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Tripartite Entanglement and Quantum Relative Entropy

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Tripartite Entanglement and Quantum Relative Entropy Tripartite entanglement and quantum relative entropy E. F. Galv˜ao1, M. B. Plenio2 and S. Virmani2 1 Centre for Quantum Computation, Clarendon Laboratory, Univ. of Oxford, Oxford OX1 3PU,UK 2 Optics Section, The Blackett Laboratory, Imperial College, London SW7 2BW, UK We establish relations between tripartite pure state entanglement and additivity properties of the bipartite relative entropy of entanglement. Our results pertain to the asymptotic limit of local ma- nipulations on a large number of copies of the state. We show that additivity of the relative entropy would imply that there are at least two inequivalent types of asymptotic tripartite entanglement. The methods used include the application of some useful lemmas that enable us to analytically calculate the relative entropy for some classes of bipartite states. I. INTRODUCTION glement, or in other words whether the set G3 = EPR AB ; EPR AC ; EPR BC ; GHZ ABC {| i | i | i | i (1)} In recent years the theory of quantum information and is an MREGS remained unanswered in [3] and [4]. The entanglement processing has developed rapidly. In the conjecture that G3 as given in eq. (1) forms an MREGS process our perception of entanglement has changed sig- has been supported by work showing that reversible nificantly. Entanglement used to be regarded just as a LOCC on this set yield Schmidt decomposable states surprising manifestation of the non-locality of quantum [3] and also a family of states discussed in [5]. Very mechanics, but today it is considered as a resource that recently, however, Wu and Zhang [6] have shown that can be exploited to implement novel quantum informa- without other effects [7], not all four-partite states can tion processing tasks at spatially separated locations [1]. be reversibly built using LOCC on the set of eleven max- As a resource, entanglement can appear in many differ- imally entangled states of two, three and four parties. ent forms and may not be available in the specific form Nevertheless, the structure of the MREGS for tripartite necessary for the chosen task. It is therefore natural systems remains unknown. to tackle the problem of the interconversion of different In addition to the developments just described, some forms of entanglement using local operations and classical relations have been established [4] between multipartite communication only (LOCC). The local concentration of pure state entanglement and a bipartite entanglement pure bipartite entanglement has already been considered measure known as the Relative Entropy of Entanglement in the asymptotic limit, i.e. when large numbers of entan- [8–10]. In this paper we strengthen these relations fur- gled pairs are available [2]. In this limit it was shown that ther, obtaining new results relating the additivity of the any partially entangled state can be reversibly converted relative entropy and the structure of the MREGS for tri- into a smaller number of maximally entangled singlet or partite states. In section II we summarize the results of EPR states. This remarkable result demonstrates that [4] and present a number of useful Lemmas that allow the entanglement of any pure bipartite state is essentially us to exploit symmetries of a quantum state to allow the equivalent to that of the singlet state. One can therefore analytic computation of the relative entropy of entangle- say that the set G2 = EPR AB containing an EPR ment. In section III we assume the working hypothe- pair between systems A{|and Bi is} a minimal reversible sis that the set G3 is an MREGS and derive a series of entanglement generating set (MREGS) for all bipartite consequences that would follow; in particular, we show pure states [3]. that the relative entropy of entanglement (with respect It is natural to ask whether there are more inequivalent to separable states) would need to be subadditive. Since forms of entanglement when one considers multi-partite to date there has been no evidence of such subadditiv- pure state entanglement in the asymptotic limit; in other ity, in section IV we adopt the alternative hypothesis of words, the problem is that of identifying an MREGS additivity and explore the consequences, in particular we for multi-partite systems. Recently it has been shown discuss implications for the cardinality of the tripartite that indeed GHZ states are inequivalent to EPR states MREGS. In section V we present some final remarks. in the asymptotic limit , i.e. there is no asymptotically reversible local procedure that allows the conversion of EPR states into GHZ states [4]. Therefore, a MREGS for II. RELATIVE ENTROPY, TRIPARTITE tripartite systems must contain at least the GHZ state ENTANGLEMENT AND SYMMETRIES and the three possible EPR’s between any two of the parties. However, the question as to whether EPR states In this section we introduce some of the notation that and GHZ states form the only kinds of tripartite entan- we will use in the remainder of the article. In the first 1 subsection we summarize the results of [4] and in the sec- that the following relationships must hold: ond subsection we present some useful Lemmas that we reg will employ later on. EX (ρij )=sij (3) S(ρA)=g + sAB + sAC (4) S(ρB)=g + sAB + sBC (5) A. Basic notation and concepts S(ρC)=g + sAC + sBC; (6) where S(ρ ) represents the Von Neumann entropy of the The relative entropy of ρ with respect to any σ is de- i reduced density matrix of party i [14]. fined as It is an open question whether all tripartite states sat- isfy the equations (3-6). Any counterexample would be S(ρ σ):=tr(ρlogρ) tr (ρlogσ)(2) k − a state which cannot be generated reversibly from the set G , representing a new kind of asymptotic tripar- This allows us to define what we mean by the Relative 3 tite entanglement. Unfortunately there are no known Entropy of Entanglement and Additivity: reg general techniques for calculating EX (ρ). In [10], 1) Relative Entropy of Entanglement: some sufficient conditions were presented under which reg For bipartite systems this entanglement measure can take EPPT(ρ)=EPPT(ρ). However, all states we investigated three different forms, ES,EPPT or END [11]. They are which obeyed these conditions also satisfy eqs.(3-6). De- defined as spite this, we were able to obtain some progress in es- tablishing relations between additivity questions and the EX (ρAB):= min S (ρAB σAB) structure of the MREGS for tripartite pure states. In σAB D(X) k ∈ particular, we present classes of states which are poten- tial candidates for violating relations (3-6). where X = S;PPT;ND and the minimum is taken over the set D of separable(S), Non-Distillable(ND), or Posi- tive Partial Transpose(PPT) density matrices [8–10,12]. B. Symmetries and continuity These measures can further be ‘regularised’ for use in discussions involving asymptotic manipulations: In this subsection we prove a number of useful Lemmas reg n that simplify the computation of the relative entropy of EX (ρAB ) := limn (1=n)minS ρAB⊗ σAB : →∞ σAB D(X) k ∈ entanglement for states that possess symmetries. In ad- dition, we state a Lemma due to Donald and Horodecki It is important to note that in the case that ρ is either concerning the continuity of the relative entropy of en- a pure state or a separable state then all the measures tanglement. reg are equal: ES(ρ)=EPPT(ρ)=END(ρ)=ES (ρ)= We begin by recalling a Lemma by Rains [10] which reg reg EPPT(ρ)=END(ρ). enables us to use local symmetries of the state ρAB to 2) Additivity: There are two major types of additivity narrow down the possible set of optimal states. Then we which will concern us in this paper: extend this Lemma to non-local symmetry operations. a) If an entanglement measure E satisfies Ereg(ρ)=E(ρ) Lemma 1 [10] If a bipartite density matrix is invariant we will say that E is an asymptotically additive measure; under a sub-group of local unitary transformations, then b) If an entanglement measure E satisfies for all ρ1,ρ2 the optimal PPT state can also be chosen to be invariant the relation E(ρ1 ρ2)=E(ρ1)+E(ρ2)thenwesay under the same sub-group. ⊗ that E is a fully additive measure. Although the proof can be found in [10] we present it The connection between the relative entropy of entan- here to clarify how this theorem can be generalized to glement and multipartite entanglement was first pointed non-local symmetry groups. out in [4], where it was shown that if two multiparty pure Proof Let there be a bipartite density matrix ρ which states can be reversibly interconverted then the relative is invariant under a sub-group of local transformations G entropy of entanglement must remain constant for any = Ui Vi , with an optimal PPT state σ. For simplicity, two parties i; j. This remarkable result can be used to de- let{ us⊗ assume} that the group is discrete (the generaliza- rive contraints that must hold if the set G3 is an MREGS tion to continuous groups is straightforward [10]). Then for tripartite pure states. In particular, suppose that we ES(ρ)isgivenby reversibly and asymptotically wish to create a tripartite pure state ΨABC between parties A, B and C, and that ES(ρ)=S(ρ σ)=S(ρ Ui ViσU† V †); (7) | i || || ⊗ i ⊗ i per output copy of ΨABC we will use g GHZ states and | i sij EPR pairs between parties i and j. Then, denoting due to the invariance of the relative entropy under uni- the reduced density matrices of parties i,j by ρij , we find tary transformations and the invariance of ρ under G.
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