  We prove that quantum thermal Gaussian input states minimize the output
entropy of the multi-mode quantum Gaussian attenuators and amplifiers that are
entanglement breaking and of the multi-mode quantum Gaussian phase
contravariant channels among all the input states with a given entropy. This is
the first time that this property is proven for a multi-mode channel without
restrictions on the input states. A striking consequence of this result is a
new lower bound on the output entropy of all the multi-mode quantum Gaussian
attenuators and amplifiers in terms of the input entropy. We apply this bound
to determine new upper bounds to the communication rates in two different
scenarios. The first is classical communication to two receivers with the
quantum degraded Gaussian broadcast channel. The second is the simultaneous
classical communication, quantum communication and entanglement generation or
the simultaneous public classical communication, private classical
communication and quantum key distribution with the Gaussian quantum-limited
attenuator.
