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

Abstract

The normalized separable states of a finite-dimensional multipartite quantum system, represented by its Hilbert space $\mathcal{H}$, form a closed convex set ${\mathcal{S}}_1$. The set ${\mathcal{S}}_1$ 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 $\mathcal{H}$, $F_V$ is the set of $\rho \in {\mathcal{S}}_1$ 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 $\mathcal{H}$. The equality $\dim \Gamma (F_V)=d(d-2)$ holds if and only if $V^{\perp }$ is spanned by a genuinely entangled vector.

14 pages

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Cited by in corpus (1)

• Three qubit separable states of length ten with unique decompositions