  We prove in the multimode scenario a fundamental relation between the Wehrl
and the von Neumann entropy, stating that the minimum Wehrl entropy among all
the quantum states with a given von Neumann entropy is achieved by thermal
Gaussian states. We also prove that thermal Gaussian input states minimize the
output von Neumann entropy of multimode quantum Gaussian attenuators,
amplifiers and phase-contravariant channels among all the input states diagonal
in some product basis and with a given entropy. This result constitutes a major
step towards the proof of the same property for generic input states, which is
still an open conjecture. This conjecture is necessary to determine the maximum
communication rates for the triple trade-off coding and broadcast communication
with the Gaussian quantum-limited attenuator. Finally, we prove that the tensor
product of n identical geometric input probability distributions minimizes the
output Shannon entropy of the n-mode thinning among all the input probability
distributions with a given entropy.
