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dc.contributor.authorDatta, Nilanjana
dc.contributor.authorRouzé, Cambyse
dc.date.accessioned2021-02-04T16:25:37Z
dc.date.available2021-02-04T16:25:37Z
dc.date.issued2020-02-05
dc.date.submitted2019-07-09
dc.identifier.issn1424-0637
dc.identifier.others00023-020-00891-8
dc.identifier.other891
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/317162
dc.description.abstractAbstract: Quantum Markov semigroups characterize the time evolution of an important class of open quantum systems. Studying convergence properties of such a semigroup and determining concentration properties of its invariant state have been the focus of much research. Quantum versions of functional inequalities (like the modified logarithmic Sobolev and Poincaré inequalities) and the so-called transportation cost inequalities have proved to be essential for this purpose. Classical functional and transportation cost inequalities are seen to arise from a single geometric inequality, called the Ricci lower bound, via an inequality which interpolates between them. The latter is called the HWI inequality, where the letters I, W and H are, respectively, acronyms for the Fisher information (arising in the modified logarithmic Sobolev inequality), the so-called Wasserstein distance (arising in the transportation cost inequality) and the relative entropy (or Boltzmann H function) arising in both. Hence, classically, the above inequalities and the implications between them form a remarkable picture which relates elements from diverse mathematical fields, such as Riemannian geometry, information theory, optimal transport theory, Markov processes, concentration of measure and convexity theory. Here, we consider a quantum version of the Ricci lower bound introduced by Carlen and Maas and prove that it implies a quantum HWI inequality from which the quantum functional and transportation cost inequalities follow. Our results hence establish that the unifying picture of the classical setting carries over to the quantum one.
dc.languageen
dc.publisherSpringer International Publishing
dc.rightsAttribution 4.0 International (CC BY 4.0)en
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/en
dc.subjectOriginal Paper
dc.titleRelating Relative Entropy, Optimal Transport and Fisher Information: A Quantum HWI Inequality
dc.typeArticle
dc.date.updated2021-02-04T16:25:37Z
prism.endingPage2150
prism.issueIdentifier7
prism.publicationNameAnnales Henri Poincaré
prism.startingPage2115
prism.volume21
dc.identifier.doi10.17863/CAM.64273
dcterms.dateAccepted2020-01-21
rioxxterms.versionofrecord10.1007/s00023-020-00891-8
rioxxterms.versionVoR
rioxxterms.licenseref.urihttp://creativecommons.org/licenses/by/4.0/
dc.identifier.eissn1424-0661
pubs.funder-project-idMunich Center for Quantum Science and Technology (DFG cluster of excellence 2111)


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Attribution 4.0 International (CC BY 4.0)
Except where otherwise noted, this item's licence is described as Attribution 4.0 International (CC BY 4.0)