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 $$\mathcal {H}$$H, form a closed convex set $$\mathcal {S}_1$$S1. The set $$\mathcal {S}_1$$S1 has two kinds of faces, induced and non-induced. An induced face, F, has the form $$F=\Gamma (F_V)$$F=Γ(FV), where V is a subspace of $$\mathcal {H}$$H, $$F_V$$FV is the set of $$\rho \in \mathcal {S}_1$$ρ∈S1 whose range is contained in V, and $$\Gamma $$Γ 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 $$d(d-2)$$d(d-2) where d is the dimension of $$\mathcal {H}$$H. The equality $$\mathrm{Dim\,}\Gamma (F_V)=d(d-2)$$DimΓ(FV)=d(d-2) holds if and only if $$V^\perp $$^V⊥ is spanned by a genuinely entangled vector.