Information Geometry and Reduced Quantum Description

by R. F. Streater; Reports on Mathematical Phys., 38, 419-436, 1996.

ABSTRACT

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 $\nabla^{\pm}$ in the unbounded case. The geometric picture of reduced quantum dynamics, described by Balian, Alhassid and Reinhardt [Phys Reports, 131, 2-146] is extended to some unbounded cases. We show that the reduced dynamics is a contraction in the Bogoliubov norm.

There is a misprint on the last line of page 429; it should read \left((1+H^\prime)X_je^{-(\lambda\alpha\beta/2)H^\prime}\right)\left(e^{-( \lambda\alpha\beta/2)H^\prime}X_k\right)\left(e^{-(1-\lambda)\alpha\beta H}\right)\left(X_kR\right)\left(e^{-(1-\alpha)\beta H}\right)\left(HR^\prime\right)

A very much stronger version of the result obtained here has subsequently been proved in The analytic quantum information manifold. First, no condition on the commutator is needed; second, we replace the condition on the state, that rho^(beta) is of trace class for all beta, by the condition that it is of trace-class for some beta less than 1. Third, we show that all Kubo moments exist, and that the free energy is an analytic function, not merely, as here, that up to the third moment exists. In a subsequent paper with Grasselli, we obtain a similar result for a range of perturbations that are form-bounded (what we call epsilon-bounded). These need not be operator bounded in the sense of the Kato-Rellich theory.