Constructions of one-way LOCC indistinguishable sets of generalized Bell states

In this paper, we mainly consider the local indistinguishability of the set of bipartite generalized Bell states (GBSs). We systematically show constructions of small sets of GBSs with cardinalities greatly smaller than d which are not distinguishable by one-way local operations and classical communication (1-LOCC) in \(d\otimes d\). The constructions, based on linear system and Vandermonde matrix, are different for odd dimensions than for even dimensions. 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 et al. (Phys Rev A 91:012329, 2015) and Wang et al. (Quantum Inf Process 15:1661, 2016). Among others, the case that d is odd of the results 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 cannot be distinguished by Fan’s method, and the minimum cardinality of 1-LOCC indistinguishable set of GBSs in \(7\otimes 7\) is 5. The case that d is even of the results shows that there exists a 1-LOCC indistinguishable set of 18 GBSs in \(100\otimes 100\).