  Alice and Bob receive a bipartite state (possibly entangled) from some finite
collection or from some subspace. Alice sends a message to Bob through a noisy
quantum channel such that Bob may determine the initial state, with zero chance
of error. This framework encompasses, for example, teleportation, dense coding,
entanglement assisted quantum channel capacity, and one-way communication
complexity of function evaluation.
  With classical sources and channels, this problem can be analyzed using graph
homomorphisms. We show this quantum version can be analyzed using homomorphisms
on non-commutative graphs (an operator space generalization of graphs).
Previously the Lov\'{a}sz $\vartheta$ number has been generalized to
non-commutative graphs; we show this to be a homomorphism monotone, thus
providing bounds on quantum source-channel coding. We generalize the Schrijver
and Szegedy numbers, and show these to be monotones as well. As an application
we construct a quantum channel whose entanglement assisted zero-error one-shot
capacity can only be unlocked by using a non-maximally entangled state.
  These homomorphisms allow definition of a chromatic number for
non-commutative graphs. Many open questions are presented regarding the
possibility of a more fully developed theory.
