Finding decompositions of a class of separable states

We consider the class of separable states which admit a decomposition ∑iAi⊗Bi with the Bi’s having independent images. We give a simple intrinsic characterization of this class of states. Given a density matrix in this class, we construct such a decomposition, which can be chosen so that the Ai’s are distinct with unit trace, and then the decomposition is unique. We relate this to the facial structure of the set of separable states.The states investigated include a class that corresponds (under the Choi–Jamiołkowski isomorphism) to the quantum channels called quantum-classical and classical-quantum by Holevo.