Local Distinguishability of Orthogonal Quantum States and Generators of SU(N)

We investigate the possibility of distinguishing a set of mutually orthogonal multipartite quantum states by local operations and classical communication (LOCC). We connect this problem with generators of SU(N) and present a new condition that is necessary for a set of orthogonal states to be locally distinguishable. We show that even in multipartite cases there exists a systematic way to check whether the presented condition is satisfied for a given set of orthogonal states. Based on the proposed checking method, we find that LOCC cannot distinguish three mutually orthogonal states in which two of them are GHZ-like states.