A gap for PPT entanglement

Let W be a finite dimensional vector space over a field with characteristic not equal to 2. Denote by VS and VA the subspaces of symmetric and antisymmetric tensors of a subspace V of W⊗W, respectively. In this paper we show that if V is generated by tensors with tensor rank 1, V=VS⊕VA and W is the...

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Vydáno v:Linear algebra and its applications Ročník 529; s. 89 - 114
Hlavní autor: Cariello, D.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Amsterdam Elsevier Inc 15.09.2017
American Elsevier Company, Inc
Témata:
Algorithms
Entanglement
Information theory
Linear algebra
Mathematical analysis
Matrices (mathematics)
Numerical analysis
Polynomials
PPT matrix
Quantum entanglement
Quantum phenomena
Separability
Sharpness
Subspaces
Symmetric tensors
Tensor rank 1
Tensors
PPT matrix
Separability
Symmetric tensors
Tensor rank 1
15B48
Entanglement
15A69
15B57
ISSN:0024-3795, 1873-1856
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Shrnutí:Let W be a finite dimensional vector space over a field with characteristic not equal to 2. Denote by VS and VA the subspaces of symmetric and antisymmetric tensors of a subspace V of W⊗W, respectively. In this paper we show that if V is generated by tensors with tensor rank 1, V=VS⊕VA and W is the smallest vector space such that V⊂W⊗W then dim⁡(VS)≥max⁡{2dim⁡(VA)dim⁡(W),dim⁡(W)2}. This result has a straightforward application to the separability problem in Quantum Information Theory: If ρ∈Mk⊗Mk≃Mk2 is separable thenrank((Id+F)ρ(Id+F))≥max{2rrank((Id−F)ρ(Id−F)),r2}, where Mn is the set of complex matrices of order n, F∈Mk⊗Mk is the flip operator, Id∈Mk⊗Mk is the identity and r is the marginal rank of ρ+FρF. We prove the sharpness of this inequality. This inequality is a necessary condition for separability. Moreover, we show that if ρ∈Mk⊗Mk is positive under partial transposition (PPT) and rank((Id+F)ρ(Id+F))=1 then ρ is separable. This result follows from Perron–Frobenius theory. We also present a large family of PPT matrices satisfying rank(Id+F)ρ(Id+F)≥r≥2r−1rank(Id−F)×ρ(Id−F). There is a possibility that a PPT matrix ρ∈Mk⊗Mk satisfying1<rank(Id+F)ρ(Id+F)<2rrank(Id−F)ρ(Id−F) exists. In this case ρ is entangled. This is a gap where we can look for PPT entanglement.
Bibliografie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2017.04.013