Local distinguishability of generalized Bell states

Abstract

We investigate the distinguishability of orthogonal generalized Bell states (GBSs) in $$d\otimes d$$ system by local operations and classical communication (LOCC), where d is a prime. We show that |S| is no more than $$d+1$$ for any l GBSs, i.e., $$|S|\le d+1$$ , where S is maximal set which is composed of pairwise noncommuting pairs in $${\varDelta } U$$ . If $$|S|\le d$$ , 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 $$|S|=d+1$$ , we do not find the answer. We conjecture that any l GBSs cannot be distinguished by one-way LOCC if $$|S|=d+1$$ . If this conjecture is right, the problem about distinguishability of GBSs with one-way LOCC is completely solved in $$d\otimes d$$ .