Boundary of the set of separable states
Lin Chen, Dragomir Ž. Djoković · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · 2015
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