  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.
