Bregman divergence as relative operator entropy

vet C ˜e — ™onvex set in — f—n—™h sp—™e —nd r ˜e — ril˜ert sp—™eF por —n oper—torE v—lued smooth fun™tion—l © X C 3 B@rAD we dene the Bregman divergence of x; y P C —s D © @x; yA Xa ©@xA ©@yA lim t3CH t I ©@y C t@x yAA ©@yA : ‚e™—ll th—t origin—lly vev fregm—n introdu™ed this ™on™ept for re—lEv—lued ™onvex fun™E tions © ‘P“F yn the selfE—djoint oper—torsD there is — st—nd—rd p—rti—l orderingD A BD if h; Ai h; Bi for —ny ve™tor in the ril˜ert sp—™eF ren™e the ™onvexity of © X C 3 B@rA m—kes sense —nd D © @x; yA ! H rem—ins trueF ‡e —re ˜—si™—lly interested in the ™—se when C is the set of positive semidenite m—tri™es of tr—™e I —nd ©@xA a @xAD where @tA a t log t is — ™ommon fun™tion dened on R C —nd @xA is dened ˜y the fun™tion—l ™—l™ulusF roweverD we —lso ™onsider ©@xA a f @xA for — gener—l smooth fun™tion f F ‚emem˜er th—t positive semidenite m—tri™es of tr—™e I —re known —s density m—triE ™es —nd pl—y ™entr—l role in qu—ntum inform—tion theory ‘TD U“F gon™erning …meg—ki9s qu—ntum rel—tive entropy gh—pF I of ‘U“ is our m—in referen™eF 1