A gap for PPT entanglement

Abstract

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.