  We prove that Gaussian thermal input states minimize the output von Neumann
entropy of the one-mode Gaussian quantum-limited attenuator for fixed input
entropy. The Gaussian quantum-limited attenuator models the attenuation of an
electromagnetic signal in the quantum regime. The Shannon entropy of an
attenuated real-valued classical signal is a simple function of the entropy of
the original signal. A striking consequence of energy quantization is that the
output von Neumann entropy of the quantum-limited attenuator is no more a
function of the input entropy alone. The proof starts from the majorization
result of De Palma et al., IEEE Trans. Inf. Theory 62, 2895 (2016), and is
based on a new isoperimetric inequality. Our result implies that geometric
input probability distributions minimize the output Shannon entropy of the
thinning for fixed input entropy. Moreover, our result opens the way to the
multimode generalization, that permits to determine both the triple trade-off
region of the Gaussian quantum-limited attenuator and the classical capacity
region of the Gaussian degraded quantum broadcast channel.
