The Golden-Thompson inequality --- historical aspects and random matrix applications

Abstract

The Golden-Thompson inequality, $\mathrm{Tr}(e^{A+B})\le \mathrm{Tr}(e^A{e}^B)$ for $A,B$ Hermitian matrices, appeared in independent works by Golden and Thompson published in 1965. Both of these were motivated by considerations in statistical mechanics. In recent years the Golden-Thompson inequality has found applications to random matrix theory. In this survey article we detail some historical aspects relating to Thompson's work, giving in particular an hitherto unpublished proof due to Dyson, and correspondence with Pólya. We show too how the $2\times 2$ case relates to hyperbolic geometry, and how the original inequality holds true with the trace operation replaced by any unitarily invariant norm. In relation to the random matrix applications, we review its use in the derivation of concentration type lemmas for sums of random matrices due to Ahlswede-Winter, and Oliveira, generalizing various classical results.

arXiv comment17 pages

References in corpus (5)

• User-friendly tail bounds for sums of random matrices
• Truncations of random unitary matrices
• Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges
• How many entries of a typical orthogonal matrix can be approximated by independent normals?
• Noncommutative martingale deviation and Poincaré type inequalities with applications

Cited by in corpus (10)

• A lower bound of quantum conditional mutual information
• Noncommutative martingale concentration inequalities
• Multivariate extensions of the Golden-Thompson inequality
• A variant of Horn's problem and derivative principle
• Graph colouring and the total Betti number
• On Araki-Type Trace Inequalities
• Free Energy Subadditivity for Symmetric Random Hamiltonians
• Golden-Thompson via pinching inequality
• Ando-Hiai and Golden-Thomspon inequalities
• High temperature convergence of the KMS boundary conditions: The Bose-Hubbard model on a finite graph