Information acquisition, scrambling, and sensitivity to errors in quantum chaos

TL;DR

This paper explores how chaos manifests in quantum systems, reviewing classical and quantum signatures of chaos, including out-of-time correlators and quantum tomography, with implications for quantum information processing.

Contribution

It provides a comprehensive review of quantum chaos signatures, including new insights into operator spreading and quantum tomography applications.

Findings

Quantum Lyapunov exponents quantified by OTOC

Signatures of chaos detectable via quantum tomography

Operator spreading characterized in Krylov subspaces

Abstract

Signatures of chaos can be understood by studying quantum systems whose classical counterpart is chaotic. However, the concepts of integrability, non-integrability and chaos extend to systems without a classical analogue. Here, we first review the classical route from order into chaos. Since nature is fundamentally quantum, we discuss how chaos manifests in the quantum domain. We briefly describe semi-classical methods, and discuss the consequences of chaos in quantum information processing. We review the quantum version of Lyapunov exponents, as quantified by the out-of-time ordered correlators (OTOC), Kolmogorov-Sinai (KS) entropy and sensitivity to errors. We then review the study of signatures of quantum chaos using quantum tomography. Classically, if we know the dynamics exactly, as we maintain a constant coarse-grained tracking of the trajectory, we gain exponentially fine-grained information about the initial condition. In the quantum setting,as we track the measurement record with fixed signal-to-noise, we gain increasing information about the initial condition. In the process, we have given a new quantification of operator spreading in Krylov subspaces with quantum state reconstruction. The study of these signatures is not only of theoretical interest but also of practical importance.