Energy-constrained private and quantum capacities of quantum channels

TL;DR

This paper develops a comprehensive theory for energy-constrained quantum and private capacities of quantum channels, establishing conditions under which these capacities can be exactly characterized and applying the results to Gaussian channels.

Contribution

It introduces a unified framework for energy-constrained capacities, derives capacity formulas under specific conditions, and applies these to Gaussian channels, providing new insights without relying on minimum output entropy conjectures.

Findings

Energy-constrained capacities are characterized by regularized coherent information.

Single-letter formulas are valid for degradable channels satisfying certain conditions.

Results recover and extend known results for quantum Gaussian channels.

Abstract

This paper establishes a general theory of energy-constrained quantum and private capacities of quantum channels. We begin by defining various energy-constrained communication tasks, including quantum communication with a uniform energy constraint, entanglement transmission with an average energy constraint, private communication with a uniform energy constraint, and secret key transmission with an average energy constraint. We develop several code conversions, which allow us to conclude non-trivial relations between the capacities corresponding to the above tasks. We then show how the regularized, energy-constrained coherent information is equal to the capacity for the first two tasks and is an achievable rate for the latter two tasks, whenever the energy observable satisfies the Gibbs condition of having a well defined thermal state for all temperatures and the channel satisfies a finite output-entropy condition. For degradable channels satisfying these conditions, we find that the single-letter energy-constrained coherent information is equal to all of the capacities. We finally apply our results to degradable quantum Gaussian channels and recover several results already established in the literature (in some cases, we prove new results in this domain). Contrary to what may appear from some statements made in the literature recently, proofs of these results do not require the solution of any kind of minimum output entropy conjecture or entropy photon-number inequality.