  We study entanglement-assisted quantum and classical communication over a
single use of a quantum channel, which itself can correspond to a finite number
of uses of a channel with arbitrarily correlated noise. We obtain
characterizations of the corresponding one-shot capacities by establishing
upper and lower bounds on them in terms of the difference of two smoothed
entropic quantities. In the case of a memoryless channel, the upper and lower
bounds converge to the known single-letter formulas for the corresponding
capacities, in the limit of asymptotically many uses of it. Our results imply
that the difference of two smoothed entropic quantities characterizing the
one-shot entanglement-assisted capacities serves as a one-shot analogue of the
mutual information, since it reduces to the mutual information, between the
output of the channel and a system purifying its input, in the asymptotic,
memoryless scenario.
