  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.
