Entanglement Entropy of a Random Pure State

I will discuss the concept of entanglement entropy of a pure state in
a bipartite system. When the pure state is chosen at random, the
associated entanglement entropy also becomes random. Using a Coulomb
gas method valid for large systems, we compute analytically the full
probability distribution of the entanglement entropy. Three different
regimes are identified 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. One of the consequences of these results is
that the entropy of a random pure state is not really maximal, as
often proclaimed in the literature based on the computation of only
the average entropy.