Orthogonality and distinguishability: Criterion for local distinguishability of arbitrary orthogonal states

We consider the relation between the orthogonality and the distinguishability of a set of arbitrary states (including multipartite states). It is shown that if a set of arbitrary states can be distinguished by local operations and classical communication (LOCC), each of the states can be written as a linear combination of product vectors such that all product vectors of one of the states are orthogonal to the other states. With this result we then prove a simple necessary condition for LOCC distinguishability of a class of orthogonal states. These conclusions may be useful in discussing the distinguishability of orthogonal quantum states further, understanding the essence of nonlocality and discussing the distillation of entanglement.