Gaussian optimizers and the additivity problem in quantum information theory

TL;DR

This paper surveys key problems in quantum information theory, focusing on recent solutions for Gaussian optimizers and their implications for the additivity conjecture, highlighting that certain properties hold for gauge-covariant Gaussian channels.

Contribution

It provides a detailed report on the partial solution of the quantum Gaussian optimizers problem and discusses the implications for the additivity conjecture in quantum information theory.

Findings

Gaussian states minimize a broad class of functionals of Gaussian channel outputs

Additivity properties hold for gauge-covariant Gaussian channels

The solution is restricted to the gauge-invariant case with a complex structure

Abstract

We give a survey of the two remarkable analytical problems of quantum information theory. The main part is a detailed report of the recent (partial) solution of the quantum Gaussian optimizers problem which establishes an optimal property of Glauber's coherent states -- a particular instance of pure quantum Gaussian states. We elaborate on the notion of quantum Gaussian channel as a noncommutative generalization of Gaussian kernel to show that the coherent states, and under certain conditions only they, minimize a broad class of the concave functionals of the output of a Gaussian channel. Thus, the output states corresponding to the Gaussian input are "the least chaotic", majorizing all the other outputs. The solution, however, is essentially restricted to the gauge-invariant case where a distinguished complex structure plays a special role. We also comment on the related famous additivity conjecture, which was solved in principle in the negative some five years ago. This refers to the additivity or multiplicativity (with respect to tensor products of channels) of information quantities related to the classical capacity of quantum channel, such as $(1\rightarrow p)$-norms or the minimal von Neumann or R\'enyi output entropies. A remarkable corollary of the present solution of the quantum Gaussian optimizers problem is that these additivity properties, while not valid in general, do hold in the important and interesting class of the gauge-covariant Gaussian channels.