  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.
