Wehrl-type entropy problem for compact connected semisimple Lie groups

This paper solves the Wehrl-type entropy problem for arbitrary compact connected semisimple Lie groups. Let $G$ be a compact connected semisimple Lie group, and let $\pi :G\to U(V_{\lambda })$ be a finite-dimensional irreducible unitary representation associated with the highest weight $\lambda$. We prove that coherent projectors are the unique minimizers of the Wehrl entropy over all density matrices on $V_{\lambda }$. They also uniquely maximize every Husimi power moment of order $p>1$. The proof uses a second-variation at the extremizer by perturbation in directions associated with Killing fields, similar to the strategy of Frank and Lieb used in \cite{FrankLieb}. Then the problem reduces to the extreme-moment property of highest-weight vectors.