  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.
