Boundary of the set of separable states

Abstract

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, S 1 , of normalized separable states in an arbitrary quantum system with finite-dimensional Hilbert space H = H 1 ⊗ H 2 ⊗ ⋯ ⊗ H n . To any subspace V ⊆ H , we associate a face F V of S 1 consisting of all states ρ ∈ S 1 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 = d 2 − 1 − ∏ ( 2 d i − 1 ) , where d i = Dim   H i and d = ∏ d i . We classify the maximal faces of S 1 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, ∂ S 1 , of S 1 is the union of all maximal faces. When d >6, it is easy to show that there exist full states on ∂ S 1 , i.e. states ρ ∈ ∂ S 1 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 , ∞ .