Construnctions of LOCC indistinguishable set of generalized Bell states (1805.09211v1)

Abstract: In this paper, we mainly consider the local indistinguishability of the set of mutually orthogonal bipartite generalized Bell states (GBSs). We construct small sets of GBSs with cardinality smaller than $d$ which are not distinguished by one-way local operations and classical communication (1-LOCC) in $d\otimes d$. The constructions, based on linear system and Vandermonde matrix, is simple and effective. The results give a unified upper bound for the minimum cardinality of 1-LOCC indistinguishable set of GBSs, and greatly improve previous results in [Zhang \emph{et al.}, Phys. Rev. A 91, 012329 (2015); Wang \emph{et al.}, Quantum Inf. Process. 15, 1661 (2016)]. The case that $d$ is odd of the results also shows that the set of 4 GBSs in $5\otimes 5$ in [Fan, Phys. Rev. A 75, 014305 (2007)] is indeed a 1-LOCC indistinguishable set which can not be distinguished by Fan's method.