On the relation between the monotone Riemannian metrics on the space of Gibbs thermal states and the linear response theory

TL;DR

This paper extends the spectral and linear response analysis of monotone Riemannian metrics on Gibbs states, linking quantum information geometry with statistical mechanics and Green's functions.

Contribution

It provides a new spectral representation of all monotone Riemannian metrics using the dynamical structure factor, connecting quantum information geometry with linear response theory.

Findings

Spectral representation of monotone Riemannian metrics derived

One-to-one correspondence established between metrics and operator monotone functions

Inequalities between different metrics demonstrated

Abstract

The proposed in J. Math. Phys. v.57,071903 (2016) analytical expansion of monotone (contractive) Riemannian metrics (called also quantum Fisher information(s)) in terms of moments of the dynamical structure factor (DSF) relative to an original intensive observable is reconsidered and extended. The new approach through the DSF which characterizes fully the set of monotone Riemannian metrics on the space of Gibbs thermal states is utilized to obtain an extension of the spectral presentation obtained for the Bogoliubov-Kubo-Mori metric (the generalized isothermal susceptibility) on the entire class of monotone Riemannian metrics. The obtained spectral presentation is the main point of our consideration. The last allows to present the one to one correspondence between monotone Riemannian metrics and operator monotone functions (which is a statement of the Petz theorem in the quantum information theory) in terms of the linear response theory. We show that monotone Riemannian metrics can be determined from the analysis of the infinite chain of equations of motion of the retarded Green's functions. Inequalities between the different metrics have been obtained as well. It is a demonstration that the analysis of information-theoretic problems has benefited from concepts of statistical mechanics and might cross-fertilize or extend both directions, and vice versa. We illustrate the presented approach on the calculation of the entire class of monotone (contractive) Riemannian metrics on the examples of some simple but instructive systems employed in various physical problems.