  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.
