Finding out all locally indistinguishable sets of generalized Bell states

TL;DR

This paper completely characterizes when sets of generalized Bell states can be distinguished using local operations in certain bipartite quantum systems, providing new necessary and sufficient conditions and solving the problem for specific dimensions.

Contribution

It introduces new practical conditions for local discrimination of GBS sets and fully solves the problem for systems of dimensions 4 and 5.

Findings

Complete characterization of local distinguishability for 4-GBS sets in c4b4c4b4c4b4c4b4c4b4

List of all locally indistinguishable 4-GBS sets in c4b4c4b4c4b4c4b4c4b4

Necessary and sufficient condition for one-way local discrimination in c5b4c5b4c5b4c5b4c5b4c5b4c5b4c5b4c5b4c5b4

Abstract

In general, for a bipartite quantum system $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ and an integer $k$ such that $4\leq k\le d$,there are few necessary and sufficient conditions for local discrimination of sets of $k$ generalized Bell states (GBSs) and it is difficult to locally distinguish $k$-GBS sets.The purpose of this paper is to completely solve the problem of local discrimination of GBS sets in some bipartite quantum systems.Firstly three practical and effective sufficient conditions are given,Fan$^{,}$s and Wang et al.$^{,}$s results [Phys Rev Lett 92, 177905 (2004); Phys Rev A 99, 022307 (2019)] can be deduced as special cases of these conditions.Secondly in $\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, a necessary and sufficient condition for local discrimination of GBS sets is provided, and a list of all locally indistinguishable 4-GBS sets is provided,and then the problem of local discrimination of GBS sets is completely solved.In $\mathbb{C}^{5}\otimes\mathbb{C}^{5}$, a concise necessary and sufficient condition for one-way local discrimination of GBS sets is obtained,which gives an affirmative answer to the case $d=5$ of the problem proposed by Wang et al.