Construnctions of LOCC indistinguishable set of generalized Bell states
Jiang-Tao Yuan, Cai-Hong Wang, Ying-Hui Yang, Shi-Jiao Geng
TL;DR
The paper addresses the problem of 1-LOCC indistinguishability of generalized Bell states in $d\otimes d$ systems by introducing $f_{GBS}(d)$, the minimum size of a 1-LOCC indistinguishable GBS set, and presenting a practical linear-system based method leveraging Vandermonde matrices. It provides several explicit constructions for odd and even dimensions that yield new, tighter upper bounds on $f_{GBS}(d)$, improving upon prior results and unifying the approach across parity of $d$. Notably, it shows that a 4-state set in $5\otimes5$ is indeed 1-LOCC indistinguishable under this framework and delivers specific bounds such as $f_{GBS}(d)\le (d+3)/2$ for odd $d$ and $f_{GBS}(d)\le 3+\lceil d/4\rceil$ for even $d$, among others. These contributions enhance understanding of the nonlocality of maximally entangled states and offer concrete tools for constructing indistinguishable GBS sets, with open questions remaining on exact values for certain dimensions.
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.