Continuity of quantum channel capacities

Understand

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.