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

We prove the longstanding conjecture stating that Gaussian thermal input states minimize the output von Neumann entropy of one-mode phase-covariant quantum Gaussian channels among all the input states with a given entropy. Phase-covariant quantum Gaussian channels model the attenuation and the noise that affect any electromagnetic signal in the quantum regime. Our result is crucial to prove the converse theorems for both the triple trade-off region and the capacity region for broadcast communication of the Gaussian quantum-limited amplifier. Our result extends to the quantum regime the Entropy Power Inequality that plays a key role in classical information theory. Our proof exploits a completely new technique based on the recent determination of the p->q norms of the quantum-limited amplifier [De Palma et al., arXiv:1610.09967]. This technique can be applied to any quantum channel.