  We consider deeply the relation between the orthogonality and the
distinguishability of a set of arbitrary states (including multi-partite
states). It is shown that if a set of arbitrary states can be distinguished by
local operations and classical communication (LOCC), \QTR{it}{\}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.
