Stochastic time-evolution, information geometry and the Cramer-Rao Bound

TL;DR

This paper explores the relationship between stochastic time-evolution, Fisher information, and the Cramer-Rao bound, establishing speed limits on observable changes and methods to detect hidden variables in stochastic systems.

Contribution

It introduces a bound on the rate of change of stochastic observables based on Fisher information, linking information geometry with dynamical constraints and hidden variable detection.

Findings

The rate of change of an observable is bounded by its variance and Fisher information.

Fisher information decreases monotonically during relaxation dynamics.

The monotonicity of Fisher information can reveal hidden variables.

Abstract

We investigate the connection between the time-evolution of averages of stochastic quantities and the Fisher information and its induced statistical length. As a consequence of the Cramer-Rao bound, we find that the rate of change of the average of any observable is bounded from above by its variance times the temporal Fisher information. As a consequence of this bound, we obtain a speed limit on the evolution of stochastic observables: Changing the average of an observable requires a minimum amount of time given by the change in the average squared, divided by the fluctuations of the observable times the thermodynamic cost of the transformation. In particular for relaxation dynamics, which do not depend on time explicitly, we show that the Fisher information is a monotonically decreasing function of time and that this minimal required time is determined by the initial preparation of the system. We further show that the monotonicity of the Fisher information can be used to detect hidden variables in the system and demonstrate our findings for simple examples of continuous and discrete random processes.