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, , of normalized separable states in an arbitrary quantum system with finite-dimensional Hilbert space . To any subspace , we associate a face FV of consisting of all states whose range is contained in V . We prove that FV is a maximal face if and only if V is a hyperplane. If V =|ψ〉⊥, where |ψ〉 is a product vector, we prove that , where and . We classify the maximal faces of 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, , of is the union of all maximal faces. When d>6, it is easy to show that there exist full states on , i.e. states 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 .
- Received February 15, 2015.
- Accepted July 29, 2015.
- © 2015 The Author(s)
Published by the Royal Society. All rights reserved.