We determine the p->q norms of the Gaussian one-mode quantum-limited
attenuator and amplifier and prove that they are achieved by Gaussian states,
extending to noncommutative probability the seminal theorem "Gaussian kernels
have only Gaussian maximizers" (Lieb in Invent Math 102(1):179-208, 1990). The
quantum-limited attenuator and amplifier are the building blocks of quantum
Gaussian channels, which play a key role in quantum communication theory since
they model in the quantum regime the attenuation and the noise affecting any
electromagnetic signal. Our result is crucial to prove the longstanding
conjecture stating that Gaussian input states minimize the output entropy of
one-mode phase-covariant quantum Gaussian channels for fixed input entropy. Our
proof technique is based on a new noncommutative logarithmic Sobolev
inequality, and it can be used to determine the p->q norms of any quantum
semigroup.Comment: Annales Henri Poincar\'e (2018