Information geometry and reduced quantum description

We consider the quantum dynamics of a system with two time-scales, having discrete time and unbounded slow variables, under a useful set of axioms. We argue that the Kubo susceptibility tensor is the appropriate generalisation of the Fisher information metric on the set of density matrices. We define the Amari connections ▽(±1) in the unbounded case. The geometric picture of reduced quantum dynamics, described by Balian, Alhassid and Reinhardt [Phys. Rep. 131, 2–146], is extended to some unbounded cases. We show that the reduced dynamics is a contraction in the Bogoliubov norm.