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Article: Dobrushin’s Ergodicity Coefficient for Markov Operators on Cones

TitleDobrushin’s Ergodicity Coefficient for Markov Operators on Cones
Authors
Keywordsordered linear space
zero error capacity
rank one matrix
noncommutative Markov chain
Markov operator
invariant measure
Dobrushin’s ergodicity coefficient
contraction ratio
consensus
quantum channel
Issue Date2014
Citation
Integral Equations and Operator Theory, 2014, v. 81, n. 1, p. 127-150 How to Cite?
Abstract© 2014, Springer Basel. Doeblin and Dobrushin characterized the contraction rate of Markov operators with respect the total variation norm. We generalize their results by giving an explicit formula for the contraction rate of a Markov operator over a cone in terms of pairs of extreme points with disjoint support in a set of abstract probability measures. By duality, we derive a characterization of the contraction rate of consensus dynamics over a cone with respect to Hopf’s oscillation seminorm (the infinitesimal seminorm associated with Hilbert’s projective metric). We apply these results to Kraus maps (noncommutative Markov chains, representing quantum channels), and characterize the ultimate contraction of the map in terms of the existence of a rank one matrix in a certain subspace.
Persistent Identifierhttp://hdl.handle.net/10722/219773
ISSN
2015 Impact Factor: 0.956
2015 SCImago Journal Rankings: 1.348

 

DC FieldValueLanguage
dc.contributor.authorGaubert, Stéphane-
dc.contributor.authorQu, Zheng-
dc.date.accessioned2015-09-23T02:57:55Z-
dc.date.available2015-09-23T02:57:55Z-
dc.date.issued2014-
dc.identifier.citationIntegral Equations and Operator Theory, 2014, v. 81, n. 1, p. 127-150-
dc.identifier.issn0378-620X-
dc.identifier.urihttp://hdl.handle.net/10722/219773-
dc.description.abstract© 2014, Springer Basel. Doeblin and Dobrushin characterized the contraction rate of Markov operators with respect the total variation norm. We generalize their results by giving an explicit formula for the contraction rate of a Markov operator over a cone in terms of pairs of extreme points with disjoint support in a set of abstract probability measures. By duality, we derive a characterization of the contraction rate of consensus dynamics over a cone with respect to Hopf’s oscillation seminorm (the infinitesimal seminorm associated with Hilbert’s projective metric). We apply these results to Kraus maps (noncommutative Markov chains, representing quantum channels), and characterize the ultimate contraction of the map in terms of the existence of a rank one matrix in a certain subspace.-
dc.languageeng-
dc.relation.ispartofIntegral Equations and Operator Theory-
dc.subjectordered linear space-
dc.subjectzero error capacity-
dc.subjectrank one matrix-
dc.subjectnoncommutative Markov chain-
dc.subjectMarkov operator-
dc.subjectinvariant measure-
dc.subjectDobrushin’s ergodicity coefficient-
dc.subjectcontraction ratio-
dc.subjectconsensus-
dc.subjectquantum channel-
dc.titleDobrushin’s Ergodicity Coefficient for Markov Operators on Cones-
dc.typeArticle-
dc.description.natureLink_to_subscribed_fulltext-
dc.identifier.doi10.1007/s00020-014-2193-2-
dc.identifier.scopuseid_2-s2.0-84922106140-
dc.identifier.volume81-
dc.identifier.issue1-
dc.identifier.spage127-
dc.identifier.epage150-

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