A Boolean analogue of Shannon's entropy and monotonicity problem

Abstract

In this article, we introduce a notion of entropy for Boolean independence analogous to the Shannon's entropy and Voiculescu's free entropy through the study of asymptotic probabilities of the set of matrix approximates. To motivate this definition, we study two random matrix models exhibiting asymptotic Boolean independence, introduced respectively by Lenczewski and by C\'ebron-Gilliers. It turns out that the asymptotic probabilities of such matrix approximations can be characterized through large deviation principles for their empirical spectral measures. We prove that the associated rate functions coincide up to a scaling factor and attain their minimum at the Rademacher distribution, which plays the role of the Gaussian measure in Boolean probability. The logarithmic integral appearing in these rate functions is therefore identified as the Boolean entropy. We further show that this entropy is maximized, though not uniquely, by the Rademacher distribution and is monotone along the Boolean central limit theorem.