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

Duan, R; Severini, S; Winter, A; (2016) On Zero-Error Communication via Quantum Channels in the Presence of Noiseless Feedback. IEEE Transactions On Information Theory , 62 (9) pp. 5260-5277. 10.1109/TIT.2016.2562580. Green open access

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 We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede.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

Type: Article
Title: On Zero-Error Communication via Quantum Channels in the Presence of Noiseless Feedback
Open access status: An open access version is available from UCL Discovery
DOI: 10.1109/TIT.2016.2562580
Publisher version: http://doi.org/10.1109/TIT.2016.2562580
Language: English
Additional information: Copyright © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
Keywords: Science & Technology, Technology, Computer Science, Information Systems, Engineering, Electrical & Electronic, Computer Science, Engineering, Quantum Information, Zero-error Capacity, Feedback, Entanglement-assisted Capacity, Reverse Shannon Theorem, Classical Capacity, States
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
URI: https://discovery.ucl.ac.uk/id/eprint/1488046
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