On Shannon entropies in ${\mu}$-deformed Segal-Bargmann analysis

TL;DR

This paper explores Shannon entropies in a ${d}$-deformed Segal-Bargmann analysis, providing explicit entropy formulas for basis elements and establishing optimal constants in reverse log-Sobolev inequalities.

Contribution

It introduces explicit Shannon entropy formulas for basis elements in the ${d}$-deformed Segal-Bargmann space and proves the optimality of coefficients in related inequalities.

Findings

Explicit formulas for Shannon entropy of basis elements

Validation of the best possible constants in reverse log-Sobolev inequalities

Enhanced understanding of entropy properties in ${d}$-deformed analysis

Abstract

We consider a ${\mu}$-deformation of the Segal-Bargmann transform, which is a unitary map from a ${\mu}$-deformed quantum configuration space onto a ${\mu}$-deformed quantum phase space (the ${\mu}$-deformed Segal-Bargmann space). Both of these Hilbert spaces have canonical orthonormal bases. We obtain explicit formulas for the Shannon entropy of some of the elements of these bases. We also consider two reverse log-Sobolev inequalities in the ${\mu}$-deformed Segal-Bargmann space, which have been proved in a previous work, and show that a certain known coefficient in them is the best possible.