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Boundary of the set of separable states

Motivated by the separability problem in quantum systems $2\otimes4$, $3\otimes3$ and $2\otimes2\otimes2$, 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 $H=H_1\otimes H_2\otimes\cdots\otimes H_n$. To any subspace $V$ of $H$ we associate a face $F_V$ of $S_1$ consisting of all states $ρ\in S_1$ whose range is contained in $V$. We prove that $F_V$ is a maximal face if and only if $V$ is a hyperplane. If $V$ is the hyperplane orthogonal to a product vector, we prove that $\dim F_V=d^2-1-\prod(2d_i-1)$, where $d_i$ is the dimension of $H_i$ and $d=\prod d_i$. We classify the maximal faces of $S_1$ in the cases $2\otimes2$ and $2\otimes3$. In particular we show that the minimum and the maximum dimension of maximal faces is 6 and 8 for $2\otimes2$, and 20 and 24 for $2\otimes3$. The boundary of $S_1$ is the union of all maximal faces. When $d>6$ we prove that there exist full states $ρ$ on the boundary, i.e., such that all partial transposes of $ρ$ (including $ρ$ itself) have rank $d$. K.-C. Ha and S.-K. Kye have recently constructed explicit such states in $2\times4$ and $3\otimes3$. In the latter case, they have also constructed a remarkable family of faces, depending on a real parameter $b>0$, $b\ne1$. Each face in the family is a 9-dimensional simplex and any interior point of the face is a full state. We construct suitable optimal entanglement witnesses (OEW) for these faces and analyze the three limiting cases $b=0,1,\infty$.