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

• URL: http://arxiv.org/abs/2412.10940v2
• Date: Mon, 23 Dec 2024 10:49:29 GMT
• Title: The Wehrl-type entropy conjecture for symmetric $SU(N)$ coherent states: cases of equality and stability
• Authors: Fabio Nicola, Federico Riccardi, Paolo Tilli,
• Abstract summary: We prove that for symmetric $SU(N)$ representations, the corresponding Wehrl-type entropy is minimized by coherent states. A sharp quantitative form of the bound by Lieb and Solovej is also proved.
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• 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.

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