  We determine the minimum Wehrl entropy among the quantum states with a given
von Neumann entropy, and prove that it is achieved by thermal Gaussian states.
This result determines the relation between the von Neumann and the Wehrl
entropies. The key idea is proving that the quantum-classical channel that
associates to a quantum state its Husimi Q representation is asymptotically
equivalent to the Gaussian quantum-limited amplifier with infinite
amplification parameter. This equivalence also permits to determine the p->q
norms of the aforementioned quantum-classical channel in the two particular
cases of one mode and p=q, and prove that they are achieved by thermal Gaussian
states. The same equivalence permits to prove that the Husimi Q representation
of a one-mode passive state (i.e. a state diagonal in the Fock basis with
eigenvalues decreasing as the energy increases) majorizes the Husimi Q
representation of any other one-mode state with the same spectrum, i.e. it
maximizes any convex functional.
