Some trace inequalities for exponential and logarithmic functions

Eric A Carlen,Elliott H Lieb

doi:10.1007/s13373-018-0123-3

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 $$F(X,Y)= X^pY^q.$$ Our first main result gives conditions on F such that $$\mathrm{Tr}[ X \log (F(Z,Y))] \le \mathrm{Tr}[X(p\log X + q \log Y)]$$ for all X, Y, Z such that $$\mathrm{Tr}Z =\mathrm{Tr}X$$ . (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 $$D(X||Y) = \mathrm{Tr}[X(\log X-\log Y])$$ , and two others, the Donald relative entropy $$D_D(X||Y)$$ , and the Belavkin–Stasewski relative entropy $$D_{BS}(X||Y)$$ . They are known to satisfy $$D_D(X||Y) \le D(X||Y)\le D_{BS}(X||Y)$$ . We prove that the Donald relative entropy provides the sharp upper bound, independent of Z on $$\mathrm{Tr}[ X \log (F(Z,Y))]$$ 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 $$D_D(X||Y)$$ and $$D_{BS}(X||Y)$$ , and show how our results about these Legendre transforms lead to new refinements of the Golden–Thompson inequality.

Highlights

Let Mn denote the set of complex n × n matrices
For X, Y ∈ Hn, X ≥ Y indicates that X − Y is positive semi-definite; i.e., in the closure of Pn, and X > Y indicates that X ∈ Pn
In particular, that the geometric mean provides a natural bridge between the pair of inequalities (1.6)

Summary

Introduction

Let Mn denote the set of complex n × n matrices. Let Pn and Hn denote the subsets of Mn consisting of strictly positive and self-adjoint matrices respectively. The resulting functional is still bounded above by the Umegaki relative entropy: that is, for all density matrices X and W , sup{Tr[X log(W −1/2 Z W −1/2)] : Z ≥ 0 Tr[Z ] ≤ 1} ≤ D(X ||W ). The goal is to show that the inequality on the right in (1.6) is a consequence of Theorem 1.6 by a simple differentiation argument This simple proof is the new feature, The statement concerning cases of equality was proved in [20]. There is a complement to Theorem 2.5 in the case Z = X that is equivalent to a result of Hiai and Petz, who formulate it differently and do not discuss extended geometric means.

Quantum relative entropy inequalities

Exponential inequalities related to the Golden–Thompson inequality

A The Peierls–Bogoliubov Inequality and the Gibbs Variational Principle

B Majorization inequalities

C Geodesics and geometric means