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

Fabio Nicola, Federico Riccardi, Paolo Tilli

TL;DR

The paper completes the Wehrl entropy conjecture for symmetric $SU(N)$ representations by proving that symmetric coherent states are the unique minimizers of the associated Wehrl-type entropy. It develops a robust framework to characterize extremizers via distribution-function stability, obtains sharp quantitative stability bounds, and translates these results into holomorphic-polynomial concentration on complex projective spaces. Additionally, it demonstrates that the extremizer structure extends to contractive inequalities in weighted Bergman spaces, highlighting a unified mechanism for equality cases across contexts. These findings provide precise concentration and stability tools with potential applications in quantum-classical entropy analysis and complex analysis on Bergman spaces.

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.

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

TL;DR

The paper completes the Wehrl entropy conjecture for symmetric representations by proving that symmetric coherent states are the unique minimizers of the associated Wehrl-type entropy. It develops a robust framework to characterize extremizers via distribution-function stability, obtains sharp quantitative stability bounds, and translates these results into holomorphic-polynomial concentration on complex projective spaces. Additionally, it demonstrates that the extremizer structure extends to contractive inequalities in weighted Bergman spaces, highlighting a unified mechanism for equality cases across contexts. These findings provide precise concentration and stability tools with potential applications in quantum-classical entropy analysis and complex analysis on Bergman spaces.

Abstract

Lieb and Solovej proved that, for the symmetric representations, the corresponding Wehrl-type entropy is minimized by symmetric coherent states. However, the uniqueness of the minimizers remained an open problem when . 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.

Paper Structure

This paper contains 7 sections, 9 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Characterization of the extremizers
  3. Applications to the maximum concentration of holomorphic polynomials
  4. Global estimates
  5. Local estimates
  6. The bound in quantitative form
  7. Extremizers of the contractive estimates in weighted Bergman spaces

Key Result

Theorem 1.1

Let $\Phi \colon [0,1] \to \mathbb{R}$ be a convex function. Then, for any density operator $\rho$ on $\mathcal{H}_M$ we have where $u_0$ is the Husimi function of any coherent state.

Theorems & Definitions (19)

  • Theorem 1.1: Lieb-Solovej inequality lieb_solovej_SU(N)
  • Theorem 1.2: full Wehrl's conjecture
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Corollary 3.1
  • Proposition 3.2
  • proof
  • ...and 9 more