Free entropy

Abstract

Free entropy was defined by Voiculescu as a generalization of classical entropy to the non-commutative context. There are several definitions of free entropy; we shall concentrate on two of them. The first is the so-called microstates entropy that measures a volume of matrices with empirical distribution approximating a given law. The second, called the microstates-free entropy, is defined via a non-commutative version of Fisher information. The classical analog of these definitions is, on one hand, the definition of the entropy of a measure ? as the volume of points whose empirical distribution approximates ?, and, on the other hand, the well-known entropy$\int {\frac{{d\mu }}{{dx}}} \log \frac{{d\mu }}{{dx}}dx$. In this classical setting, Sanov’s theorem shows that these two entropies are equal. The free analog statement is still open but we shall give in this section bounds to compare the microstates and the microstates-free entropies. The ideas come from [37, 55, 56] but we shall try to simplify the proof to hopefully make it more accessible to non-probabilists (the original proof uses Malliavin calculus but we shall here give an elementary version of the few properties of Malliavin calculus we need). In the following, we consider only laws of self-adjoint variables (i.e., $A_i^* = A_i {\rm for }1 \le i \le m$). We do not loose generality since any operator can be decomposed as the sum of two self-adjoint operators.