Construnctions of LOCC indistinguishable set of generalized Bell states

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.