From joint convexity of quantum relative entropy to a concavity theorem of Lieb

Tropp, Joel A. (2012) From joint convexity of quantum relative entropy to a concavity theorem of Lieb. Proceedings of the American Mathematical Society, 140 (5). pp. 1757-1760. ISSN 0002-9939. doi:10.1090/S0002-9939-2011-11141-9. https://resolver.caltech.edu/CaltechAUTHORS:20120515-094709707

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Abstract

This paper provides a succinct proof of a 1973 theorem of Lieb that establishes the concavity of a certain trace function. The development relies on a deep result from quantum information theory, the joint convexity of quantum relative entropy, as well as a recent argument due to Carlen and Lieb.

Item Type:

Article

Related URLs:

URL	URL Type	Description
http://dx.doi.org/10.1090/S0002-9939-2011-11141-9	DOI	Article
http://www.ams.org/journals/proc/2012-140-05/S0002-9939-2011-11141-9/	Publisher	Article
https://arxiv.org/abs/1101.1070	arXiv	Discussion Paper

ORCID:

Author	ORCID
Tropp, Joel A.	0000-0003-1024-1791

Additional Information:

© 2012 American Mathematical Society. Received by the editors January 2, 2011 and, in revised form, January 4, 2011. The author thanks Eric Carlen for a very illuminating discussion of matrix convexity theorems, including the paper [CL08], as well as comments on an early draft of this paper. Edward Effros contributed insights on quantum information theory, and Elliott Lieb emphasized the equivalences among concavity theorems. This work has been supported in part by ONR awards N00014-08-1-0883 and N00014-11-1-0025, AFOSR award FA9550-09-1-0643, and a Sloan Fellowship. The research was performed while the author attended the IPAM Fall 2010 program on optimization.

Funders:

Funding Agency	Grant Number
Office of Naval Research (ONR)	N00014-08-1-0883
Office of Naval Research (ONR)	N00014-11-1- 0025
Air Force Office of Scientific Research (AFOSR)	FA9550-09-1-0643
Alfred P. Sloan Foundation	UNSPECIFIED

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Classification Code:

MSC (2010): Primary 52A41

DOI:

10.1090/S0002-9939-2011-11141-9

Record Number:

CaltechAUTHORS:20120515-094709707

Persistent URL:

https://resolver.caltech.edu/CaltechAUTHORS:20120515-094709707

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No commercial reproduction, distribution, display or performance rights in this work are provided.

ID Code:

31462

Collection:

CaltechAUTHORS

Deposited By:

Jason Perez

Deposited On:

15 May 2012 21:24

Last Modified:

09 Nov 2021 19:53

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