Dimension formula for induced maximal faces of separable states and genuine entanglement

The normalized separable states of a finite-dimensional multipartite quantum system, represented by its Hilbert space ${\cal H}$, form a closed convex set ${\cal S}_1$. The set ${\cal S}_1$ has two kinds of faces, induced and non-induced. An induced face, $F$, has the form $F=\Gamma(F_V)$, where $V$ is a subspace of ${\cal H}$, $F_V$ is the set of $\rho\in{\cal S}_1$ whose range is contained in $V$, and $\Gamma$ is a partial transposition operator. Such $F$ is a maximal face if and only if $V$ is a hyperplane. We give a simple formula for the dimension of any induced maximal face. We also prove that the maximum dimension of induced maximal faces is equal to $d(d-2)$ where $d$ is the dimension of ${\cal H}$. The equality $\dim\Gamma(F_V)=d(d-2)$ holds if and only if $V^\perp$ is spanned by a genuinely entangled vector.