KAM-Stability for Conserved Quantities in Finite-Dimensional Quantum Systems

TL;DR

This paper demonstrates that conserved quantities in finite-dimensional quantum systems exhibit robustness to small perturbations, akin to classical KAM theory, with a new resummation technique extending quantum Zeno dynamics.

Contribution

It introduces a novel perturbation series resummation method that characterizes the robustness of conserved quantities in quantum systems, drawing an analogy with classical KAM theory.

Findings

Conserved quantities can be stable under small perturbations in quantum systems.

Fragile symmetries are sensitive to perturbations, leading to large deviations.

The new resummation approach generalizes quantum Zeno dynamics.

Abstract

We show that for any finite-dimensional quantum systems the conserved quantities can be characterized by their robustness to small perturbations: for fragile symmetries small perturbations can lead to large deviations over long times, while for robust symmetries their expectation values remain close to their initial values for all times. This is in analogy with the celebrated Kolmogorov-Arnold-Moser (KAM) theorem in classical mechanics. To prove this remarkable result, we introduce a resummation of a perturbation series, which generalizes the Hamiltonian of the quantum Zeno dynamics.