Lieb's simple proof of concavity of Tr A^p K^* B^(1-p) K and remarks on related inequalities

Details Lieb's simple proof of concavity of Tr A^p K^* B^(1-p) K and remarks on related inequalities Lieb's simple proof of concavity of Tr A^p K^* B^(1-p) K and remarks on related inequalities

2004-04-21

arxiv.org/abs/quant-ph/0404126v4

International Jour. Quant. Info 3, 570--590 (2005); erratum 4, 747-748 (2006)

Physics

Quantum Physics

Version 3 contains Appendix C, which corrects some flaws in the presentation of the proof of the main theorem

Scientific paper

A simple, self-contained proof is presented for the concavity of the map (A,B) --> Tr(A^p K^* B^(1-p) K). The author makes no claim to originality; this note gives Lieb's original argument in its simplest, rather than its most general, form. A sketch of the chain of implications from this result to concavity of A --> Tr e^[K + log(A)] is then presented. An independent elementary proof is given for the joint convexity of the map (A,B,X) --> Tr \int X^* (A+ uI)^{-1} X (B+ uI)^{-1} du which plays a key role in entropy inequalities.

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