Gaussian optimizers for entropic inequalities in quantum information

TL;DR

This paper reviews the progress on quantum Gaussian optimizer conjectures, which identify Gaussian states as optimal solutions for key quantum information optimization problems involving Gaussian channels, crucial for quantum communication.

Contribution

It surveys the current state of the art in proving the quantum Gaussian optimizer conjectures, connecting functional analysis, probability, and quantum information theory.

Findings

Quantum Gaussian states are conjectured to be optimal for several quantum channel problems.

The conjectures relate to fundamental inequalities like Entropy Power Inequality and Young's inequality.

Quantum Gaussian channels are essential for determining maximum communication rates in quantum optics.

Abstract

We survey the state of the art for the proof of the quantum Gaussian optimizer conjectures of quantum information theory. These fundamental conjectures state that quantum Gaussian input states are the solution to several optimization problems involving quantum Gaussian channels. These problems are the quantum counterpart of three fundamental results of functional analysis and probability: the Entropy Power Inequality, the sharp Young's inequality for convolutions, and the theorem "Gaussian kernels have only Gaussian maximizers." Quantum Gaussian channels play a key role in quantum communication theory: they are the quantum counterpart of Gaussian integral kernels and provide the mathematical model for the propagation of electromagnetic waves in the quantum regime. The quantum Gaussian optimizer conjectures are needed to determine the maximum communication rates over optical fibers and free space. The restriction of the quantum-limited Gaussian attenuator to input states diagonal in the Fock basis coincides with the thinning, which is the analog of the rescaling for positive integer random variables. Quantum Gaussian channels provide then a bridge between functional analysis and discrete probability.