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

Abstract

Let $|\psi\rangle\langle \psi|$ 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 $\rho_1=Tr_{\mathbb{C}^s}(|\psi\rangle\langle \psi |)$ 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 $\rho- {1\mkern -4mu{\rm l}}/d^2$ converges almost surely to the Marcenko-Pastur law with parameter $c^2$ as $d\rightarrow \infty$ and $s/d \rightarrow c$. As an application, we prove that the state $\rho$ is separable generically if $\rho_1, \rho_2$ are PPT entangled.