@article {duan_zero-error_2016,
title = {On {Zero}-{Error} {Communication} via {Quantum} {Channels} in the {Presence} of {Noiseless} {Feedback}},
journal = {IEEE Transactions on Information Theory},
volume = {62},
number = {9},
year = {2016},
month = {sep},
pages = {5260{\textendash}5277},
abstract = {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{\textquoteright}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 We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede.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.},
keywords = {bipartite equivocation graph, Bipartite graph, Capacity planning, Choi-Kraus operators, coding theorem, commutative bipartite graph, Electronic mail, encoding, entanglement-assisted capacity, feedback, feedback-assisted capacity, feedback-assisted zero-error capacity, information theory, noiseless feedback channel, quantum channels, quantum communication, quantum entanglement, Quantum information, Receivers, Shannon{\textquoteright}s zero-error communication, telecommunication channels, unlimited quantum capacity, zero-error capacity},
issn = {0018-9448},
doi = {10.1109/TIT.2016.2562580},
author = {Duan, R. and Severini, S. and Winter, A.}
}