Quantifying operator spreading and chaos in Krylov subspaces with quantum state reconstruction

TL;DR

This paper investigates operator spreading and quantum chaos in many-body systems using quantum tomography, revealing that information gain correlates with chaos and offers a new operational perspective on operator spreading.

Contribution

It introduces a method to quantify operator spreading via quantum tomography, demonstrating its effectiveness over Krylov complexity in detecting quantum chaos.

Findings

Operator spreading increases with chaos degree.

Information gain sharply rises transitioning from integrable to nonintegrable dynamics.

Quantum tomography provides a more consistent chaos indicator than Krylov complexity.

Abstract

We study operator spreading in many-body quantum systems by its potential to generate an informationally complete measurement record in quantum tomography. We adopt continuous weak measurement tomography for this purpose. We generate the measurement record as a series of expectation values of an observable evolving under the desired dynamics, which can show a transition from integrability to complete chaos. We find that the amount of operator spreading, as quantified by the fidelity in quantum tomography, increases with the degree of chaos in the system. We also observe a remarkable increase in information gain when the dynamics transitions from integrable to nonintegrable. We find our approach in quantifying operator spreading is a more consistent indicator of quantum chaos than Krylov complexity as the latter may correlate/anti-correlate or show no explicit behavior with the level of chaos in the dynamics. We support our argument through various metrics of information gain for two models: the Ising spin chain with a tilted magnetic field and the Heisenberg XXZ spin chain with an integrability-breaking field. Our paper gives an operational interpretation for operator spreading in quantum chaos.