  Noisy quantum channels may be used in many information carrying applications.
We show that different applications may result in different channel capacities.
Upper bounds on several of these capacities are proved. These bounds are based
on the coherent information, which plays a role in quantum information theory
analogous to that played by the mutual information in classical information
theory. Many new properties of the coherent information and entanglement
fidelity are proved. Two non-classical features of the coherent information are
demonstrated: the failure of subadditivity, and the failure of the pipelining
inequality. Both properties arise as a consequence of quantum entanglement, and
give quantum information new features not found in classical information
theory. The problem of a noisy quantum channel with a classical observer
measuring the environment is introduced, and bounds on the corresponding
channel capacity proved. These bounds are always greater than for the
unobserved channel. We conclude with a summary of open problems.
