We determine the p->q norms of the one-mode quantum-limited attenuator and amplifier, and prove that they are achieved by Gaussian states. The proof technique is completely new: it is based on a new logarithmic Sobolev inequality, and it can be used to determine the p->q norms of any quantum semigroup. Our result extends to noncommutative probability the seminal theorem "Gaussian kernels have only Gaussian maximizers" [Lieb, Invent. Math. 102, 179 (1990)], stating that Gaussian functions achieve the p->q norms of Gaussian integral kernels.
We then exploit our result to 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. This 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. This result also extends to the quantum regime the Entropy Power Inequality that plays a key role in classical information theory. The proof technique can be applied to any quantum channel.