  The entanglement-assisted classical capacity of a noisy quantum channel is
the amount of information per channel use that can be sent over the channel in
the limit of many uses of the channel, assuming that the sender and receiver
have access to the resource of shared quantum entanglement, which may be used
up by the communication protocol. We show that this capacity is given by an
expression parallel to that for the capacity of a purely classical channel:
i.e., the maximum, over channel inputs $\rho$, of the entropy of the channel
input plus the entropy of the channel output minus their joint entropy, the
latter being defined as the entropy of an entangled purification of $\rho$
after half of it has passed through the channel. We calculate
entanglement-assisted capacities for two interesting quantum channels, the
qubit amplitude damping channel and the bosonic channel with
amplification/attenuation and Gaussian noise. We discuss how many independent
parameters are required to completely characterize the asymptotic behavior of a
general quantum channel, alone or in the presence of ancillary resources such
as prior entanglement. In the classical analog of entanglement assisted
communication--communication over a discrete memoryless channel (DMC) between
parties who share prior random information--we show that one parameter is
sufficient, i.e., that in the presence of prior shared random information, all
DMC's of equal capacity can simulate one another with unit asymptotic
efficiency.
