Bogoliubov's metric as a global characteristic of the family of metrics in the Hilbert algebra of observables

We comparatively analyze a one-parameter family of bilinear complex functionals with the sense of "deformed" Wigner–Yanase–Dyson scalar products on the Hilbert algebra of operators of physical observables. We establish that these functionals and the corresponding metrics depend on the deformation parameter and the extremal properties of the Kubo–Martin–Schwinger and Wigner–Yanase metrics in quantum statistical mechanics. We show that the Bogoliubov–Kubo–Mori metric is a global (integral) characteristic of this family. It occupies an intermediate position between the extremal metrics and has the clear physical sense of the generalized isothermal susceptibility. We consider the example for the $SU(2)$ algebra of observables in the simplest model of an ideal quantum spin paramagnet.