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arXiv:quant-ph/0106148 · Physical Review Letters

Distinguishability of Bell States

Sibasish Ghosh A, Guruprasad Kar, Anirban Roy, Aditi Sen, Ujjwal Sen

Published Dec 13, 2001• 258 citations
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Abstract

More than two multipartite orthogonal states cannot always be discriminated if only local operations and classical communication (LOCC) are allowed. We show that four Bell states cannot be discriminated by LOCC, even probabilistically, using the separability properties of a four-party unlockable bound entangled state. Using an existing inequality among the measures of entanglement, we show that any three Bell states cannot be discriminated with certainty by LOCC. Exploiting the inequality, we calculate the distillable entanglement of a certain class of 4 multiply sign in circle 4 mixed states.

DOI
10.1103/physrevlett.87.277902

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More than two multipartite orthogonal states cannot always be discriminated if only local operations and classical communication (LOCC) are allowed. We show that four Bell states cannot be discriminated by LOCC, even probabilistically, using the separability properties of a four-party unlockable bound entangled state. Using an existing inequality among the measures of entanglement, we show that any three Bell states cannot be discriminated with certainty by LOCC. Exploiting the inequality, we calculate the distillable entanglement of a certain class of 4 multiply sign in circle 4 mixed states.
We show that the four states $a|00〉+b|11〉, \overline{b}|00〉\ensuremath{-}\ifmmode \bar{a}\else \={a}\fi{}|11〉, c|01〉+d|10〉,$ and $\overline{d}|01〉\ensuremath{-}\overline{c}|10〉$ cannot be discriminated with certainty if only local operations and classical communication (LOCC) are allowed and if only a single copy is provided, except in the case when they are simply ∣00〉,∣11〉,∣01〉,|00〉, |11〉, |01〉,∣00〉,∣11〉,∣01〉, and ∣10〉|10〉∣10〉 (in which case they are trivially distinguishable with LOCC). We go on to show that there exists a continuous range of values of a,b,c,a, b, c,a,b,c, and d such that even three states among the above four are not locally distinguishable, if only a single copy is provided. The proof follows from the fact that logarithmic negativity is an upper bound of distillable entanglement.
In $2\ensuremath{\bigotimes}2$, more than two orthogonal Bell states with a single copy can never be discriminated with certainty if only local operations and classical communication (LOCC) are allowed. We show here that more than ddd numbers of pairwise orthogonal maximally entangled states in $d\ensuremath{\bigotimes}d$, which are in canonical form, used by Bennett et al. [Phys. Rev. Lett. 70, 1895 (1993)], can never be discriminated with certainty by LOCC, when single copies of the states are provided. Interestingly we show here that all orthogonal maximally entangled states, which are in canonical form, can be discriminated with certainty by LOCC if and only if two copies of each of the states are provided. We provide here a conjecture regarding the highly nontrivial problem of local distinguishability of any ddd or fewer numbers of pairwise orthogonal maximally entangled states in $d\ensuremath{\bigotimes}d$ (in the single copy case).
There exist sets of multipartite orthogonal states which can not be distinguished by local operation and classical communication (LOCC). The simplest example is the set of four Bell states. The problem of characterizing a set in terms of distinguishability is a hard problem in general. For maximally entangled states, some results are known. Local cloning of entangled states by supplying necessary entanglement in the blank state is another idea that was introduced very recently. Interestingly there are sets of orthogonal entangled states which can be distinguished locally but can not be copied by local operation (even if necessary entanglement is supplied in the blank state) contrary to the fact that happens in the case of global operation.
Entangled states can help in quantum-state discrimination by local operations and classical communication (LOCC). For example, a Bell state is necessary (and sufficient) to perfectly discriminate a set of either three or four Bell states by LOCC. In this paper, we consider the task of LOCC discrimination of the states of noisy Bell ensembles, where a given ensemble consists of the states obtained by mixing the Bell states with an arbitrary two-qubit state with nonzero probabilities. It is proved that a Bell state is required for optimal discrimination by LOCC, even though the ensembles do not contain, in general, any maximally entangled state and, in specific instances, any entangled state.
We show that there exist sets of three mutually orthogonal ddd-dimensional maximally entangled states which cannot be perfectly distinguished using one-way local operations and classical communication (LOCC) for arbitrarily large values of ddd. This contrasts with several well-known families of maximally entangled states, for which any three states can be perfectly distinguished. We then show that two-way LOCC is sufficient to distinguish these examples. We also show that any three mutually orthogonal ddd-dimensional maximally entangled states can be perfectly distinguished using measurements with a positive partial transpose (PPT) and can be distinguished with one-way LOCC with high probability. These results circle around the question of whether there exist three maximally entangled states which cannot be distinguished using the full power of LOCC; we discuss possible approaches to answer this question.
Abstract It is an interesting topic how much entanglement resources are necessary to distinguish locally indistinguishable orthogonal quantum states by local operations and classical communication (LOCC). Using only one ebit of entanglement some locally indistinguishable orthogonal product bases can be distinguished by LOCC [2019 Phys. Rev. A 99 012343]. However, it is still an open question whether one ebit of entanglement is sufficient for distinguishing a locally indistinguishable set of orthogonal maximally entangled states. We focus on the generalized Bell states (GBSs), and firstly find that any four GBSs in a 4 ⊗ 4 quantum system are always distinguished by one-way LOCC using one ebit of entanglement. Then we consider a general case, i.e., d ⊗ d quantum system ( d is even), and show that the set L of any GBSs can be distinguished by one-way LOCC using one ebit of entanglement if S A \ S L is nonempty, where S A is the admissible solutions set of L and S L is the genuinely nonadmissible solutions set of L . This criterion is very simple and convenient since only some congruence equations are calculated. Finally, we find a curious and counterintuitive case, i.e., in a 3 ⊗ 3 quantum system there exist four GBSs which cannot be distinguished by one-way LOCC using one ebit of entanglement, although the dimension of the subsystem is less than four. These results can help us better understand the role of entanglement as a resource in local state discrimination problems.