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Physical Review A

Distinguishing bipartite states by local operations and classical communication

Heng Fan

Published Jan 31, 2007• 56 citations
Publisher

Abstract

We present a general method for distinguishing bipartite states by local operations and classical communication. It unifies several available methods together. This method is applicable for arbitrary bipartite states, i.e., it works not only for entangled states but also for separable states. We also study the local distinguishability for maximally entangled states which are in an arbitrary basis.

DOI
10.1103/physreva.75.014305

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We present a general method for distinguishing bipartite states by local operations and classical communication. It unifies several available methods together. This method is applicable for arbitrary bipartite states, i.e., it works not only for entangled states but also for separable states. We also study the local distinguishability for maximally entangled states which are in an arbitrary basis.
In the topic of perfect local distinguishability of orthogonal multipartite quantum states, most results obtained so far pertain to bipartite systems whose subsystems are of specific dimensions. In contrast very few results for bipartite systems whose subsystems are of arbitrary dimensions, are known. This is because a rich variety of (algebraic or geometric) structure is exhibited by different sets of orthogonal states owing to which it is difficult to associate some common property underlying them all, i.e., a common property that would play a crucial role in the local distinguishability of these states. In this Rapid Communication, I propose a framework for the distinguishability by one-way local operations and classical communication (1-LOCC) of sets of orthogonal bipartite states in a ${d}_{A}\ensuremath{\bigotimes}{d}_{B}$ bipartite system, where dA,dB{d}_{A},{d}_{B}dA​,dB​ are the dimensions of both subsytems, labeled as AAA and BBB. I show that if the ithi\mathrm{th}ith party (where i=A,Bi=A,Bi=A,B) can initiate a 1-LOCC protocol to perfectly distinguish among a set of orthogonal bipartite states, then the information of the existence of such a 1-LOCC protocol lies in a subspace of ${d}_{i}\ifmmode\times\else\texttimes\fi{}{d}_{i}$ Hermitian matrices, denoted by ${\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$, and that the method to extract this information (of the existence of this 1-LOCC protocol) from ${\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$ depends on the value of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$. In this way one can give sweeping results for the 1-LOCC (in)distinguishability of all sets of orthogonal bipartite states corresponding to certain values of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$. Thus I propose that the value of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$ gives the common underlying property based on which sweeping results for the 1-LOCC (in)distinguishability of orthogonal bipartite quantum states can be made.
Local distinguishability of quantum states is a basic problem in quantum information theory. For an arbitrary group of entangled states in d ⊗ d bipartite system, Singal presented a subspace related to these states, and then he determined that the group of states can be distinguished by one-way local operations and classical communication(1-LOCC) if this subspace contains a maximally Abelian subspace. In this paper according to Singal's results, we present an algorithm to distinguish orthogonal bipartite states in d ⊗ d system based on 1-LOCC; we gives a sufficient condition to judge whether generalized orthogonal quantum states in 4 ⊗ 4 can be distinguished. Finally, we show an corresponding example to verify the conclusion.
Two types of results are presented for distinguishing pure bipartite quantum states using local operations and classical communications. We examine sets of states that can be perfectly distinguished, in particular showing that any three orthogonal maximally entangled states in C3⊗C3 form such a set. In cases where orthogonal states cannot be distinguished, we obtain upper bounds for the probability of error using LOCC taken over all sets of k orthogonal states in Cn⊗Cm. In the process of proving these bounds, we identify some sets of orthogonal states for which perfect distinguishability is not possible.
It has recently been shown that the work extractable from correlated bipartite quantum systems under an appropriate protocol can be used to distinguish entanglement from classical correlation. A natural question is now whether it can be generalized to multipartite systems. In this paper, we devise a protocol to distinguish the GHZ, the W, and separable states in terms of the thermodynamically extractable work under local operations and classical communication, and compare the results with those obtained from Mermin's inequalities.