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SUMMARY:Quantum f-divergences in von Neumann algebras - Fumio Hiai (Tohoku
  University )
DTSTART:20180726T134500Z
DTEND:20180726T143000Z
UID:TALK108397@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:This talk is a comprehensive survey on recent developments of 
 quantum divergences in general von Neumann algebras\, including standard f
 -divergences\, maximal f-divergences\, and R\\&#39\;enyi type divergences\
 , whose mathematical backgrounds are Haagerup&#39\;s L^p-spaces and Araki&
 #39\;s relative modular operator. Standard f-divergences were formerly stu
 died by Petz in a bit more general form with name quasi-entropy\, whose mo
 st familiar one is the relative entropy initiated by Umegaki and extended 
 to general von Neumann algebras by Araki. We extend Kosaki&#39\;s variatio
 nal expression of the relative entropy to an arbitrary standard f-divergen
 ce\, from which most important properties of standard f-divergences follow
  immediately. We also go into standard R\\&#39\;enyi divergences (as a var
 iation of standard f-divergences) in some detail\, and touch briefly sandw
 iched R\\&#39\;enyi divergences in von Neumann algebras\, which have recen
 tly been developed by Jen\\v cov\\&#39\;a and Berta-Scholz-Tomamichel. Fin
 ally\, we treat maximal f-divergences and discuss their definition\, integ
 ral expression\, and comparison with standard f-divergences. This talk is 
 dedicated to the memory of D\\&#39\;enes Petz.
LOCATION:Seminar Room 1\, Newton Institute
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