  We initiate the study of zero-error communication via quantum channels when
the receiver and sender have at their disposal a noiseless feedback channel of
unlimited quantum capacity, generalizing Shannon's zero-error communication
theory with instantaneous feedback.
  We first show that this capacity is a function only of the linear span of
Choi-Kraus operators of the channel, which generalizes the bipartite
equivocation graph of a classical channel, and which we dub "non-commutative
bipartite graph". Then we go on to show that the feedback-assisted capacity is
non-zero (with constant activating noiseless communication) if and only if the
non-commutative bipartite graph is non-trivial, and give a number of equivalent
characterizations. This result involves a far-reaching extension of the
"conclusive exclusion" of quantum states [Pusey/Barrett/Rudolph, Nature Phys.
8:475-478].
  We then present an upper bound on the feedback-assisted zero-error capacity,
motivated by a conjecture originally made by Shannon and proved later by
Ahlswede. We demonstrate this bound to have many good properties, including
being additive and given by a minimax formula. We also prove that this quantity
is the entanglement-assisted capacity against an adversarially chosen channel
from the set of all channels with the same Choi-Kraus span, which can also be
interpreted as the feedback-assisted unambiguous capacity. The proof relies on
a generalization of the "Postselection Lemma" [Christandl/Koenig/Renner, PRL
102:020504] that allows to reflect additional constraints, and which we believe
to be of independent interest.
  We illustrate our ideas with a number of examples, including
classical-quantum channels and Weyl diagonal channels, and close with an
extensive discussion of open questions.
