TY - JOUR
T1 - On {Zero}-{Error} {Communication} via {Quantum} {Channels} in the {Presence} of {Noiseless} {Feedback}
JF - IEEE Transactions on Information Theory
Y1 - 2016
A1 - Duan, R.
A1 - Severini, S.
A1 - Winter, A.
KW - bipartite equivocation graph
KW - Bipartite graph
KW - Capacity planning
KW - Choi-Kraus operators
KW - coding theorem
KW - commutative bipartite graph
KW - Electronic mail
KW - encoding
KW - entanglement-assisted capacity
KW - feedback
KW - feedback-assisted capacity
KW - feedback-assisted zero-error capacity
KW - information theory
KW - noiseless feedback channel
KW - quantum channels
KW - quantum communication
KW - quantum entanglement
KW - Quantum information
KW - Receivers
KW - Shannon's zero-error communication
KW - telecommunication channels
KW - unlimited quantum capacity
KW - zero-error capacity
AB - 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 We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede.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.
VL - 62
ER -