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Chinese Physics B

Deterministically distinguishing a remote Bell state

Zhiguo Zhao, Weimin Peng, Yong-Gang Tan

Published Jan 1, 2011• 1 citation
Publisher

Abstract

It has been proven that, with a single copy provided, the four Bell states cannot be distinguished by local operations and classical communications (LOCC). Traditionally, a Bell basis projective measurement is needed to distinguish the four Bell states, which is usually carried out with a local interference between two particles. This paper presents an interesting protocol that allows two remote parties to distinguish four Bell states deterministically. We prove that our protocol of distinguishing remote Bell states is beyond LOCC.

DOI
10.1088/1674-1056/20/1/010307

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It has been proven that, with a single copy provided, the four Bell states cannot be distinguished by local operations and classical communications (LOCC). Traditionally, a Bell basis projective measurement is needed to distinguish the four Bell states, which is usually carried out with a local interference between two particles. This paper presents an interesting protocol that allows two remote parties to distinguish four Bell states deterministically. We prove that our protocol of distinguishing remote Bell states is beyond LOCC.
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.
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.
Measuring an entangled state of two particles is crucial to many quantum communication protocols. Yet Bell-state distinguishability using a finite apparatus obeying linear evolution and local measurement is theoretically limited. We extend known bounds for Bell-state distinguishability in one and two variables to the general case of entanglement in nnn two-state variables. We show that at most ${2}^{n+1}\ensuremath{-}1$ classes out of 4n{4}^{n}4n hyper-Bell states can be distinguished with one copy of the input state. With two copies, complete distinguishability is possible. We present optimal schemes in each case.
We present a complete characterization for the local distinguishability of orthogonal $2\ensuremath{\bigotimes}3$ pure states except for some special cases of three states. Interestingly, we find there is a large class of four or three states that are indistinguishable by local projective measurements and classical communication (LPCC), but can be perfectly distinguished by LOCC. That indicates the ability of LOCC for discriminating $2\ensuremath{\bigotimes}3$ states is strictly more powerful than that of LPCC, which is strikingly different from the case of multiqubit states. We also show that classical communication plays a crucial role for local distinguishability by constructing a class of $m\ensuremath{\bigotimes}n$ states which require at least $2\phantom{\rule{0.2em}{0ex}}\mathrm{min}{m,n}\ensuremath{-}2$ rounds of classical communication in order to achieve a perfect local discrimination.
It is well known that orthogonal quantum states can be distinguished perfectly. However, if we assume that these orthogonal quantum states are shared by spatially separated parties, the distinguishability of these shared quantum states may be completely different. We show that a set of linearly independent quantum states ${({U}_{m,n}\ensuremath{\bigotimes}I){\ensuremath{\rho}}^{AB}({U}_{m,n}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\bigotimes}I){}}_{m,n=0}^{d\ensuremath{-}1}$, where Um,n{U}_{m,n}Um,n​ are generalized Pauli matrices, cannot be discriminated deterministically or probabilistically by local operations and classical communication. On the other hand, any lll maximally entangled states from this set are locally distinguishable if $l(l\ensuremath{-}1)\ensuremath{\le}2d$. The explicit projecting measurements are obtained to locally discriminate these states. As an example, we show that four Werner states are locally indistinguishable.
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).