Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State

TL;DR

This paper analytically investigates the probability distribution of Renyi entropies in large bipartite quantum systems, revealing phase transitions at specific critical points linked to Coulomb gas charge density changes.

Contribution

It introduces a Coulomb gas approach to derive the distribution of Renyi entropies and identifies two phase transitions affecting the distribution's shape.

Findings

Identifies two critical points where the distribution shape changes.

Links phase transitions to Coulomb gas charge density phenomena.

Verifies results with Monte Carlo simulations.

Abstract

Using a Coulomb gas method, we compute analytically the probability distribution of the Renyi entropies (a standard measure of entanglement) for a random pure state of a large bipartite quantum system. We show that, for any order q>1 of the Renyi entropy, there are two critical values at which the entropy's probability distribution changes shape. These critical points correspond to two different transitions in the corresponding charge density of the Coulomb gas: the disappearance of an integrable singularity at the origin and the detachement of a single-charge drop from the continuum sea of all the other charges. These transitions respectively control the left and right tails of the entropy's probability distribution, as verified also by Monte Carlo numerical simulations of the Coulomb gas equilibrium dynamics.