Mutual information and bose-einstein condensation

In this work we study an ideal bosonic quantum field system at finite temperature, and in a canonical and a grand canonical ensemble. For a simple spatial partition we derive the corresponding mutual information, a quantity that measures the total amount of information of one of the parts about the other. In order to find it, we first derive the von Neumann entropy that corresponds to the spatially separated subsystem (i.e. the geometric entropy) and then we subtract its extensive part which coincides with the thermal entropy of the subsystem. In the framework of the grand canonical description, we examine the influence of the underlying Bose-Einstein condensation on the behaviour of the mutual information, and we find that its derivative with respect to the temperature possesses a finite discontinuity at exactly the critical temperature.