%0 Journal Article
%J IEEE Transactions on Information Theory
%D 2016
%T On {Zero}-{Error} {Communication} via {Quantum} {Channels} in the {Presence} of {Noiseless} {Feedback}
%A Duan, R.
%A Severini, S.
%A Winter, A.
%K bipartite equivocation graph
%K Bipartite graph
%K Capacity planning
%K Choi-Kraus operators
%K coding theorem
%K commutative bipartite graph
%K Electronic mail
%K encoding
%K entanglement-assisted capacity
%K feedback
%K feedback-assisted capacity
%K feedback-assisted zero-error capacity
%K information theory
%K noiseless feedback channel
%K quantum channels
%K quantum communication
%K quantum entanglement
%K Quantum information
%K Receivers
%K Shannon's zero-error communication
%K telecommunication channels
%K unlimited quantum capacity
%K zero-error capacity
%X 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 We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede.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.
%B IEEE Transactions on Information Theory
%V 62
%P 5260–5277
%8 sep
%R 10.1109/TIT.2016.2562580