A gap for PPT entanglement

Let W be a finite dimensional vector space over a field with characteristic not equal to 2. Denote by V-S and V-A the subspaces of symmetric and antisymmetric tensors of a subspace V of W circle times W, respectively. In this paper we show that if V is generated by tensors with tensor rank 1, V = V-s circle plus V-A and W is the smallest vector space such that V subset of W circle times W then dim(V-S) >= max{2dim(V-A)/dim(W),dim(W)/2} This result has a straightforward application to the separability problem in Quantum Information Theory: If rho is an element of M-k circle times M-k similar or equal to M-k(2) is separable then rank((Id F)rho(Id + F)) >= max{2/r rank((Id - F)rho(Id - F)), r/2} where M-n, is the set of complex matrices of order n, F epsilon M-k circle times M-k is the flip operator, Id epsilon M-k circle times M-k is the identity and r is the marginal rank of rho+ F rho F. We prove the sharpness of this inequality. This inequality is a necessary condition for separability. Moreover, we show that if p epsilon M-k circle times M-k is positive under partial transposition (PPT) and rank((Id + F)rho(Id + F)) =1 then rho is separable. This result follows from Perron Frobenius theory. We also present a large family of PPT matrices satisfying rank(Id + F)rho(Id + F) >= r >= F) 2/r-1rank (Id - F) X rho(Id + F). There is a possibility that a PPT matrix rho epsilon M-k epsilon M-k satisfying 1 < rank(Id + F)rho(Id+ F)rho(Id + F) < 2/r-1rank (Id - F)rho(Id + F) exists.In this case rho is entangled. This is a gap where we can look for PPT entanglement. (C) 2017 Elsevier Inc. All rights reserved.