Free relative entropy for measures and a corresponding perturbation theory

Abstract

Voiculescu's single variable free entropy is generalized in two different ways to the free relative entropy for compactly supported probability measures on the real line. The one is introduced by the integral expression and the other is based on matricial (or microstates) approximation; their equivalence is shown based on a large deviation result for the empirical eigenvalue distribution of a relevant random matrix. Next, the perturbation theory for compactly supported probability measures via free relative entropy is developed on the analogy of the perturbation theory via relative entropy. When the perturbed measure via relative entropy is suitably arranged on the space of selfadjoint matrices and the matrix size goes to infinity, it is proven that the perturbation via relative entropy on the matrix space approaches asymptotically to that via free relative entropy. The whole theory can be adapted to probability measures on the unit circle.