# 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.

## Key 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.