  The entanglement-assisted classical 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
classical communication task. That is, we consider how quickly the rates of
entanglement-assisted codes converge to the entanglement-assisted classical
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
thus completely characterize the convergence towards the entanglement-assisted
classical capacity when the number of channel uses increases. Our results also
apply to entanglement-assisted quantum communication, due to the equivalence
between entanglement-assisted classical and quantum communication established
by the teleportation and super-dense coding protocols.
