Rényi Divergences as Weighted Non-commutative Vector-Valued L<inf>p</inf> -Spaces

Berta, M; Scholz, VB; Tomamichel, M

Rényi Divergences as Weighted Non-commutative Vector-Valued L<inf>p</inf> -Spaces

Berta, M Scholz, VB Tomamichel, M

Permalink

Publication Type:: Journal Article
Citation:: Annales Henri Poincare, 2018, 19 (6), pp. 1843 - 1867
Issue Date:: 2018-06-01

Open Access

Copyright Clearance Process

Recently Added
In Progress
Open Access

This item is open access.

Adobe PDF

Download Accepted Manuscript VersionAdobe PDF (295.69 kB)

View on publisher's site

View statistics

Full metadata record

Field	Value	Language
dc.contributor.author	Berta, M	en_US
dc.contributor.author	Scholz, VB	en_US
dc.contributor.author	Tomamichel, M https://orcid.org/0000-0001-5410-3329	en_US
dc.date.issued	2018-06-01	en_US
dc.identifier.citation	Annales Henri Poincare, 2018, 19 (6), pp. 1843 - 1867	en_US
dc.identifier.issn	1424-0637	en_US
dc.identifier.uri	http://hdl.handle.net/10453/129759
dc.description.abstract	© 2018, Springer International Publishing AG, part of Springer Nature. We show that Araki and Masuda’s weighted non-commutative vector-valued Lp-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=p2. 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→{12,1,∞} 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 Lp-spaces and an Araki–Lieb–Thirring inequality for states on von Neumann algebras.	en_US
dc.relation.ispartof	Annales Henri Poincare	en_US
dc.relation.isbasedon	10.1007/s00023-018-0670-x	en_US
dc.subject.classification	Mathematical Physics	en_US
dc.title	Rényi Divergences as Weighted Non-commutative Vector-Valued L<inf>p</inf> -Spaces	en_US
dc.type	Journal Article
utslib.citation.volume	6	en_US
utslib.citation.volume	19	en_US
utslib.for	0105 Mathematical Physics	en_US
utslib.for	0202 Atomic, Molecular, Nuclear, Particle and Plasma Physics	en_US
pubs.embargo.period	Not known	en_US
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology/School of Computer Science
pubs.organisational-group	/University of Technology Sydney/Strength - QSI - Centre for Quantum Software and Information
utslib.copyright.status	open_access
pubs.issue	6	en_US
pubs.publication-status	Published	en_US
pubs.volume	19	en_US

Abstract:

© 2018, Springer International Publishing AG, part of Springer Nature. We show that Araki and Masuda’s weighted non-commutative vector-valued Lp-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=p2. 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→{12,1,∞} 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 Lp-spaces and an Araki–Lieb–Thirring inequality for states on von Neumann algebras.

Please use this identifier to cite or link to this item:

http://hdl.handle.net/10453/129759