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

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 $$\mathcal {S}_1$$ S 1 . The set $$\mathcal {S}_1$$ S 1 has two kinds of faces, induced and non-induced. An induced face, F, has the form $$F=\Gamma (F_V)$$ F = Γ ( F V ) , where V is a subspace of $$\mathcal {H}$$ H , $$F_V$$ F V is the set of $$\rho \in \mathcal {S}_1$$ ρ ∈ S 1 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 $$\mathrm{Dim\,}\Gamma (F_V)=d(d-2)$$ Dim Γ ( F V ) = d ( d - 2 ) holds if and only if $$V^\perp$$ V ⊥ is spanned by a genuinely entangled vector. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

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

, Volume 14 (9) – Jun 26, 2015
16 pages

/lp/springer_journal/dimension-formula-for-induced-maximal-faces-of-separable-states-and-4kT80ipRKv
Publisher
Springer US
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-015-1051-8
Publisher site
See Article on Publisher Site

### Abstract

The normalized separable states of a finite-dimensional multipartite quantum system, represented by its Hilbert space $$\mathcal {H}$$ H , form a closed convex set $$\mathcal {S}_1$$ S 1 . The set $$\mathcal {S}_1$$ S 1 has two kinds of faces, induced and non-induced. An induced face, F, has the form $$F=\Gamma (F_V)$$ F = Γ ( F V ) , where V is a subspace of $$\mathcal {H}$$ H , $$F_V$$ F V is the set of $$\rho \in \mathcal {S}_1$$ ρ ∈ S 1 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 $$\mathrm{Dim\,}\Gamma (F_V)=d(d-2)$$ Dim Γ ( F V ) = d ( d - 2 ) holds if and only if $$V^\perp$$ V ⊥ is spanned by a genuinely entangled vector.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Jun 26, 2015

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