Speed-Fisher Information: Chaos and Irreversibility in Classical and Quantum Dynamics

Abstract

We propose a unified perspective on classical and quantum chaos based on the sensitivity of a system's stationary states to slow driving. We probe this sensitivity via the system's susceptibility to the average protocol speed, which we call the ``speed-Fisher information,"and relate it to irreversible entropy production in the system. We show that chaotic dynamics manifests as a divergence of the speed-Fisher information with the protocol time, and that this response is controlled by the perturbation's low-frequency spectral weight. This approach to chaos applies to both classical and quantum Hamiltonian systems, and naturally extends to non-Hamiltonian classical flows. Additionally, we identify a quantum regime at times shorter than the Planckian timescale, in which the speed-Fisher information exhibits a uniquely quantum scaling. We illustrate this framework with simple classical and quantum examples