A gap for PPT entanglement

Let $W$ be a finite dimensional vector space over a field with characteristic not equal to 2. Denote by $\text{Sym}(V)$ and $\text{Skew-Sym}(V)$ the subspaces of symmetric and skew-symmetric tensors of a subspace $V$ of $W\otimes W$, respectively. In this paper we show that if $V$ is generated by tensors with tensor rank 1, $V=\text{Sym}(V)\oplus\text{Skew-Sym}(V)$ and $W$ is the smallest vector space such that $V\subset W\otimes W$ then $\dim(\text{Sym}(V))\geq\max\{\frac{2\dim(\text{Skew-Sym}(V))}{\dim(W)}, \frac{\dim(W)}{2}\}$. This result has a straightforward application to the separability problem in Quantum Information Theory: If $\rho\in M_k\otimes M_k\simeq M_{k^2}$ is separable then $\text{rank}(Id+F)\rho(Id+F)\geq\text{max}\{ \frac{2}{r}\text{rank}(Id-F)\rho(Id-F), \frac{r}{2}\},$ where $F\in M_k\otimes M_k$ is the flip operator, $Id\in M_k\otimes M_k$ is the identity and $r$ is the marginal rank of $\rho+F\rho F$. We prove the sharpness of this inequality. Moreover, we show that if $\rho\in M_k\otimes M_k$ is positive under partial transposition (PPT) and $\text{rank }(Id+F)\rho(Id+F)=1$ then $\rho$ is separable. This result follows from Perron-Frobenius theory. We also present a large family of PPT matrices satisfying $\text{rank}(Id+F)\rho(Id+F)\geq r\geq \frac{2}{r-1} \text{rank}(Id-F)\rho(Id-F)$. There is a possibility that an entangled PPT matrix $\rho\in M_k\otimes M_k$ satisfying $1<\text{rank}(Id+F)\rho (Id+F)<\frac{2}{r} \text{rank}(Id-F)\rho (Id-F)$ exists. However, the family referenced above shows that finding one shall not be trivial.