Zero-error information theory

Körner, János; Orlitsky, Alon

doi:10.1109/18.720537

Cited by 202 publications

(195 citation statements)

References 113 publications

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“…Given a discrete memoryless channel, it was defined a capacity for transmitting information with an error probability equal to zero. The so called zero-error information theory [4] found applications in areas like graph theory, combinatorics, and computer science.…”

Section: Introductionmentioning

confidence: 99%

Quantum Zero-Error Capacity

Medeiros

Assis

2005

Int. J. Quantum Inform.

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Abstract. The zero-error capacity of quantum channels was defined as the least upper bound of rates at which classical information is transmitted through a quantum channel with probability of error equal to zero. This paper investigates some properties of input states used to attain the zero-error capacity of quantum channels. Initially, we reformulate the problem of finding the zero-error capacity in the language of graph theory. We use this alternative definition to prove that the zero-error capacity of any quantum channel is reached by using only pure states.

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Section: Introductionmentioning

confidence: 99%

Quantum Zero-Error Capacity

Medeiros

Assis

2005

Int. J. Quantum Inform.

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“…From the information theory point of view, the graphs fulfilling the sharp inequality describing channels facilitate an advantage in the zero-error communication [7]. It is interesting that the exact value of the Shannon capacity of C 7 is still unknown.…”

Section: Theorem 12 (Shannon [9]mentioning

confidence: 99%

“…In particular we give a formal definition of the Shannon capacity. For a deeper discussion of this subject we refer the reader to [7,9,8].…”

Section: Introduction and Basic Definitionsmentioning

confidence: 99%

On The Independence Number Of Some Strong Products Of Cycle-Powers

Jurkiewicz

Kubale

Ocetkiewicz

2015

Foundations of Computing and Decision Sciences

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Abstract. In the paper we give some theoretical and computational results on the third strong power of cycle-powers, for example, we have found the independence numbers α((C 2 10 ) ⊠3 ) = 30 and α((C 4 14 ) ⊠3 ) = 14. A number of optimizations have been introduced to improve the running time of our exhaustive algorithm used to establish the independence number of the third strong power of cycle-powers. Moreover, our results establish new exact values and/or lower bounds on the Shannon capacity of noisy channels.

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“…It is well-known that when communicating by block codes over a discrete memoryless channel at rate below the capacity, the error probability goes to zero exponentially in the block length, and while it is one of the major open problems of information theory to characterize the tradeoff between rate and error exponent in general, we have by now a fairly good understanding of it. However, if the error probability is required to vanish faster than exponential, or equivalently is required to be zero exactly (at least in the case of finite alphabets), we enter the strange and much less understood realm of zero-error information theory [37,45], which concerns asymptotic combinatorial problems, most of which are unsolved and are considered very difficult. There are a couple of exceptions to this rather depressing state of affairs, one having been already identified by Shannon in his founding paper [45], namely the discrete memoryless channel N (y|x) assisted by instantaneous noiseless feedback, whose capacity is given by the fractional packing number of a bipartite graph Γ representing the possible transitions N (y|x) > 0.…”

Section: Zero-error Communication Assisted By Noiseless Quantum Fementioning

confidence: 99%

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

Duan

Severini

Winter

2016

IEEE Trans. Inform. Theory

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We initiate the study of zero-error communication via quantum channels when the receiver and 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 a function only 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 [Pusey/Barrett/Rudolph, Nature Phys. 8(6):475-478, 2012].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 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 feedbackassisted unambiguous capacity. The proof relies on a generalization of the "Postselection Lemma" (de Finetti reduction) [Christandl/König/Renner, Phys. Rev. Lett. 102:020504, 2009] 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. a A preliminary version of this paper was presented as a poster at QIP 2012, 12-16 December 2011

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Zero-error information theory

Cited by 202 publications

References 113 publications

Quantum Zero-Error Capacity

Quantum Zero-Error Capacity

On The Independence Number Of Some Strong Products Of Cycle-Powers

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

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