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Published May 2012 | Published + Submitted
Journal Article Open

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

Tropp, Joel A.


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.

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.

Attached Files

Submitted - 1101.1070.pdf

Published - Tropp2012p18146P_Am_Math_Soc.pdf


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August 19, 2023
August 19, 2023