Reasonable fermionic quantum information theories require relativity

Nicolai Friis1, 2, ∗

1Institute for Theoretical Physics, University of Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria2Institute for Quantum Optics and Quantum Information,

Austrian Academy of Sciences, Technikerstraße 21a, A-6020 Innsbruck, Austria(Dated: February 17, 2015)

We show that any abstract quantum information theory based on anticommuting operators mustbe supplemented by a superselection rule deeply rooted in relativity. While quantum informationmay be encoded in the Fock space generated by such operators, the unrestricted fermionic theoryhas a peculiar feature: Pairs of bipartition marginals of pure states need not have identical spectra.This leads to an ambiguous definition of the entropy of entanglement. We prove that this problemis removed by a superselection rule that arises from Lorentz invariance and no-signalling.

PACS numbers: 03.67.Mn, 05.30.-d, 11.10.-z

Introduction. A significant virtue of quantum infor-mation theory lies in its abstraction from the physicalcontext. Dealing with problems purely on the level of aHilbert space HAB , its subsystems A and B, and opera-tions thereon, without reference to the specific physicalimplementation, provides a level of freedom and gener-ality that is highly desirable. Statements can be madefor all Hilbert spaces of a certain type. For instance,quantum information processing with qubits can typi-cally be investigated without reference to their imple-mentation — although examples exist, where specifyingthe encoding of the qubit is relevant for an abstract prob-lem, see, e.g., Ref. [1]. From the theorist’s point of view,it is hence clearly advantageous to favor an abstract per-spective, while the particular realization of the quantumsystems to which the Hilbert space in question is asso-ciated remains a secondary concern. The basic ingre-dients for an abstract quantum information theory arepure states on the total Hilbert space, |ψ 〉 ∈ H, and re-duced states with respect to some bipartition (A|B), i.e.,ρA(B) = TrB(A)

(|ψ 〉〈ψ |

). With this, one may already

study correlation measures such as the mutual informa-tion, and, most importantly, the entropy of entanglementE(|ψ 〉) = S(ρA) = S(ρB), where S(ρ) = −Tr

(ρ ln(ρ)

).

Besides multi-qubit systems, quantum harmonic oscilla-tors are prominent examples for successful abstraction.

Based on the commutation relations, [ai, a†j ] = δij and

[ai, aj ] = 0, a bosonic Fock space is constructed, thatprovides a playground for quantum optics, irrespectiveof the particular realization, be it as optical modes, su-perconducting L − C circuits, or vibrational degrees offreedom, to name only a few examples.

Here, another type of Hilbert space — the fermionicFock space — will be considered. That is, the basic alge-bra is based on anticommuting operators, rather thancommuting ones. Absent physical interpretation, onemay yet work with such a Hilbert space, identify its sub-systems, and their correlations. In other words, one may

∗ nicolai.friis@uibk.ac.at

attempt to construct an abstract fermionic quantum in-formation theory, see, e.g., Refs. [2–4]. However, as weshall show here, the physically unrestricted theory suffersfrom a disconcerting malady: The marginals ρA and ρBof bipartite pure states may not have matching spectra,leaving the typical notion of entropy of entanglement in astate of ambiguity, since S(ρA) 6= S(ρB). Depending onthe choice of subsystem, different amounts of entangle-ment would be attributed to the system. This problemdoes not occur in theories with a natural tensor prod-uct structure, like bosonic modes or qubits, where theSchmidt decomposition guarantees symmetric marginalentropies for pure states. For fermions, on the otherhand, mappings to a tensor product space, i.e., to qubits,do not generally preserve the structure of the subsys-tems [5], and the issue persists.

We demonstrate here that this problem can be over-come by imposing a superselection rule (SSR) that for-bids coherent superpositions of even and odd numbersof fermions. Although the problem of asymmetric purestate marginals is thus removed, it seems rather artificialto enforce such a restriction within the abstract theory.Only once the model is embedded in a physical context,in this case relativistic field theory, does the SSR arisenaturally from Lorentz invariance and the constraints ofsignalling. With this, we provide an alternative viewon the connection between abstract quantum informa-tion theory and relativistic quantum field theory, argu-ing that the latter is indeed necessary for the reasonableconstruction of the former. This work hence adds a newfacette to the discussion of informational constructionsof quantum theory (see, e.g., Refs. [6–8]), by introducingan information-theoretic aspect of SSRs — a fascinatingtopic in its own right (see, e.g., Refs. [9–17] for a selectionof literature).

In the following, we will first outline the constructionof the fermionic Fock space, as well as of the pure andmixed states in such a Hilbert space. To understandthe origin of the problem described above, we will thendiscuss the subtleties involved in forming subsystemsof fermionic modes, and give an example for a purestate that features marginals with different entropies.

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Finally, the role and the origin of SSRs are discussed,and we show how the problem can be disposed of. Tohighlight the intrinsically different character of fermionicmodes and qubits, we supplement our discussion withan Appendix, where an example for a pure state thatsatisfies the SSR, but still cannot be consistently mappedto a multi-qubit state, is presented.

Fermionic Fock space. Let us consider a system of n

fermionic modes with mode operators bk and b†k for (k =1, . . . , n), which satisfy the anticommutation relations

{bi , b†j } = δij , {bi , bj } = 0 , (1a)

for all i, j. The vacuum state is annihilated by all bk, i.e.,bk ||0〉〉 = 0 ∀ k, and the purpose of the double-lined no-tation for the state vectors will become apparent shortly.

The creation operators b†k populate the vacuum with sin-

gle fermions, that is, b†k ||0〉〉 = ||1k 〉〉. When two, or more,fermions are created, the corresponding tensor productof single-particle states needs to be antisymmetrized dueto the indistinguishability of the particles. We use theconvention

b†kb†k′ ||0〉〉 = ||1k 〉〉 ∧ ||1k′〉〉 = ||1k 〉〉 ||1k′〉〉 , (2)

where we use the double-lined notation to imply the an-tisymmetrized wedge product “∧” between single-modestate vectors with particle content (as opposed to thestandard notation | · 〉 | · 〉 = | · 〉 ⊗ | · 〉). With thisdefinition at hand, and postponing possible physical re-strictions, one may write arbitrary pure states on theFock space as

||Ψ〉〉 = γ0 || 0 〉〉+

n∑i=1

γi ||1i 〉〉+∑j,k

γjk ||1j 〉〉 ||1k 〉〉+ . . . .

(3)

where the complex coefficients γ0, γi, γjk, . . . are chosensuch that the state is normalized. Similarly, mixedstate density operators can be written as convex sumsof projectors on such pure states. For more details onthis notation and the fermionic Fock space, see, e.g.,Refs. [5, 18].

Partitioning the Hilbert space. To construct aquantum information theory, it is then necessary to es-tablish a meaningful notion of subsystems. Since the par-ticle number in the Fock space need not be fixed, we willconsider entanglement between different modes. How-ever, due to the antisymmetrization, the Fock space isnot naturally equipped with a tensor product structurewith respect to the individual mode subspaces. Thesesubspaces may nonetheless be defined by invoking con-sistency conditions [5] that ensure that the expectationvalues of all local observables OA (i.e., as in, operatorspertaining only to the modes of the subspace A) yieldthe same result for the global state ρAB , and for the cor-responding local reduced states ρA = TrB(ρAB), i.e.,

FIG. 1. Inconsistency between fermions and qubits:The global n-mode fermionic state ρAB may be mapped toan isomorphic n-qubit state ρAB . The marginals of ρAB , e.g.,ρA = TrB(ρAB), are well-defined by Eq. (4), and may also bemapped to isomorphic qubit states (e.g., ρA ↔ ρ ′A). However,as shown in Ref. [5], it is in general impossible to match allthe marginals of the n-mode state to all marginals of the n-qubit state, ρ ′A 6= ρA = TrB(ρAB). An example for a statefeaturing this problem is given in the Appendix.

〈OA〉ρAB= 〈OA〉ρA . (4)

This procedure uniquely defines the mode subspacemarginals of any global state, i.e., the partial trace op-eration, via

Trk(b†µ1

. . . b†µib†k || 0 〉〉〈〈 0 || bk bν1 . . . bνj

)= b†µ1

. . . b†µi|| 0 〉〉〈〈 0 || bν1 . . . bνj . (5)

That is, operators corresponding to modes that are be-ing traced out are anticommuted towards the vacuumprojector, before being removed.

At this point, it is helpful to understand the differ-ences between fermionic modes and qubits. For any fixednumber n, the fermionic n-mode Fock space is isomor-phic to an n-qubit space. A widely known example forsuch an isomorphism is the Jordan-Wigner transforma-tion, (see, e.g., [2]). Such mappings generally do not com-mute with the procedure of partial tracing [5], since localmode operators are mapped to global qubit operations.In other words, it is generally not possible to establishisomorphisms between a fermionic n-mode state and ann-qubit state in such a way that also all of the respec-tive fermionic marginals are isomorphic to their qubitcounterparts. An illustration of this problem is shownin Fig. 1. Consequently, the (quantum) correlations be-tween n fermionic modes may generally not be identifiedwith those of the isomorphic n-qubit states.

In spite of this inequivalence, the partial trace, andhence the subsystems and their entropies remain well de-fined for fermionic modes. Moreover, the construction ofthe density operators and its marginals is based solelyon the algebraic structure of Eqs. (1), together with therequirement that the expectation values of subsystem ob-servables yield the same result when evaluated using ei-ther the global states or the corresponding marginals. Noother assumptions are required for a consistent definition

3

of the subsystems, and their total correlations. For in-stance, the mutual information IAB , a measure of theoverall correlation between subsystems A and B, is inthis context already well defined by the expression

IAB(ρAB) = S(ρA) + S(ρB) − S(ρAB) , (6)

where ρA(B) = TrB(A)(ρAB), and S(ρ) = −Tr(ρ ln(ρ)

)is the von Neumann entropy of the density operator ρ.But, as we shall elaborate on shortly, the same cannotbe said for genuine quantum correlations, i.e., entan-glement. Consider, for instance, the non-superselectedtwo-mode pure state given by

||ψ〉〉 = γ0 ||0〉〉+ γk ||1k 〉〉+ γk′ ||1k′〉〉+ γkk′ ||1k 〉〉||1k′〉〉 .(7)

According to the prescription of Eq. (5), the single-modereduced states can be quickly checked to be

ρk = Trk′(||ψ〉〉〈〈ψ ||

)=(|γ0|2 + |γk′ |2

)||0〉〉〈〈0|| (8a)

+(|γk|2 + |γkk′ |2

)||1k 〉〉〈〈1k ||

+[(γ0γ∗k + γk′γ

∗kk′)||0〉〉〈〈1k ||+ H. c.

],

ρk′ = Trk(||ψ〉〉〈〈ψ ||

)=(|γ0|2 + |γk|2

)||0〉〉〈〈0|| (8b)

+(|γk′ |2 + |γkk′ |2

)||1k′〉〉〈〈1k′||

+[(γ0γ∗k′ − γkγ∗kk′

)||0〉〉〈〈1k′||+ H. c.

],

where the symmetry between the subsystems isbroken by the different relative signs within the off-diagonal elements. The eigenvalues of the two reducedstates do not match in general. For example, whenγ0 = γk = γk′ = γkk′ = 1/2, the mode k appears tobe in a pure state (with eigenvalues 0 and 1), whereasthe state of the mode k′ is maximally mixed (botheigenvalues are 1/2). Normally, the entropy of thesubsystem of a pure state would be considered as anentanglement measure. Here, depending on the choiceof subsystem, one would either conclude that the overallstate is maximally entangled, or not entangled at all.This problem is not limited to pure states. It persistsfor mixed states, where the entropy of entanglement isof central importance for the entanglement of formation.Such an ambiguity in the definition of entanglement is ofcourse highly undesirable. One possibility to resolve theissue, would be to change the definition of entanglement,and work with at a non-symmetric quantity. On theother hand, such a drastic step may not be required, ifone is willing to embed the abstract fermionic quantuminformation theory in a physical framework. As weshall show in the following, a reasonable definition ofentanglement between fermionic modes is obtained wheninvoking an additional physical principle—the spin-statistics connection, which itself arises from (special)relativity.

Invoking relativity—superselection rules. NoSSRs have been introduced up to this point. Note that

the term SSR may refer to different restrictions. For in-stance, they may arise from fundamental symmetries ofthe system, such as parity [9], or charge conservation [10–12]. Alternatively, effective (or generalized) SSRs orig-inate from practical limitations, such as particle num-ber conservation due to energy constraints, see, e.g.,Refs. [13–15, 19, 20]. Both type of restrictions may beformalized as constraints on the observables, see, e.g.,Ref. [3, 4].

Here, we will formulate such constraints in a different,but equivalent way, as restrictions on the components γ0,γi, γjk, . . . [see Eq. (3)], of pure state decompositionswith respect to the Fock basis. In particular, we willconsider any coherent superpositions of even and oddnumbers of fermionic excitations as unphysical. Theargument that we will advocate to defend this positionis based on the well-known spin-statistics connection,relating the anticommutation relation algebra to thetransformation properties associated to objects ofhalf-integer spin. Recall that, a priori, we have madeno assumptions on the physical realization of theanticommutation relations of Eq. (1a). Nonetheless,as an inevitable consequence of this anticommutationrelation algebra, the excitations of the mode operatorsmust satisfy Fermi-Dirac statistics. At this stage, wemay surrender some level of abstraction and provide aphysical context. When we place the fermionic quan-tum information theory in the context of relativisticfield theory, the spin-statistics connection follows fromLorentz invariance (see, e.g., Refs. [21] and [22, pp. 52]).Thus, we must interpret the fermionic excitations of ourtheory as particles of half-integer spin. The correspond-ing states switch their sign when subjected to spatialrotations by 2π. If a superposition of even and oddnumbers of fermions was permitted in such a physicaltheory, it would entail that rotations of our referenceframe by 2π would change the relative sign of thesecontributions within the superposition. Put bluntly, anobserver spinning around once, could switch a remotequantum system between two orthogonal states atwill. Since such instantaneous signals are forbiddenin relativity, one arrives at the aforementioned SSR.Crucially, this line of reasoning is intimately tied torelativity. On the other hand, as we will show next, thethus imposed constraint provides the essential ingredientto guarantee a meaningful quantum (information) theorybased on anticommuting operators.

Superselection rules & symmetric pure statemarginals. We shall now provide the main technicalstatement of this work, and its proof.

Theorem. The marginals ρA(B) = TrB(A)

(||ψ〉〉〈〈ψ ||

)of any bipartition (A|B) of a pure state ||ψ〉〉 in thefermionic Fock space have the same spectrum, if ||ψ〉〉satisfies the SSR prohibiting superpositions of even andodd numbers of fermions, i.e.,

||ψ〉〉 satifies SSR ⇒ spetr(ρA) = spetr(ρB) ∀ (A|B).

4

Proof. To show this, let us consider a pure state ||ψNeven 〉〉in an n-mode fermionic Fock space, where N ={1, 2, . . . , n} denotes the set of modes, and without lossof generality we have chosen ||ψNeven 〉〉 to be a superposi-tion of states with even numbers of fermions. The set Nis then partitioned into the subsets M = {µi|µi,j ∈ N :µi 6= µj if i 6= j; i, j = 1, 2, . . . ,m < n} and MC = N\M ,such that N = M ∪MC and M ∩MC = ∅. With respectto this bipartition, we may write the state ||ψNeven 〉〉 in thepure state decomposition

||ψNeven 〉〉 = γ0 ||0〉〉 +

m∑i=1

γµi ||1µi 〉〉||ψMC

µi,odd 〉〉 (9)

+

m∑i,j=1j>i

γµiµj||1µi〉〉||1µj

〉〉||ψMC

µiµj ,even 〉〉 + . . .

+ γµ1...µm ||1µ1 〉〉 . . . ||1µm 〉〉||ψMC

µ1...µm,odd 〉〉 ,

where γµi, . . . , γµ1...µm

∈ C, 〈〈ψNeven ||ψNeven 〉〉 = 1, andwithout loss of generality we have here selected m to

be odd. The states ||ψMC

µi...µk,even(odd)〉〉 contain only even

(odd) numbers of excitations, and only in modes fromthe set MC . Any sign changes that may occur whenrewriting a given state in such a decomposition canbe absorbed into the γ-coefficients. Adhering to the“outside-in” tracing rule of Eq. (5), we note that thestate has been brought to a form, where the partial traceover MC is achieved by simply removing all projectors

||ψMC

µi...µk〉〉〈〈ψMC

µi...µk|| pertaining to MC from the projec-

tor on ||ψNeven 〉〉, without incurring any additional signflips. On the other hand, if we trace over the modes inthe set M instead, anticommuting the operators corre-sponding to modes in M towards the vacuum projectorin the process, we may generate sign changes. However,for the superselected state, all the nonzero contributionsto the partial trace over M are generated from elementssuch as

TrM(|γµi|2 b†µi

||ψMC

µi,odd 〉〉〈〈ψMC

µi,odd || bµi

)= |γµi

|2 ||ψMC

µi,odd 〉〉〈〈ψMC

µi,odd || . (10)

There, the parity of the number of anticommutations to-wards ||0〉〉 from the left, is the same as the parity of thenumber of anticommutations towards 〈〈0|| from the right.In other words, once the state has been brought to theform of Eq. (9), the partial trace can be carried out as ifoperating on a tensor product of Hilbert spaces, i.e., asif HN = HM ⊗HMC . In particular, this implies that thetwo reduced states on HM and HMC , which are isomor-phic to the corresponding m-mode and (n−m)-mode re-duced fermionic states, respectively, have the same spec-trum. For any bipartition of the Fock space, a decom-position with this property can be found, although, ingeneral, no decomposition exists that simultaneously ac-complishes the required task for all bipartitions at once.An example is presented in the Appendix. We hence con-clude that the SSR forbidding superpositions of even and

odd numbers of fermions, guarantees that the spectra ofthe marginals for any bipartition are pairwise identical.An analogous argument applies if the initial state is asuperposition of only odd numbers of fermions, or if mis even, and the proof therefore applies without restric-tion.

Discussion. Within quantum information theoryand quantum computation, discussing problems in anabstract context has proven to be very useful. However,when attempting a similar approach to a quantuminformation theory based on a fermionic Fock spaceone encounters difficulties. As we have shown, anunrestricted fermionic model features pure states withnon-symmetric bipartitions. That is, pairs of reducedstates across bipartitions need not have the samespectra, which is very problematic for the definitionof entanglement. As we have shown, this problem isremoved, when superpositions of even and odd numbersof fermions are forbidden. Nonetheless, the removal of aninconvenience appears to be a rather weak justificationfor the introduction of a restriction of generality withinthe abstract model. On the other hand, when placingthe fermionic system within the physical frameworkof a relativistic quantum field theory, the SSR followsnaturally from the requirements of Lorentz invarianceand the constraints of signalling.

We hence argue that, in contrast to bosonic or qudit-based variants, any fermionic quantum informationtheory must be seen as (part of) a relativistic quantumfield theory. This strong hint at the inseparability ofquantum information theory and the theory of relativityis rather surprising, but may provide deeper insightinto constructions of quantum theory based on informa-tional principles, see, e.g., Refs. [6–8]. Moreover, ourresults provide fresh insight into the debate of entan-glement in systems of indistinguishable particles (see,e.g., [19, 20]) in general, and questions of entanglementbetween fermionic modes [23] specifically. Finally, itwill be of significant interest to see to what extentsimulations of fermions, for instance in superconductingmaterials [24], or in graphene [25, 26], can capture thebehaviour of superpositions of different particle numbers.

Acknowledgements. We are grateful to Sergey Filippovfor valuable discussions, and for bringing the problemof non-symmetric pure state marginals to our attention.We are also grateful to Vedran Dunjko, Geza Giedke,Michalis Skotiniotis, and Sterling Archer for their en-lightening comments. We acknowledge funding by theAustrian Science Fund (FWF) through the SFB FoQuS:F4012.

5

Appendix: Superselected fermionic pure stateinequivalent to qubits state

Here, we shall present an example for a pure stateof 4 fermionic modes that satisfies the superselection rule(SSR) that forbids superpositions of even and odd num-bers of fermions. We explicitly compute the marginals ofthis state and show that the subsystem spectra match forany bipartition. We further prove that, nonetheless, thisstate and its marginals do not admit a consistent map-ping to a 4-qubit state (and its marginals). The most gen-eral pure state for four fermionic modes, labelled 1, 2, 3, 4,that only contains even numbers of excitations, may bewritten as

||ψ(4)even 〉〉 = α0 ||0〉〉 + α12 ||11 〉〉||12 〉〉 + α13 ||11 〉〉||13 〉〉

+ α14 ||11 〉〉||14 〉〉 + α23 ||12 〉〉||13 〉〉

+ α24 ||12 〉〉||14 〉〉 + α34 ||13 〉〉||14 〉〉

+ α1234 ||11 〉〉||12 〉〉||13 〉〉||14 〉〉 , (A.1)

where |α0|2 + |α12|2 + |α13|2 + |α14|2 + |α23|2 + |α24|2 +|α34|2 + |α1234|2 = 1. First, let us consider the biparti-tions into two subsets containing one and three modes,respectively. For instance, let us consider the bipartition

(1|2, 3, 4). We may decompose the state ||ψ(4)even 〉〉 into

terms containing excitations in mode 1, and those whichdo not contain such terms, i.e., we write

||ψ(4)even 〉〉 = γ1 ||φ1 〉〉 + γ¬1 ||φ¬1 〉〉 , (A.2)

where |γ1|2 + |γ¬1|2 = 1, b†1b1 ||φ1 〉〉 = ||φ1 〉〉, andb1 ||φ¬1 〉〉 = 0. Since 〈〈φ1 ||φ¬1 〉〉 = 0, the marginals withrespect to the bipartition (1|2, 3, 4) are then easily ob-tained as

ρ1 = Tr2,3,4(||ψ(4)

even 〉〉〈〈ψ(4)even ||

)(A.3a)

= |γ1|2 ||11 〉〉〈〈11 || + |γ¬1|2 ||0〉〉〈〈0|| ,

ρ2,3,4 = Tr1(||ψ(4)

even 〉〉〈〈ψ(4)even ||

)(A.3b)

= |γ1|2 ||φ1 〉〉〈〈φ1 || + |γ¬1|2 ||φ¬1 〉〉〈〈φ¬1 || ,

where ||φ1 〉〉〈〈φ1 || = Tr1(||φ1 〉〉〈〈φ1 ||

). Because ||ψ(4)

even 〉〉contains only even numbers of fermions, the same istrue for ||φ1 〉〉 and ||φ¬1 〉〉, whereas ||φ1 〉〉 contains onlyodd numbers of fermions. Hence we may concludethat 〈〈φ1 ||φ¬1 〉〉 = 0, and it is thus easy to see that themarginals have the same spectrum. The same argumentgoes through for the bipartitions (2|1, 3, 4), (3|1, 2, , 4)and (4|1, 2, 3).

Let us now turn to the bipartitions into pairs of modes,

starting with (1, 2|3, 4). For the marginals we find

ρ1,2 = Tr3,4(||ψ(4)

even 〉〉〈〈ψ(4)even ||

)(A.4a)

= peven1,2 ρeven1,2 + podd1,2 ρodd1,2

ρ3,4 = Tr1,2(||ψ(4)

even 〉〉〈〈ψ(4)even ||

)(A.4b)

= peven3,4 ρeven3,4 + podd3,4 ρodd3,4

where the reduced state density operators in the evenand odd subspaces are given by

peven1,2 ρeven1,2 =(|α0|2 + |α34|2

)||0〉〉〈〈0|| (A.5a)

+(|α12|2 + |α1234|2

)||11 〉〉||12 〉〉〈〈12 ||〈〈11 ||

+[(α0α

∗12 + α34α

∗1234

)||0〉〉〈〈12 ||〈〈11 ||+ H.c.

],

podd1,2 ρodd1,2 =

(|α13|2 + |α14|2

)||11 〉〉〈〈11 || (A.5b)

+(|α23|2 + |α24|2

)||12 〉〉〈〈12 ||

+[(α13α

∗23 + α14α

∗24

)||11 〉〉〈〈12 ||+ H.c.

],

for the subspace of modes 1 and 2. Similarly, for 3 and 4we obtain

peven3,4 ρeven3,4 =(|α0|2 + |α12|2

)||0〉〉〈〈0|| (A.6a)

+(|α34|2 + |α1234|2

)||13 〉〉||14 〉〉〈〈14 ||〈〈13 ||

+[(α0α

∗34 + α12α

∗1234

)||0〉〉〈〈14 ||〈〈13 ||+ H.c.

],

podd3,4 ρodd3,4 =

(|α13|2 + |α23|2

)||13 〉〉〈〈13 || (A.6b)

+(|α14|2 + |α24|2

)||14 〉〉〈〈14 ||

+[(α13α

∗14 + α23α

∗24

)||13 〉〉〈〈14 ||+ H.c.

].

For the superselected state the even and odd subspacesdecouple. We may therefore compare the characteristicpolynomials for the even and odd subspaces separately.A simple computation reveals that both peven1,2 ρeven1,2 andpeven3,4 ρeven3,4 yield the characteristic polynomial

det(peven1,2 ρeven1,2 − λ1

)= det

(peven3,4 ρeven3,4 − λ1

)= λ2 − λ

(|α0|2 + |α12|2 + |α34|2 + |α1234|2

)+ (α0α1234)(α0α1234)∗ + (α12α34)(α12α34)∗

− (α0α1234)(α12α34)∗ − (α0α1234)∗(α12α34) . (A.7)

Similarly, for the odd subspace we find

det(podd1,2 ρ

odd1,2 − λ1

)= det

(podd3,4 ρ

odd3,4 − λ1

)= λ2 − λ

(|α13|2 + |α14|2 + |α23|2 + |α24|2

)+ (α13α24)(α13α24)∗ + (α14α23)(α14α23)∗

− (α13α24)(α14α23)∗ − (α13α24)∗(α14α23) . (A.8)

6

We hence find that the eigenvalues of the marginals ρ1,2and ρ3,4 coincide. Some straightforward computationalong the same lines confirm that this is also the casefor the bipartitions (1, 3|2, 4) and (1, 4|2, 3). Now, theinteresting aspect of this insight pertains to the fact thatthe 4-mode system may not be consistently mapped to a4-qubit state. For the latter, the matching subsystemspectra would be guaranteed by the Schmidt decomposi-tion. Recall that the consistency conditions (see Ref. [5])for partial tracing demand that the numerical value ofthe expectation values of any “local” operator (in thesense of mode-subspaces) is independent of being evalu-ated for the overall state, or for the corresponding localreduced state. These consistency conditions then fix therelative signs of different contributions from matrix el-ements of the total state to the matrix elements of thereduced states. For the example state at hand, the result-ing off-diagonal matrix elements of all 2-mode vs. 2-modebipartitions are collected in Table I.

Now, one wonders, whether the pure state ||ψ(4)even 〉〉

and its marginals can be faithfully represented as a

4-qubit state |ψ(4)even 〉 ∈ (C2)⊗4. Here, we will call

such a mapping faithful, if all the diagonal matrix ele-

ments of |ψ(4)even 〉〈ψ(4)

even | and its marginals with respect tothe computational basis match the diagonal elements of

||ψ(4)even 〉〉〈〈ψ(4)

even || with respect to the Fock basis. For theoff-diagonal elements we impose a slightly weaker con-dition, i.e., that the absolute values of the off-diagonals(with respect to the respective bases) match. This corre-sponds to demanding that measurements in the Fock ba-sis are reproduced, and that the marginals have the samespectra. These conditions imply that a faithful mappingfrom the Fock basis of four fermionic modes to the com-putational basis of four qubits must be of the form

||0〉〉 7→ eiφ0 |0000 〉 , (A.9a)

||11 〉〉||12 〉〉 7→ eiφ12 |1100 〉 , (A.9b)

||11 〉〉||13 〉〉 7→ eiφ13 |1010 〉 , (A.9c)

||11 〉〉||14 〉〉 7→ eiφ14 |1001 〉 , (A.9d)

||12 〉〉||13 〉〉 7→ eiφ23 |0110 〉 , (A.9e)

||12 〉〉||14 〉〉 7→ eiφ24 |0101 〉 , (A.9f)

||13 〉〉||14 〉〉 7→ eiφ34 |0011 〉 , (A.9g)

||11 〉〉||12 〉〉 ||13 〉〉||14 〉〉 7→ eiφ1234 |1111 〉 . (A.9h)

Performing the mapping of (A.9) for the state

of Eq. (A.1), i.e., ||ψ(4)even 〉〉 7→ |ψ(4)

even 〉, and tak-ing the partial traces for the qubits as usual,we obtain the off-diagonal elements that are tobe compared with those in Table I. For instance,

comparing | 〈0000 |Tr3,4(|ψ(4)

even 〉〈ψ(4)even |

)|1100 〉 | with

| 〈〈0|| ρ1,2 ||11 〉〉||12 〉〉 | we get the condition

|ei(φ0−φ12)α0α∗12 + ei(φ34−φ1234)α34α

∗1234|

= |α0α∗12 + α34α

∗1234| . (A.10)

i, j 〈〈0 || ρi,j ||1i 〉〉||1j 〉〉 〈〈1i || ρi,j ||1j 〉〉

1, 2 α0α∗12 + α34α

∗1234 α13α

∗23 + α14α

∗24

3, 4 α0α∗34 + α12α

∗1234 α13α

∗14 + α23α

∗24

1, 3 α0α∗13 − α24α

∗1234 α14α

∗34 − α12α

∗23

2, 4 α0α∗24 − α13α

∗1234 α12α

∗14 − α23α

∗34

1, 4 α0α∗14 + α23α

∗1234 −α12α

∗24 − α13α

∗34

2, 3 α0α∗23 + α14α

∗1234 α12α

∗13 + α24α

∗34

TABLE I. The off-diagonal matrix elements of the reductions

ρi,j = Tr¬i,j(||ψ(4)

even 〉〉〈〈ψ(4)even ||

)are shown.

Since this must hold independently of the values of α0,α12, α34, and α1234, we arrive at

φ12 + φ34 − φ0 − φ1234 = 2n1 π , (A.11)

where n1 ∈ Z. Similarly, the other off-diagonals fromTable I provide the conditions

φ13 + φ24 − φ0 − φ1234 = (2n2 + 1)π , (A.12a)

φ14 + φ23 − φ0 − φ1234 = 2n3 π , (A.12b)

φ23 + φ14 − φ13 − φ24 = 2n4 π , (A.12c)

φ12 + φ34 − φ14 − φ23 = (2n5 + 1)π , (A.12d)

φ13 + φ24 − φ12 − φ34 = 2n6 π , (A.12e)

with n2,3,4,5,6 ∈ Z. These conditions cannot all be met atthe same time. This can be seen, for instance, by com-bining (A.11) with (A.12b), and comparing to (A.12d),which results in

2(n1 − n3) = (2n5 + 1) , (A.13)

which cannot be satisfied, since the left-hand side is aneven integer for all n1, n3 ∈ Z, while the right-hand side isan odd integer for all n5 ∈ Z. We hence conclude that thestate of Eq. (A.1) cannot be consistently mapped to a 4-qubit state, even though it satisfies the SSR, and despitethe fact that for any of its bipartitions the respectivemarginals have the same spectra.

7

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