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

Fabio Nicola, Federico Riccardi, Paolo Tilli

TL;DR

The paper addresses the stability of the Lieb–Solovej Wehrl-entropy bound for symmetric $SU(N)$ coherent states. It introduces an elementary, geometry-based strategy that reexpresses the Wehrl-type entropy as a function on a high-dimensional unit sphere and performs a second-variation analysis around the coherent-state manifold, avoiding complicated level-set measures. The main results include a explicit quadratic deficit bound for $C^2$ convex entropies, followed by an extension to arbitrary convex entropies through smoothing and limiting arguments. The approach highlights the role of holomorphic polynomials, reproducing kernels, and differential-geometric structure in proving stability with explicit constants.

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.

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

TL;DR

The paper addresses the stability of the Lieb–Solovej Wehrl-entropy bound for symmetric coherent states. It introduces an elementary, geometry-based strategy that reexpresses the Wehrl-type entropy as a function on a high-dimensional unit sphere and performs a second-variation analysis around the coherent-state manifold, avoiding complicated level-set measures. The main results include a explicit quadratic deficit bound for convex entropies, followed by an extension to arbitrary convex entropies through smoothing and limiting arguments. The approach highlights the role of holomorphic polynomials, reproducing kernels, and differential-geometric structure in proving stability with explicit constants.

Abstract

We consider the problem of the stability (with sharp exponent) of the Lieb--Solovej inequality for symmetric 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 , for some suitable , and on some explicit (and somewhat surprising) computations.

Paper Structure

This paper contains 8 sections, 8 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Notation and rephrasing of the problem in terms of holomorphic polynomials
  3. Notation
  4. From the original setting to the unit sphere
  5. Preliminaries of differential geometry
  6. Proof of Theorem \ref{['th:main theorem on the sphere']} when $\Phi$ is of class $C^2$
  7. Proof Theorem \ref{['th:main theorem on the sphere']} for an arbitrary convex function
  8. Technical lemmas

Key Result

Theorem 2.2

For every convex function $\Phi \colon [0,1] \to \mathbb{R}$ that is strictly convex in some interval $(a,1)$ with $a \in (0,1)$, there exists a constant $c>0$ such that for every density matrix $\rho$ on $\mathcal{H}_{M}$ we have where $u_0$ is the Husimi function of any coherent state.

Theorems & Definitions (20)

  • Remark 2.1
  • Theorem 2.2: NRT_wehrl_SU(N)
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 10 more