  Now, the known ensembles of orthogonal states which are distinguishable by
local operators and classical communication (LOCC) satisfy the condition that
the sum of Schmidit numbers of the orthogonal states is not bigger than the
dimensions of the whole space. A natural question is whether an arbitary
ensembles of LOCC-distinguishable orthogonal states satisfies the condition. We
first show that, in this paper, the answer is positive. Then we generalize it
into multipartite systems, and show that a necessary condition for
LOCC-distinguishability of multipartite orthogonal quantum states is that the
sum of the least numbers of the product states (For bipartite system, the least
number of product states is Schmidit number) of the orthogonal states is not
bigger than the dimensions of the Hilbert space of the multipartite system.
This necessary condition is very simple and general, and one can get many cases
of indistinguishability by it. It means that the least number of the product
states acts an important role in distinguishablity of states, and implies that
the least number of the product states may be an good manifestion of quantum
nonlocality in some sense. In fact, entanglement emphases the "amount" of
nonlocality, but the least number of the product states emphases the types of
nonlocality. For example, the known W states and GHZ states have different
least number of the product states, and are different in type.
