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

TL;DR

This paper demonstrates that for certain classes of independent states, the PPT square conjecture holds generically, showing that entanglement swapping leads to separable states under typical conditions in high dimensions.

Contribution

It proves the PPT square conjecture holds generically for classes of states obtained via entanglement swapping, with spectral convergence to the Marcenko-Pastur law.

Findings

Spectral distribution of the induced state converges to Marcenko-Pastur law.

States are generically separable if the initial states are PPT entangled.

Results apply in high-dimensional asymptotic regimes.

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.