Abstract. Let S k be the set of separable states on B(C m ⊗ C n ) 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 S k admits a subset V k such that V k is dense and open in S k , and such that each state in V k 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 V k . 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 m ⊗ C n ) that preserve entanglement and separability.

Abstract. Any continuous, transitive, piecewise monotonic map is determined up to a binary choice by its dimension module with the associated finite sequence of generators. The dimension module by itself determines the topological entropy of any transitive piecewise monotonic map, and determines any transitive unimodal map up to conjugacy. For a transitive piecewise monotonic map which is not essentially injective, the associated dimension group is a direct sum of simple dimension groups, each with a unique state.

We consider the class of separable states which admit a decomposition i A i ⊗ B i with the B i '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 A i '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.

It was conjectured that the structural physical approximation (SPA) of an optimal entanglement witness is separable (or equivalently, that the SPA of an optimal positive map is entanglement breaking). This conjecture was disproved, first for indecomposable maps and more recently for decomposable maps. The arguments in both cases are sketched along with important related results. This review includes background material on topics including entanglement witnesses, optimality, duality of cones, decomposability, and the statement and motivation for the SPA conjecture so that it should be accessible for a broad audience.

Abstract. For each piecewise monotonic map τ of [0, 1], we associate a pair of C*-algebras F τ and O τ and calculate their K-groups. The algebra F τ is an AI-algebra. We characterize when F τ and O τ are simple. In those cases, F τ has a unique trace, and O τ is purely infinite with a unique KMS state. In the case that τ is Markov, these algebras include the Cuntz-Krieger algebras O A , and the associated AF-algebras F A . Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and β-transformations. For the case of interval exchange maps and of β-transformations, the C*-algebra O τ coincides with the algebras defined by Putnam and Katayama-Matsumoto-Watatani, respectively.