Local distinguishability of four multipartite orthogonal product states

In this paper, we give the local distinguishability of four orthogonal product states (OPSs) on multipartite and high-dimension systems. To characterize the different orthogonal relationships between four multipartite OPSs, a new concept, i.e., the vector of the numbers of pairwise orthogonal relations, is proposed. We classify four tripartite OPSs, any two of which are orthogonal only on one subsystem, into three categories by using the vectors of the numbers of pairwise orthogonal relations. We prove that any four tripartite OPSs belonging to two of the three categories can be perfectly distinguished by local operations and classical communication (LOCC). Meanwhile, we find an interesting phenomenon that two sets of four tripartite OPSs belonging to the remaining one of the three categories have different local distinguishability even if they have the same orthogonal graph. This means that the orthogonal graph is not a sufficient condition for determining the local distinguishability of a set of OPSs. Additionally, for the remaining category, we specify the conditions under which four tripartite OPSs can be locally distinguished and the conditions under which four tripartite OPSs cannot be perfectly distinguished by LOCC. On the other hand, we prove that any four tripartite OPSs with seven or more pairwise orthogonal relations can be perfectly distinguished by LOCC. Furthermore, we prove that any four OPSs with six or more pairwise orthogonal relations can be perfectly distinguished by LOCC on four-, five-, and six-partite systems.