Quantum analogues of exponential sensitivity: from Loschmidt echo to Krylov complexity

TL;DR

This paper reviews quantum analogues of classical chaos indicators, focusing on the Loschmidt echo, OTOCs, and Krylov complexity, highlighting their roles in understanding quantum chaos.

Contribution

It provides a pedagogical overview of three key quantum quantities—Loschmidt echo, OTOCs, and Krylov complexity—that serve as analogues to classical exponential sensitivity.

Findings

Discusses how these quantities capture quantum chaos behavior

Highlights recent research developments in quantum chaos indicators

Provides a comparative understanding of different quantum chaos measures

Abstract

One of the fundamental manifestations of classical chaos is exponential sensitivity to initial conditions that is, two trajectories starting from nearly identical initial states diverge exponentially over time. This behavior is quantified by the Lyapunov exponents. Due to the unitary nature of quantum mechanics, such exponential divergence is elusive in quantum systems. As a result, several alternative quantities have been proposed and studied in recent years to capture analogous behavior. In this article, we present a pedagogical overview of three such quantities that have been the focus of intense research in recent years: the Loschmidt echo, out-of-time-order correlators (OTOCs), and Krylov complexity.