On Zero-Error Communication via Quantum Channels in the Presence of Noiseless Feedback

TitleOn Zero-Error Communication via Quantum Channels in the Presence of Noiseless Feedback
Publication TypeJournal Article
Year of Publication2016
AuthorsDuan, R, Severini, S, Winter, A
JournalIEEE Transactions on Information Theory
Date Published09/2016
Keywordsbipartite 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's zero-error communication, telecommunication channels, unlimited quantum capacity, zero-error capacity

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 We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede.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.

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