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

TL;DR

This paper proves that Gaussian states maximize the p->q norms of quantum-limited Gaussian channels, extending classical results to the quantum domain and aiding in understanding quantum communication limits.

Contribution

It establishes that Gaussian states are the maximizers for the p->q norms of quantum-limited attenuators and amplifiers, extending classical Gaussian kernel results to quantum channels.

Findings

Gaussian states achieve the p->q norms of quantum-limited channels.

The result supports the conjecture that Gaussian inputs minimize output entropy.

Introduces a new noncommutative logarithmic Sobolev inequality.

Abstract

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.