Local distinguishability of generalized Bell states

We investigate the distinguishability of orthogonal generalized Bell states (GBSs) in d circle times d system by local operations and classical communication (LOCC), where d is a prime. We show that vertical bar S vertical bar is no more than d + 1 for any l GBSs, i.e., vertical bar S vertical bar <= d + 1, where S is maximal set which is composed of pairwise noncommuting pairs in Delta U. If vertical bar S vertical bar <= d, then the l GBSs can be distinguished by LOCC according to our main Theorem. Compared with the results (Fan in Phys Rev Lett 92:177905, 2004; Tian et al. in Phys Rev A 92:042320, 2015), our result is more general. It can determine local distinguishability of l(> k) GBSs, where k is the number of GBSs in Fan's and Tian's results. Only for vertical bar S vertical bar = d + 1, we do not find the answer. We conjecture that any l GBSs cannot be distinguished by one-way LOCC if vertical bar S vertical bar = d + 1. If this conjecture is right, the problem about distinguishability of GBSs with one-way LOCC is completely solved in d circle times d.