03324nas a2200445 4500008003900000022001400039245010700053210006900160260000800229300001600237490000700253520193200260653003302192653002002225653002202245653002502267653001902292653003202311653002002343653001302363653003502376653001302411653003102424653004202455653002302497653003102520653002102551653002602572653002502598653002402623653001402647653003902661653003102700653003102731653002402762100001302786700001702799700001402816856004802830 2016 d a0018-944800aOn {Zero}-{Error} {Communication} via {Quantum} {Channels} in the {Presence} of {Noiseless} {Feedback}0 aZero Error Communication via Quantum Channels in the Presence of csep a5260–52770 v623 aWe 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 We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede.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, R.1 aSeverini, S.1 aWinter, A uhttps://grupsderecerca.uab.cat/giq/node/870