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

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.