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The normalized separable states of a finite-dimensional multipartite quantumsystem, 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 andnon-induced. An induced face, $F$, has the form $F=\Gamma(F_V)$, where $V$ is asubspace of ${\cal H}$, $F_V$ is the set of $\rho\in{\cal S}_1$ whose range iscontained in $V$, and $\Gamma$ is a partial transposition operator. Such $F$ isa maximal face if and only if $V$ is a hyperplane. We give a simple formula forthe dimension of any induced maximal face. We also prove that the maximumdimension of induced maximal faces is equal to $d(d-2)$ where $d$ is thedimension of ${\cal H}$. The equality $\dim\Gamma(F_V)=d(d-2)$ holds if andonly if $V^\perp$ is spanned by a genuinely entangled vector.

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