Uniform finite-dimensional approximation of basic capacities of energy-constrained channels

TL;DR

This paper demonstrates that energy-constrained infinite-dimensional quantum channels can be approximated by finite-dimensional ones with controllable error, enabling continuity analysis of their capacities.

Contribution

The authors establish a uniform finite-dimensional approximation for basic capacities of energy-constrained quantum channels, facilitating continuity proofs.

Findings

Finite-dimensional approximation of channels with minimal eigenvalues

Smaller dimension for energy-limited channels

Continuity bounds for capacities based on energy constraints

Abstract

We consider energy-constrained infinite-dimensional quantum channels from a given system (satisfying a certain condition) to any other systems. We show that dealing with basic capacities of these channels we may assume (accepting arbitrarily small error $\varepsilon$) that all channels have the same finite-dimensional input space -- the subspace corresponding to the $m(\varepsilon)$ minimal eigenvalues of the input Hamiltonian. We also show that for the class of energy-limited channels (mapping energy-bounded states to energy-bounded states) the above result is valid with substantially smaller dimension $m(\varepsilon)$. The uniform finite-dimensional approximation allows to prove the uniform continuity of the basic capacities on the set of all quantum channels with respect to the strong (pointwise) convergence topology. For all the capacities we obtain continuity bounds depending only on the input energy bound and the energy-constrained-diamond-norm distance between quantum channels (generating the strong convergence on the set of quantum channels).