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Quantum conditional relative entropy and quasi-factorization of the relative entropy

Capel, Ángela and Lucia, Angelo and Pérez-García, David (2018) Quantum conditional relative entropy and quasi-factorization of the relative entropy. Journal of Physics A: Mathematical and Theoretical, 51 (48). Art. No. 484001. ISSN 1751-8113. doi:10.1088/1751-8121/aae4cf. https://resolver.caltech.edu/CaltechAUTHORS:20181101-102812455

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Abstract

The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub-σ-algebras. In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasi-factorization results for it. As an illustration of their potential, we use one of them to obtain a positive log-Sobolev constant for the heat-bath dynamics with product fixed point.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1088/1751-8121/aae4cfDOIArticle
https://arxiv.org/abs/1804.09525arXivDiscussion Paper
ORCID:
AuthorORCID
Capel, Ángela0000-0001-6713-6760
Lucia, Angelo0000-0003-1709-1220
Pérez-García, David0000-0003-2990-791X
Additional Information:© 2018 IOP Publishing Ltd. Received 30 April 2018; Accepted 27 September 2018; Accepted Manuscript online 27 September 2018; Published 1 November 2018. The authors would like to thank David Sutter, Ivan Bardet and Andreas Bluhm for very helpful conversations. AC and DPG acknowledge support from MINECO (grants MTM2014-54240-P and MTM2017-88385-P), from Comunidad de Madrid (grant QUITEMAD+-CM, ref. S2013/ICE-2801), and the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 648913). AC is partially supported by a La Caixa-Severo Ochoa grant (ICMAT Severo Ochoa project SEV-2011-0087, MINECO). AL acknowledges financial support from the European Research Council (ERC Grant Agreement no 337603), the Danish Council for Independent Research (Sapere Aude) and VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059). This work has been partially supported by ICMAT Severo Ochoa project SEV-2015-0554 (MINECO).
Funders:
Funding AgencyGrant Number
Ministerio de Economía, Industria y Competitividad (MINECO)MTM2014-54240-P
Ministerio de Economía, Industria y Competitividad (MINECO)MTM2017-88385-P
Comunidad de MadridS2013/ICE-2801
European Research Council (ERC)648913
Centro de Excelencia Severo OchoaSEV-2011-0087
European Research Council (ERC)337603
Danish Council for Independent ResearchUNSPECIFIED
Villum Foundation10059
Ministerio de Economía, Industria y Competitividad (MINECO)SEV-2015-0554
Issue or Number:48
DOI:10.1088/1751-8121/aae4cf
Record Number:CaltechAUTHORS:20181101-102812455
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20181101-102812455
Official Citation:Ángela Capel et al 2018 J. Phys. A: Math. Theor. 51 484001
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:90558
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:01 Nov 2018 17:39
Last Modified:12 Jul 2022 19:52

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