We initiate the study of zero-error communication via quantum channels when the receiver and the 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 only a function 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 (allowing for a constant amount of 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. We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede.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 that this bound to have many good properties, including being additive and given by a minimax formula. We also prove a coding theorem showing 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 (de Finetti reduction) that allows to reflect additional constraints, and which we believe to be of independent interest. This capacity is a relaxation of the feedback-assisted zero-error capacity; however, we have to leave open the question of whether they coincide in general. 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.

10abipartite equivocation graph10aBipartite graph10aCapacity planning10aChoi-Kraus operators10acoding theorem10acommutative bipartite graph10aElectronic mail10aencoding10aentanglement-assisted capacity10afeedback10afeedback-assisted capacity10afeedback-assisted zero-error capacity10ainformation theory10anoiseless feedback channel10aquantum channels10aquantum communication10aquantum entanglement10aQuantum information10aReceivers10aShannon's zero-error communication10atelecommunication channels10aunlimited quantum capacity10azero-error capacity1 aDuan, Runyao1 aSeverini, Simone1 aWinter, Andreas uhttps://grupsderecerca.uab.cat/giq/node/87000616nas a2200157 4500008003900000022001400039245010300053210006900156260001200225300001600237490000700253100001700260700002100277700002000298856014000318 2013 d a1557-965400aZero-Error Communication via Quantum Channels, Noncommutative Graphs, and a Quantum Lovász Number0 aZeroError Communication via Quantum Channels Noncommutative Grap c02/2013 a1164 - 11740 v591 aDuan, Runyao1 aSeverini, Simone1 aWinter, Andreas uhttps://grupsderecerca.uab.cat/giq/publications/zero-error-communication-quantum-channels-noncommutative-graphs-and-quantum-lov%C3%A1sz