Gaussian states minimize the output entropy of the one-mode quantum attenuator

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.