Second-order coding rates for entanglement-assisted communication

The entanglement-assisted capacity of a quantum channel is known to provide the formal quantum generalization of Shannon´s classical channel capacity theorem, in the sense that it admits a single-letter characterization in terms of the quantum mutual information and does not increase in the presence of a noiseless quantum feedback channel from receiver to sender. In this work, we investigate second-order asymptotics of the entanglement-assisted communication task. That is, we consider how quickly the rates of entanglement-assisted codes converge to the entanglement-assisted capacity of a channel as a function of the number of channel uses and the error tolerance. We define a quantum generalization of the mutual information variance of a channel in the entanglement-assisted setting. For covariant channels, we show that this quantity is equal to the channel dispersion, and characterizes the convergence towards the entanglement-assisted capacity when the number of channel uses increases. More generally, we prove that the Gaussian approximation for a second-order coding rate is achievable for all quantum channels.