An elementary approach to Wehrl-type entropy bounds in quantitative form

TL;DR

This paper provides an elementary proof for the stability of the Lieb--Solovej inequality for symmetric $SU(N)$ coherent states, reformulating Wehrl-type entropy on the unit sphere and performing explicit computations.

Contribution

It introduces a simplified, elementary approach to proving the stability of a key entropy inequality in quantum mechanics.

Findings

Proved sharp stability bounds for the Lieb--Solovej inequality.

Reformulated Wehrl-type entropy as a function on the unit sphere.

Performed explicit computations to support the proof.

Abstract

We consider the problem of the stability (with sharp exponent) of the Lieb--Solovej inequality for symmetric $SU(N)$ coherent states, which was obtained only recently by the authors. Here, we propose an elementary proof of this result, based on reformulating the Wehrl-type entropy as a function defined on the unit sphere in $\mathbb{C}^d$, for some suitable $d$, and on some explicit (and somewhat surprising) computations.