F. Verstraete, J. I. Cirac, J. I. Latorre, E. Rico and M. M. Wolf, Phys. Rev. Lett.94, 140601 (2005).

[35] F. Verstraete, J. J. Garcıa-Ripoll and J. I. Cirac, Phys. Rev. Lett. 93, 207204 (2004).

[36] F. Verstraete, D. Porras, and J. I. Cirac, Phys. Rev. Lett. 93, 227205 (2004)

[37] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).

[38] G. Vidal, Phys. Rev. Lett. 93, 040502 (2004).

[39] R. F. Werner, Phys. Rev. A 40, 4277 (1989).

[40] S. R. White, Phys. Rev. Lett. 69, 2863 (1992), Phys. Rev. B 48, 10345 (1993).

[41] E. P. Wigner, Group theory and its application to the quantum, mechanics of atomicspectra, New York: Academic Press (1959).

[42] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).

Acknowledgements

A l’Alvar y el super Calico Electronico.A la Maria, perque el present es sempre nomes un principi.

A l’Alex, per tot el que hem apres.Gracies.

A mama, Ceci y Silvia.A Julia, Marcin y Vasili, autores intelectuales de mi lado cientıfico.

A Alfonso, por los errores: los tuyos o los mıos?A Ignacio, es solo el comienzo.

A Juanjo y Diego, por los consejos y la infinita paciencia.A Belen, por la psicologıa (¡que guay!).

A Roman y David, por los piropos y los chocolates.A Juan y Carlitos, por la direccion artıstica y el soporte technicolor.

Gracias.

An Michael, fur seinen guten Rat und nutzlichen Diskussionen.Danke.

To Sir Toby: when I grow up, I want to be like you.To all of you, Vasudhaiva Kutumbakam.

Thank you.

Al Lucas, simplement per ser un gat.

m.

65

66

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Physik Department Matrix Product Formalism

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