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Monotonicity of quantum relative entropy and recoverability

Berta, Mario and Lemm, Marius and Wilde, Mark M. (2015) Monotonicity of quantum relative entropy and recoverability. Quantum Information and Computation, 15 (15-16). pp. 1333-1354. ISSN 1533-7146. https://resolver.caltech.edu/CaltechAUTHORS:20160301-081841151

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Abstract

The relative entropy is a principal measure of distinguishability in quantum information theory, with its most important property being that it is non-increasing with respect to noisy quantum operations. Here, we establish a remainder term for this inequality that quantifies how well one can recover from a loss of information by employing a rotated Petz recovery map. The main approach for proving this refinement is to combine the methods of [Fawzi and Renner, 2014] with the notion of a relative typical subspace from [Bjelakovic and Siegmund-Schultze, 2003]. Our paper constitutes partial progress towards a remainder term which features just the Petz recovery map (not a rotated Petz map), a conjecture which would have many consequences in quantum information theory. A well known result states that the monotonicity of relative entropy with respect to quantum operations is equivalent to each of the following inequalities: strong subadditivity of entropy, concavity of conditional entropy, joint convexity of relative entropy, and monotonicity of relative entropy with respect to partial trace. We show that this equivalence holds true for refinements of all these inequalities in terms of the Petz recovery map. So either all of these refinements are true or all are false.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dl.acm.org/citation.cfm?id=2871383PublisherArticle
http://arxiv.org/abs/1412.4067arXivDiscussion Paper
ORCID:
AuthorORCID
Berta, Mario0000-0002-0428-3429
Additional Information:© 2015 Rinton Press. We are especially grateful to Rupert Frank for many discussions on the topic of this paper. We thank the anonymous referees for many suggestions that helped to improve the paper. We acknowledge additional discussions with Siddhartha Das, Nilanjana Datta, Omar Fawzi, Renato Renner, Volkher Scholz, Kaushik P. Seshadreesan, Marco Tomamichel, and Michael Walter. MMW acknowledges support from startup funds from the Department of Physics and Astronomy at LSU, the NSF under Award No. CCF-1350397, and the DARPA Quiness Program through US Army Research Office award W31P4Q-12-1-0019.
Group:IQIM, Institute for Quantum Information and Matter
Funders:
Funding AgencyGrant Number
Louisiana State University (LSU)UNSPECIFIED
NSFCCF-1350397
Army Research Office (ARO)W31P4Q-12-1-0019
Issue or Number:15-16
Record Number:CaltechAUTHORS:20160301-081841151
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160301-081841151
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:64891
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:01 Mar 2016 23:27
Last Modified:03 Oct 2019 09:42

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