The generalized Wehrl entropy bound in quantitative form
Rupert L. Frank, Fabio Nicola, Paolo Tilli
TL;DR
This work provides a sharp, quantitative stability result for Wehrl entropy: if a density operator $\rho$ on $L^2(\mathbb{R})$ nearly minimizes Wehrl entropy, then $\rho$ is close to a coherent-state projector in trace norm, with the deficit controlled by $c_* D[\rho]^2$ where $D[\rho]$ is the distance to the coherent-state family and $c_*$ is explicit. The authors extend this stability to generalized Wehrl entropies via convex $\Phi$, establishing a similar $D[\rho]^2$-bound with an explicit constant. They present two proofs: a direct approach based on a detailed Fock-space analysis and a second proof for pure states leveraging a stability version of the Faber–Krahn inequality and majorization ideas. As a key application, they derive a sharp quantitative log-Sobolev inequality for entire functions in the Fock space, with equality cases matching Gaussian-type optimizers, and show how these results interrelate via the Bargmann transform and the coherent-state transform.
Abstract
Lieb and Carlen have shown that mixed states with minimal Wehrl entropy are coherent states. We prove that mixed states with almost minimal Wehrl entropy are almost coherent states. This is proved in a quantitative sense where both the norm and the exponent are optimal and the constant is explicit. We prove a similar bound for generalized Wehrl entropies. As an application, a sharp quantitative form of the log-Sobolev inequality for functions in the Fock space is provided.