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Erschienen in:
01.09.2015
Dimension formula for induced maximal faces of separable states and genuine entanglement
Erschienen in: Quantum Information Processing | Ausgabe 9/2015
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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 \(\mathrm{Dim\,}\Gamma (F_V)=d(d-2)\) holds if and only if \(V^\perp \) is spanned by a genuinely entangled vector.