Continuity of quantum channel capacities

TL;DR

This paper demonstrates that various quantum channel capacities are continuous functions of the channel, showing that small changes in the channel lead to small changes in its communication capabilities, with bounds depending on the channel's output dimension.

Contribution

It establishes the continuity of classical, quantum, and private capacities of quantum channels, providing explicit bounds and considering capacities with additional classical communication.

Findings

Capacities are continuous with respect to the diamond norm.

Bounds depend linearly on the number of channel copies and the output dimension.

Capacities with free classical communication are continuous on the interior of the capacity set.

Abstract

We prove that a broad array of capacities of a quantum channel are continuous. That is, two channels that are close with respect to the diamond norm have correspondingly similar communication capabilities. We first show that the classical capacity, quantum capacity, and private classical capacity are continuous, with the variation on arguments epsilon apart bounded by a simple function of epsilon and the channel's output dimension. Our main tool is an upper bound of the variation of output entropies of many copies of two nearby channels given the same initial state; the bound is linear in the number of copies. Our second proof is concerned with the quantum capacities in the presence of free backward or two-way public classical communication. These capacities are proved continuous on the interior of the set of non-zero capacity channels by considering mutual simulation between similar channels.