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Conference Paper: Markov operators on cones and non-commutative consensus

TitleMarkov operators on cones and non-commutative consensus
Authors
Issue Date2013
Citation
2013 European Control Conference, ECC 2013, 2013, p. 2693-2700 How to Cite?
AbstractThe analysis of classical consensus algorithms relies on contraction properties of Markov matrices with respect to the Hilbert semi-norm (infinitesimal version of Hilbert's projective metric) and to the total variation norm. We generalize these properties to the case of operators on cones. This is motivated by the study of 'non-commutative consensus', i.e., of the dynamics of linear maps leaving invariant cones of positive semi-definite matrices. Such maps appear in quantum information (Kraus maps), and in the study of matrix means. We give a characterization of the contraction rate of an abstract Markov operator on a cone, which extends classical formulæ obtained by Dœblin and Dobrushin in the case of Markov matrices. In the special case of Kraus maps, we relate the absence of contraction to the positivity of the 'zero-error capacity' of a quantum channel. We finally show that a number of decision problems concerning the contraction rate of Kraus maps reduce to finding a rank one matrix in linear spaces satisfying certain conditions and discuss complexity issues. © 2013 EUCA.
Persistent Identifierhttp://hdl.handle.net/10722/219736

 

DC FieldValueLanguage
dc.contributor.authorGaubert, Stephane-
dc.contributor.authorQu, Zheng-
dc.date.accessioned2015-09-23T02:57:50Z-
dc.date.available2015-09-23T02:57:50Z-
dc.date.issued2013-
dc.identifier.citation2013 European Control Conference, ECC 2013, 2013, p. 2693-2700-
dc.identifier.urihttp://hdl.handle.net/10722/219736-
dc.description.abstractThe analysis of classical consensus algorithms relies on contraction properties of Markov matrices with respect to the Hilbert semi-norm (infinitesimal version of Hilbert's projective metric) and to the total variation norm. We generalize these properties to the case of operators on cones. This is motivated by the study of 'non-commutative consensus', i.e., of the dynamics of linear maps leaving invariant cones of positive semi-definite matrices. Such maps appear in quantum information (Kraus maps), and in the study of matrix means. We give a characterization of the contraction rate of an abstract Markov operator on a cone, which extends classical formulæ obtained by Dœblin and Dobrushin in the case of Markov matrices. In the special case of Kraus maps, we relate the absence of contraction to the positivity of the 'zero-error capacity' of a quantum channel. We finally show that a number of decision problems concerning the contraction rate of Kraus maps reduce to finding a rank one matrix in linear spaces satisfying certain conditions and discuss complexity issues. © 2013 EUCA.-
dc.languageeng-
dc.relation.ispartof2013 European Control Conference, ECC 2013-
dc.titleMarkov operators on cones and non-commutative consensus-
dc.typeConference_Paper-
dc.description.natureLink_to_subscribed_fulltext-
dc.identifier.scopuseid_2-s2.0-84893256309-
dc.identifier.spage2693-
dc.identifier.epage2700-

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