Boundary of the set of separable states

Understand

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 $\rho\in S_1$ 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 $d=\prod d_i$.