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

Abstract

Let $|ψ\rangle\langle ψ|$ be a random pure state on $\mathbb{C}^{d^2}\otimes \mathbb{C}^s$, where $ψ$ is a random unit vector uniformly distributed on the sphere in $\mathbb{C}^{d^2}\otimes \mathbb{C}^s$. Let $ρ_1$ be random induced states $ρ_1=Tr_{\mathbb{C}^s}(|ψ\rangle\langle ψ|)$ whose distribution is $μ_{d^2,s}$; and let $ρ_2$ be random induced states following the same distribution $μ_{d^2,s}$ independent from $ρ_1$. Let $ρ$ be a random state induced by the entanglement swapping of $ρ_1$ and $ρ_2$. We show that the empirical spectrum of $ρ- {1\mkern -4mu{\rm l}}/d^2$ 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 $ρ$ is separable generically if $ρ_1, ρ_2$ are PPT entangled.

20 pages, 3 figures