The PPT square conjecture holds generically for some classes of independent states

Abstract

Let $|\psi ⟩⟨\psi |$ be a random pure state on ${\mathbb{C}}^{d^2}\otimes {\mathbb{C}}^s$, where $\psi$ is a random unit vector uniformly distributed on the sphere in ${\mathbb{C}}^{d^2}\otimes {\mathbb{C}}^s$. Let ${\rho }_1$ be random induced states ${\rho }_1=T{r}_{{\mathbb{C}}^s}(|\psi ⟩⟨\psi |)$ whose distribution is ${\mu }_{d^2,s}$; and let ${\rho }_2$ be random induced states following the same distribution ${\mu }_{d^2,s}$ independent from ${\rho }_1$. Let $\rho$ be a random state induced by the entanglement swapping of ${\rho }_1$ and ${\rho }_2$. We show that the empirical spectrum of $\rho -1\mathrm{l}/d^2$ converges almost surely to the Marcenko-Pastur law with parameter $c^2$ as $d\to \infty$ and $s/d\to c$. As an application, we prove that the state $\rho$ is separable generically if ${\rho }_1,{\rho }_2$ are PPT entangled.

20 pages, 3 figures