Kolmogorov-Arnold-Moser Stability for Conserved Quantities in Finite-Dimensional Quantum Systems.

Area of Science:

• Quantum mechanics
• Classical mechanics
• Dynamical systems

Background:

• Conserved quantities are fundamental in physics.
• Understanding the stability of symmetries under perturbation is crucial.
• The Kolmogorov-Arnold-Moser (KAM) theorem describes stability in classical Hamiltonian systems.

Purpose of the Study:

• To characterize conserved quantities in finite-dimensional quantum systems based on their robustness to perturbations.
• To establish an analogy between quantum symmetry robustness and the KAM theorem in classical mechanics.

Main Methods:

• Introduction of a novel resummation technique for perturbation series.
• Generalization of the Hamiltonian for quantum Zeno dynamics.

Main Results:

• Demonstration that conserved quantities can be classified as either robust or fragile.
• Fragile symmetries exhibit large deviations under small perturbations over long times.
• Robust symmetries maintain their expectation values close to initial values indefinitely.

Conclusions:

• The robustness of conserved quantities provides a new characterization for quantum symmetries.
• This quantum characterization is analogous to the KAM theorem in classical mechanics.
• The developed methods offer insights into the long-term behavior of quantum systems under perturbation.