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Rényi divergences as weighted non-commutative vector valued L_p-spaces

Berta, Mario and Scholz, Volkher B. and Tomamichel, Marco (2018) Rényi divergences as weighted non-commutative vector valued L_p-spaces. Annales Henri Poincaré, 19 (6). pp. 1843-1867. ISSN 1424-0637. https://resolver.caltech.edu/CaltechAUTHORS:20170817-111159578

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Abstract

We show that Araki and Masuda’s weighted non-commutative vector-valued L_p-spaces (Araki and Masuda in Publ Res Inst Math Sci Kyoto Univ 18:339–411, 1982) correspond to an algebraic generalization of the sandwiched Rényi divergences with parameter α = p/2. Using complex interpolation theory, we prove various fundamental properties of these divergences in the setup of von Neumann algebras, including a data-processing inequality and monotonicity in α. We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases α → {1/2,1,∞} leading to minus the logarithm of Uhlmann’s fidelity, Umegaki’s relative entropy, and the max-relative entropy, respectively. As a contribution that might be of independent interest, we derive a Riesz–Thorin theorem for Araki–Masuda L_p-spaces and an Araki–Lieb–Thirring inequality for states on von Neumann algebras.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/s00023-018-0670-xDOIArticle
https://rdcu.be/bbCOWPublisherFree ReadCube access
https://arxiv.org/abs/1608.05317arXivDiscussion Paper
ORCID:
AuthorORCID
Berta, Mario0000-0002-0428-3429
Tomamichel, Marco0000-0001-5410-3329
Additional Information:© 2018 Springer International Publishing AG, part of Springer Nature. First Online: 17 March 2018. We thank Anna Jenčová for pointing out a mistake in Eq. (9) in a previous version of this manuscript [16]. MB and MT thank the Department of Physics at Ghent University, and MB and VBS thank the School of Physics at University of Sydney for their hospitality while part of this work was done. VBS is supported by the EU through the ERC Qute. MT is funded by an ARC Discovery Early Career Researcher Award (DECRA) fellowship and acknowledges support from the ARC Centre of Excellence for Engineered Quantum Systems (EQUS).
Group:IQIM, Institute for Quantum Information and Matter
Funders:
Funding AgencyGrant Number
European Research Council (ERC)UNSPECIFIED
Australian Research CouncilUNSPECIFIED
Issue or Number:6
Classification Code:Mathematics Subject Classification: Primary 81P45, 81R15, 94A17. Secondary 46L52
Record Number:CaltechAUTHORS:20170817-111159578
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170817-111159578
Official Citation:Berta, M., Scholz, V.B. & Tomamichel, M. Ann. Henri Poincaré (2018) 19: 1843. https://doi.org/10.1007/s00023-018-0670-x
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:80558
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:17 Aug 2017 18:19
Last Modified:03 Oct 2019 18:32

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