  We give a sufficient condition that an operator sum representation of a
separable quantum channel in terms of product operators is the unique product
representation for that channel, and then provide examples of such channels for
any number of parties. This result has implications for efforts to determine
whether or not a given separable channel can be exactly implemented by local
operations and classical communication. By the Choi-Jamiolkowski isomorphism,
it also translates to a condition for the uniqueness of product state ensembles
representing a given quantum state. These ideas follow from considerations
concerning whether or not a subspace spanned by a given set of product
operators contains at least one additional product operator.
