Statistical distribution of quantum entanglement for a random bipartite state

TL;DR

This paper analytically derives the probability distribution of entanglement measures for random bipartite quantum states, revealing phase transitions in the distribution through Coulomb gas techniques and confirming results with simulations.

Contribution

It introduces a Coulomb gas approach to compute the full entanglement entropy distribution and identifies phase transitions in the distribution for large quantum systems.

Findings

Three regimes in entropy distribution with phase transitions.

Identification of critical points related to charge density changes.

Analytical results confirmed by Monte Carlo simulations.

Abstract

We compute analytically the statistics of the Renyi and von Neumann entropies (standard measures of entanglement), for a random pure state in a large bipartite quantum system. The full probability distribution is computed by first mapping the problem to a random matrix model and then using a Coulomb gas method. We identify three different regimes in the entropy distribution, which correspond to two phase transitions in the associated Coulomb gas. The two critical points correspond to sudden changes in the shape of the Coulomb charge density: the appearance of an integrable singularity at the origin for the first critical point, and the detachement of the rightmost charge (largest eigenvalue) from the sea of the other charges at the second critical point. Analytical results are verified by Monte Carlo numerical simulations. A short account of some of these results appeared recently in Phys. Rev. Lett. {\bf 104}, 110501 (2010).