2nd Reading May 13,2019 11:36 WSPC/1664-3607 319-BMS 1950008 Bulletin of Mathematical Sciences Vol. 1, No. 2 (2019) 1950008 (39 pages) c The Author(s) ⃝ DOI: 10.1142/S1664360719500085 Some trace inequalities for exponential and logarithmic functions , , Eric A. Carlen∗ ‡ and Elliott H. Lieb† § ∗Department of Mathematics, Hill Center, Rutgers University 110 Frelinghuysen Road Piscataway, NJ 08854-8019, USA †Departments of Mathematics and Physics Jadwin Hall, Princeton University Washington Road, Princeton, NJ 08544, USA ‡
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[email protected] Received 8 October 2017 Accepted 24 April 2018 Published 14 May 2019 Communicated by Ari Laptev 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 )=XpY q . 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 TrZ =TrX.(NotethatZ is absent from the right side of the inequal- ity.) 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 Bull. Math. Sci. Downloaded from www.worldscientific.com just X, Y alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum rel- ative entropy D(X Y )=Tr[X(log X log Y ]), and two others, the Donald relative ∥ − entropy D (X Y ), and the Belavkin–Stasewski relative entropy D (X Y ).