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arXiv:2512.04245

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

Fabio Nicola, Federico Riccardi, Paolo Tilli

math-phmath.FAquant-phSubmitted Dec 3, 2025• Announced Dec 5, 2025• 0 citations
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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.

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21 pages

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We consider the problem of the stability (with sharp exponent) of the Lieb--Solovej inequality for symmetric SU(N)SU(N)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 Cd\mathbb{C}^dCd, for some suitable ddd, and on some explicit (and somewhat surprising) computations.
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The Wehrl entropy is the entropy of the probability distribution in phase space corresponding to the Husimi function in terms of coherent states. We explain the significance of the Wehrl entropy in quantum information theory, and present the theory behind the Lieb conjecture, which states that, in finite dimensions, the minimum Wehrl entropy occurs for Bloch coherent states. This was proven by Lieb and Solovej in 2012. We present the theory behind coherent states, with a particular emphasis on Bloch coherent states, and give a geometrical representation of quantum states as points on a sphere. Using this representation, we identify spherical arrangements of 2–9 points that maximize the Wehrl entropy locally. We conjecture that these maxima are in fact global. Furthermore, we investigate how the maximally entangled symmetric states are related to the states corresponding to maximal Wehrl entropy.
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