Optimal conclusive discrimination of two nonorthogonal pure product multipartite states through local operations

Abstract

We consider one copy of a quantum system prepared in one of two non-orthogonal pure product states of multipartite distributed among separated parties. We show that there exist protocols which achieve optimal conclusive discrimination by means of local operations and classical communications(LOCC) as good as by global measurements. Also, we show a protocol which minimizes the average number of local operations. Our result implies that two pure product multipartite states might not have the non-local property although more than two can have. 03.65.It is a key point that it is impossible to discriminate perfectly between non-orthogonal quantum states if only one copy is provided. Generally, orthogonal states may be distinguished with certainty by means of global operations since quantum information of orthogonality may be encoded in entanglement which may not be extracted by LOCC operations. Recently, Walgate et.al.[1] demonstrated that any two orthogonal multipartite pure states can be distinguished perfectly by only LOCC operations. Virmani et.al. [2] utilized their result to show that optimal discrimination of two non-orthogonal pure states can also be obtained by LOCC in the sense of inclusive discrimination. They also mentioned conclusive discrimination of certain regimes. Bennett et.al. [3] demonstrated that there exist bases of orthogonal product pure states which can not be locally reliably distinguished despite the fact that each state in the basis contains no entanglement. So product pure states may also have non-local property. In this note, we consider the problem of conclusive discrimination of two non-orthogonal product pure states by LOCC operations and show that the optimal disrimination achieved by global measurements can also be obtained by LOCC. This may implies that two product pure multipartite states might not have non-local property.The problem of identifying two non-orthogonal states has been considered by Ivanovic [4], Dieks [5], Peres [6], Jaeger and Shimony [7] by global measurements. Since we utilize their results especially Jaeger and Shimony's, we recall their results in brief and point out some properties which is implicit in their results. Suppose one copy of quantum state prepared as |p > or |q >, generally non-orthogonal, occurring with probabilities r and s respectively, where r + s = 1. The conclusive discrimination means that our measurement on the copy gives three outcomes which allow us to determine the prior state is |p > or |q > with certainty or "don't know". The optimization of conclusive discrimination is to obtain the maximal probability of decisive outcomes.The procedures suggested by Dieks [5], Peres [6], Jaeger and Shimony [7] are the following: prepare an auxiliary system in a state |s 0 > and choose a unitary evolution which yieldswhere < p 1 |q 1 >= 0, < s 1 |s 2 >= 0 and |q 2 > is identical with |p 2 > except for a phase factor. The measurement on the auxiliary system gives two outcomes: if |s 1 > occurs, a measurement between |p 1 > and |q 1 > gives the prior s...

Asymptotically perfect discrimination in the local-operation-and-classical-communication paradigm

“…Ref. [7][8][9][10][11]) or to the more general class of stochastic LOCC measurements (or separable measurements [17]), cf. e.g.…”

Quantum nonlocality without entanglement depending on nonzero prior probabilities in optimal unambiguous discrimination

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“…Our result means that sets of linearly independent product states can be classified into three types in terms of optimal UD: Type I where NLWE does not occur regardless of nonzero prior probabilities (e.g., two multipartite pure states 6 , 32 , 46 – 48 ), Type II where NLWE occurs regardless of nonzero prior probabilities (e.g., domino states 5 , indistinguishable product basis 44 , and Example 2 ), and Type III where NLWE occurs depending on nonzero prior probabilities (e.g. Example 1 ).…”

“…The optimization of UD is to minimize the probability of obtaining inconclusive results. Whereas optimal UD of two multipartite pure states can always be achieved using LOCC 32 , 46 – 48 , optimal UD of more than two multipartite states is not in general. A typical example is “indistinguishable product basis” presented by Duan et al 44 , in which all states cannot be unambiguously discriminated using LOCC.…”

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