Abstract
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) = (XYq)-Y-p. Our first main result gives conditions on F such that Tr[X log(F(Z, Y))] <= Tr[X(p logX + q log Y)] for all X, Y, Z such that TrZ = TrX. (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 vertical bar vertical bar Y) = Tr[X(logX - log Y]), and two others, the Donald relative entropy D-D(X vertical bar vertical bar Y), and the Belavkin-Stasewski relative entropy D-BS(X vertical bar vertical bar Y). They are known to satisfy D-D(X vertical bar vertical bar Y) <= D(X vertical bar vertical bar Y) <= D-BS(X vertical bar vertical bar Y). We prove that the Donald relative entropy provides the sharp upper bound, independent of Z on Tr[X log(F(Z, Y))] in a number of cases in which F(Z, Y) is homogeneous of degree 1 in Z and -1 in Y. We also investigate the Legendre transforms in X of D-D(X vertical bar vertical bar Y) and D-BS(X vertical bar vertical bar Y), and show how our results for these lead to new refinements of the Golden-Thompson inequality.