The Wehrl-type entropy conjecture for symmetric $SU(N)$ coherent states: cases of equality and stability

TL;DR

This paper proves that symmetric coherent states are the unique minimizers of Wehrl-type entropy for symmetric $SU(N)$ representations, establishing the conjecture's cases of equality and stability, with applications to holomorphic polynomials.

Contribution

It completes the proof of the Wehrl entropy conjecture for symmetric $SU(N)$ representations by showing uniqueness of minimizers and establishing stability results.

Findings

Symmetric coherent states are the only minimizers of Wehrl entropy.

A sharp quantitative bound for the Wehrl entropy is established.

Applications include a Faber-Krahn inequality for holomorphic polynomials.

Abstract

Lieb and Solovej proved that, for the symmetric $SU(N)$ representations, the corresponding Wehrl-type entropy is minimized by symmetric coherent states. However, the uniqueness of the minimizers remained an open problem when $N\geq 3$. In this note, we complete the proof of the Wehrl entropy conjecture for such representations by showing that symmetric coherent states are, in fact, the only minimizers. We also provide an application to the maximum concentration of holomorphic polynomials and deduce a corresponding Faber-Krahn inequality. A sharp quantitative form of the bound by Lieb and Solovej is also proved.