Boundary of the set of separable states

Motivated by the separability problem in quantum systems 2 ⊗ 4, 3 ⊗ 3 and 2 ⊗ 2 ⊗ 2, we study the maximal (proper) faces of the convex body, S₁, of normalized separable states in an arbitrary quantum system with finite-dimensional Hilbert space H = H₁ ⊗ H₂ ⊗ ··· ⊗ H_n. To any subspace V ⊆H, we associate a face F_V of S₁ consisting of all states ρ ∈ S₁ whose range is contained in V. We prove that F_V is a maximal face if and only if V is a hyperplane. If V = | ψ〉 where |ψ〉 is a product vector, we prove that Dim F_V = d² - 1 - Π(2d_i - 1), where d_i = Dim H_i and d = Πd_i. We classify the maximal faces of S₁ in the cases 2 ⊗ 2 and 2 ⊗ 3. In particular, we show that the minimum and the maximum dimension of maximal faces is 6 and 8 for 2 ⊗ 2, and 20 and 24 for 2 ⊗ 3. The boundary, ∂S₁, of S₁ is the union of all maximal faces. When d > 6, it is easy to show that there exist full states on ∂S₁, i.e. states ρ ∈ ∂S₁ such that all partial transposes of ρ (including ρ itself) have rank d. Ha and Kye have recently constructed explicit such states in 2 ⊗ 4 and 3 ⊗ 3. In the latter case, they have also constructed a remarkable family of faces, depending on a real parameter b > 0, b ≠ 1. Each face in the family is a nine-dimensional simplex, and any interior point of the face is a full state. We construct suitable optimal entanglement witnesses for these faces and analyse the three limiting cases b = 0,1, ∞.