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We determine the p->q norms of the Gaussian one-mode quantum-limitedattenuator and amplifier and prove that they are achieved by Gaussian states,extending to noncommutative probability the seminal theorem "Gaussian kernelshave only Gaussian maximizers" (Lieb in Invent Math 102(1):179-208, 1990). Thequantum-limited attenuator and amplifier are the building blocks of quantumGaussian channels, which play a key role in quantum communication theory sincethey model in the quantum regime the attenuation and the noise affecting anyelectromagnetic signal. Our result is crucial to prove the longstandingconjecture stating that Gaussian input states minimize the output entropy ofone-mode phase-covariant quantum Gaussian channels for fixed input entropy. Ourproof technique is based on a new noncommutative logarithmic Sobolevinequality, and it can be used to determine the p->q norms of any quantumsemigroup.

The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have Gaussian Maximizers