Energy-constrained diamond norm with applications to the uniform continuity of continuous variable channel capacities

TL;DR

This paper investigates an energy-constrained version of the diamond norm for quantum channels, aiming to establish continuity properties of channel capacities and related quantities in infinite-dimensional systems.

Contribution

It introduces and analyzes an energy-constrained diamond norm, demonstrating its utility for proving continuity of quantum channel capacities and semigroups.

Findings

Energy-constrained diamond norm is well-suited for infinite-dimensional systems.

Proves continuity of quantum channel capacities under energy constraints.

Applies to the stability analysis of quantum dynamical semigroups.

Abstract

The channels, and more generally superoperators acting on the trace class operators of a quantum system naturally form a Banach space under the completely bounded trace norm (aka diamond norm). However, it is well-known that in infinite dimension, the norm topology is often "too strong" for reasonable applications. Here, we explore a recently introduced energy-constrained diamond norm on superoperators (subject to an energy bound on the input states). Our main motivation is the continuity of capacities and other entropic quantities of quantum channels, but we also present an application to the continuity of one-parameter unitary groups and certain one-parameter semigroups of quantum channels.