TY - JOUR
AU1 - Carlen, Eric
AU2 - Lieb, Elliott
AB - Bull. Math. Sci. https://doi.org/10.1007/s13373-018-0123-3 Some trace inequalities for exponential and logarithmic functions 1 2 Eric A. Carlen · Elliott H. Lieb Received: 8 October 2017 / Revised: 17 April 2018 / Accepted: 24 April 2018 © The Author(s) 2018 Abstract Consider a function F ( X, Y ) of pairs of positive matrices with values in the p q positive matrices such that whenever X and Y commute F ( X, Y ) = X Y . Our first main result gives conditions on F such that Tr[ X log( F ( Z , Y ))]≤ Tr[ X ( p log X + q log Y )] for all X, Y, Z such that Tr Z = 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 logarith- mic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki 
TI - Some trace inequalities for exponential and logarithmic functions
JF - Bulletin of Mathematical Sciences
DO - 10.1007/s13373-018-0123-3
DA - 2018-05-29
UR - https://www.deepdyve.com/lp/springer-journals/some-trace-inequalities-for-exponential-and-logarithmic-functions-FQfDfz06AY
SP - 1
EP - 40
VL - OnlineFirst
IS - 
DP - DeepDyve
ER -