The generalized Wehrl entropy bound in quantitative form

TL;DR

This paper establishes a quantitative relationship between Wehrl entropy and coherence in quantum states, providing explicit bounds and extending to generalized Wehrl entropies with applications to log-Sobolev inequalities.

Contribution

It introduces a precise quantitative bound linking Wehrl entropy to state coherence and extends the result to generalized Wehrl entropies with explicit constants.

Findings

Almost minimal Wehrl entropy implies almost coherent states.

Derived a sharp quantitative log-Sobolev inequality for Fock space functions.

Extended Wehrl entropy bounds to generalized versions.

Abstract

Lieb and Carlen have shown that mixed states with minimal Wehrl entropy are coherent states. We prove that mixed states with almost minimal Wehrl entropy are almost coherent states. This is proved in a quantitative sense where both the norm and the exponent are optimal and the constant is explicit. We prove a similar bound for generalized Wehrl entropies. As an application, a sharp quantitative form of the log-Sobolev inequality for functions in the Fock space is provided.