Abstract
Chaotic systems are highly sensitive to a small perturbation and are ubiquitous throughout the biological sciences, the physical sciences, and even the social sciences. Taking this as the underlying principle, we construct an operational notion for quantum chaos. Namely, we demand that the future state of a many-body isolated quantum system is sensitive to past multitime operations on a small subpart of that system. By “sensitive,” we mean that the resultant states from two different perturbations cannot easily be transformed into each other. That is, the pertinent quantity is the complexity of the effect of the perturbation within the final state. From this intuitive metric, which we call the butterfly-flutter fidelity, we use the language of multitime quantum processes to identify a series of operational conditions on chaos; in particular, the scaling of the spatiotemporal entanglement. Our criteria already contain the routine notions, as well as the well-known diagnostics for quantum chaos. This includes the Peres-Loschmidt echo, dynamical entropy, tripartite mutual information, and local-operator entanglement. We hence present a unified framework for these existing diagnostics within a single structure. We also go on to quantify how several mechanisms, such as evolution generated from random circuits, lead to quantum chaos. Our work paves the way to systematically study many-body dynamical phenomena such as many-body localization, measurement-induced phase transitions, and Floquet dynamics.
- Received 16 January 2023
- Revised 12 December 2023
- Accepted 19 December 2023
DOI:https://doi.org/10.1103/PRXQuantum.5.010314
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Chaos refers to systems that are highly sensitive to small changes. This can happen in a variety of fields, from economics and ecology to meteorology and physics. In this paper, we identify the principle underlying quantum chaos, a historically difficult problem to broach.
Imagine that you have a large system, like a clock, and you make a small change to one part of it, such as slightly adjusting the position of the attached pendulum. In a chaotic system, if the clock used a double pendulum, for example, such a small change would have a big impact on the future behavior of the clock, leading to it reading a wildly varying time. Quantum mechanically, this becomes difficult to translate: Schrödinger’s equation is linear, and so distances between wave functions are preserved as an isolated system evolves in time. Instead, in this work we identify that quantum chaos constitutes a sensitivity to a past sequence of local operations on a quantum system. A strong type of entanglement within the quantum process underlies this operational criterion, which contains the routine notions, as well as the well-known diagnostics for quantum chaos.
We also identify several mechanisms that can lead to quantum chaos and show how our principle fits into the current landscape of research on this topic. Essentially, we provide a comprehensive understanding of chaos in quantum systems, including how it arises and how it can be detected. This work may help researchers better understand and study a wide range of problems in quantum systems, including exotic phenomena such as many-body localization.
