  We study the local indistinguishability problem of quantum states. By
introducing an easily calculated quantity, non-commutativity, we present an
criterion which is both necessary and sufficient for the local
indistinguishability of a complete set of pure orthogonal product states. A
constructive distinguishing procedure to obtain the concrete local measurements
and classical communications is given. The non-commutativity of ensembles can
be also used to characterize the quantumness for classical-quantum or
quantum-classical correlated states.
