# Energy-constrained diamond norms and their use in quantum information theory

## TL;DR

This paper introduces energy-constrained diamond norms for quantum channels, showing they induce strong convergence and provide continuity bounds for quantum information measures, enhancing analysis of infinite-dimensional channels.

## Contribution

It establishes that energy-constrained diamond norms generate strong convergence and offers new continuity bounds for quantum channel capacities.

## Key findings

- Energy-constrained diamond norms induce strong convergence.
- Continuity bounds for quantum channel capacities are derived.
- These bounds imply uniform continuity of information characteristics.

## Abstract

We consider the family of energy-constrained diamond norms on the set of Hermitian-preserving linear maps (superoperators) between Banach spaces of trace class operators. We prove that any norm from this family generates the strong (pointwise) convergence on the set of all quantum channels (which is more adequate for describing variations of infinite-dimensional channels than the diamond norm topology). We obtain continuity bounds for information characteristics (in particular, classical capacities) of energy-constrained quantum channels (as functions of a channel) with respect to the energy-constrained diamond norms which imply uniform continuity of these characteristics with respect to the strong convergence topology.