Isomorphism of nonlocal sets of orthogonal product states in bipartite quantum systems

To characterize nonlocal sets of orthogonal product states (OPSs) in a given bipartite quantum system, we propose two new concepts, i.e., orthogonal relation matrix of a set of OPSs and isomorphism of different sets of OPSs. In ℂ3⊗ℂ3 quantum system, we find that two sets of OPSs, which are isomorphic to a nonlocal set of OPSs, are locally indistinguishable. In ℂ5⊗ℂ5 quantum system, we construct a new nonlocal set of OPSs according to the orthogonal relation matrix of a set of OPSs that cannot be perfectly distinguished by local operations and classical communication (LOCC). These results show a phenomenon that a set of OPSs is locally indistinguishable if its orthogonal relation matrix is equivalent to the orthogonal relation matrix of another set of OPSs that cannot be perfectly distinguished by LOCC in a given quantum system, and the orthogonal relation matrix of a nonlocal set of OPSs can be used to construct a new nonlocal set of OPSs. On the other hand, we give the geometric interpretation of nonlocal sets with isomorphism relation by orthogonal graphs (OGs). Our results, which can be used to classify nonlocal sets of bipartite OPSs, improve the theory of local distinguishability and deepen people’s understanding of the structures of nonlocal sets.