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

Understand

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.