2017/09/16 by Eric A. Carlen, Elliott H. Lieb, Carlen, Eric A. +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1709.05450
In version 2 we have added two 1994 references, one to Ando and Hiai and one to Hiai
arxiv created 2018/04/30 · arxiv updated 2018/05/02
Consider a function F(X,Y) of pairs of positive matrices with values in the positive matrices such that whenever X and Y commute F(X,Y)= XpYq. Our first main result gives conditions on F such that \rm Tr[ X log (F(Z,Y))] ≤ \rm Tr[X(plog X + q log Y)] for all X,Y,Z such that \rm Tr Z = \rm Tr X. (Note that Z is absent from the right side of the inequality.) We give several examples of functions F to which the theorem applies. Our theorem allows us to give simple proofs of the well known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables X,Y,Z instead of just X,Y alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy D(X||Y) = \rm Tr [X(log X-log Y]), and two others, the Donald relative entropy DD(X||Y), and the Belavkin-Stasewski relative entropy DBS(X||Y). They are known to satisfy DD(X||Y) ≤ D(X||Y)≤ DBS(X||Y). We prove that the Donald relative entropy provides the sharp upper bound, independent of Z, on \rm Tr[ X log (F(Z,Y))] in a number of cases in which (Z,Y) is homogeneous of degree 1 in Z and -1 in Y. We also investigate the Legendre transforms in X of DD(X||Y) and DBS(X||Y), and show how our results for these lead to new refinements of the Golden-Thompson inequality.