Unique decompositions, faces, and automorphisms of separable states

Let Sk be the set of separable states on B(C ⊗ C) admitting a representation as a convex combination of k pure product states, or fewer. If m > 1, n > 1, and k ≤ max (m,n), we show that Sk admits a subset Vk such that Vk is dense and open in Sk, and such that each state in Vk has a unique decomposition as a convex combination of pure product states, and we describe all possible convex decompositions for a set of separable states that properly contains Vk. In both cases we describe the associated faces of the space of separable states, which in the first case are simplexes, and in the second case are direct convex sums of faces that are isomorphic to state spaces of full matrix algebras. As an application of these results, we characterize all affine automorphisms of the convex set of separable states, and all automorphisms of the state space of B(C⊗C) that preserve entanglement and separability.