The theory of quantum communication is far richer and less well understood than that of classical communication, which is now best seen as a special case of quantum communication. For example, quantum channels have multiple distinct capacities, depending on what one is trying to use them for, and what auxiliary resources are brought into play. These include

C, the unassisted classical capacity, ie the capacity for sending classical bits through the channel, with the help of quantum encoder and decoder;

Q, the unassisted quantum capacity, ie the capacity for sending intact qubits through the channel, with the help of a quantum encoder and decoder;

C_E, the classical capacity assisted by unlimited prior entanglement between sender and receiver; and

Q_2, the quantum capacity assisted by unlimited 2-way classical communication between sender and receiver.

More generally, one can consider the generalized capacity for one quantum channel to simulate another, with or without the help of auxiliary resources. Generalizing in another way, any nonlocal interaction, described for example by an interaction Hamiltonian, may be viewed as a bidirectional quantum channel, with capacities for generating entanglement, performing forward and backward communication, or simulating other interactions.

Ideally one would like to express all these capacities in terms of a small number of channel or interaction parameters. This is probably not possible in general, but considerable progress has been achieved in the entanglement assisted setting, where sender and receiver share unlimited prior entanglement. In this setting, quantum channels can be described by a single parameter, C_E, which is the natural generalization the single capacity of classical information theory, and is described by an analogous additive entropic expression, the quantum mutual input:output entropy, optimized over channel inputs. A noisy channel's C_E determines not only its entanglement-assisted classical capacity, but also its entanglement-assisted quantum capacity (which is exactly half C_E), and the amount of noiseless classical communication required to simulate the channel in the entanglement-assisted setting (the "quantum reverse Shannon theorem").

In the unassisted setting a quantum channel's classical capacity is described by another simple entropic expression, the Holevo quantity, which is suspected of being additive. The unassisted quantum capacity (which is equal to the quantum capacity assisted by forward communication only) is closely related to another, less well behaved, entropic expression, the coherent information, which is known to be nonadditive.

Still less well understood is the quantum capacity assisted by two way classical communication, which is known to be nonadditive and for which there is no candidate for a simple entropic expression.

We review the current state of knowledge and recent progress in the theory of quantum channels and interactions and their capacities.