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arXiv:2202.07060 · PRX Quantum

Scrambling Dynamics and Out-of-Time-Ordered Correlators in Quantum Many-Body Systems

Shenglong Xu, Brian G. Swingle

Published Jan 22, 2024• 141 citations
arXivHTMLPDF

Abstract

This tutorial article introduces the physics of quantum information scrambling in quantum many-body systems. The goals are to understand how to precisely quantify the spreading of quantum information and how causality emerges in complex quantum systems. We introduce a general framework to study the dynamics of quantum information, including detection and decoding. We show that the dynamics of quantum information is closely related to operator dynamics in the Heisenberg picture, and, under certain circumstances, can be precisely quantified by the so-called out-of-time-ordered correlator (OTOC). The general behavior of the OTOC is discussed based on several toy models, including the Sachdev-Ye-Kitaev model, random circuit models, and Brownian models, in which the OTOC is analytically tractable. We introduce numerical methods, including exact diagonalization and tensor network methods, to calculate the OTOC for generic quantum many-body systems. We also survey current experimental schemes for measuring the OTOC in various quantum simulators. Published by the American Physical Society 2024

DOI
10.1103/prxquantum.5.010201

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This tutorial article introduces the physics of quantum information scrambling in quantum many-body systems. The goals are to understand how to precisely quantify the spreading of quantum information and how causality emerges in complex quantum systems. We introduce a general framework to study the dynamics of quantum information, including detection and decoding. We show that the dynamics of quantum information is closely related to operator dynamics in the Heisenberg picture, and, under certain circumstances, can be precisely quantified by the so-called out-of-time-ordered correlator (OTOC). The general behavior of the OTOC is discussed based on several toy models, including the Sachdev-Ye-Kitaev model, random circuit models, and Brownian models, in which the OTOC is analytically tractable. We introduce numerical methods, including exact diagonalization and tensor network methods, to calculate the OTOC for generic quantum many-body systems. We also survey current experimental schemes for measuring the OTOC in various quantum simulators. Published by the American Physical Society 2024
A closed quantum system never forgets its initial state, but the encoded information can get scrambled and become inaccessible without measuring a large fraction of all the system degrees of freedom. This scrambling can be diagnosed by studying the spatial spreading of initially local operators under the Heisenberg time evolution, and the decay of the out-of-time-ordered correlators (OTOC). What insights can OTOCs provide to understand the dynamics of quantum many-body systems? What are the characteristic behaviors of OTOCs during the time evolution? How is information scrambling affected by the dissipation in open quantum many-body systems? We first study slow scrambling in many-body localized systems via calculating various correlators, two-point retarded correlators and OTOCs. Comparing with retarded correlators, OTOCs provide more information about the dynamics. We find that disorder slows and partially halts the onset of information scrambling. Instead of ballistic spreading, propagation of information forms a logarithmic light cone. Next, we study the finite-size scaling of OTOCs at late times in generic thermalizing quantum many-body systems. When energy is conserved, the late-time saturation value of the OTOC of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We also study information scrambling in open quantum many-body systems. We define a dissipative version of OTOC and study its behaviors in a prototypical chaotic quantum chain with dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the information light cone can only reach a finite distance even when the effect of overall decay is removed. Finally, we construct a family of local Hamiltonians for understanding the asymmetric information scrambling. Our models live on a one-dimensional lattice and exhibit asymmetric butterfly light cone between the left and right spatial directions.
Abstract Recent advances in quantum information science have shed light on the intricate dynamics of quantum many-body systems, for which quantum information scrambling is a perfect example. Motivated by considerations of the thermodynamics of quantum information, this perspective aims at synthesizing key findings from several pivotal studies and exploring various aspects of quantum scrambling. We consider quantifiers such as the out-of-time-ordered correlator (OTOC) and the quantum mutual information, their connections to thermodynamics, and their role in understanding chaotic vs . integrable quantum systems. With a focus on representative examples, we cover a range of topics, including the thermodynamics of quantum information scrambling, and the scrambling dynamics in quantum gravity models such as the Sachdev-Ye-Kitaev (SYK) model. Examining these diverse approaches enables us to highlight the multifaceted nature of quantum information scrambling and its significance in understanding the fundamental aspects of quantum many-body dynamics at the intersection of quantum mechanics and thermodynamics.
Out-of-time-order correlators (OTOCs) can be used to probe how quickly a quantum system scrambles information when the initial conditions of the dynamics are changed. In sufficiently large quantum systems, one can extract from the OTOC the quantum analog of the Lyapunov coefficient that describes the timescale on which a classical chaotic system becomes scrambled. OTOCs have been applied only to a very limited number of toy models, such as the Sachdev-Ye-Kitaev model connected with black hole information scrambling, but they could find much wider applicability for information scrambling in quantum systems that allow comparison with experiments. The vibrations of polyatomic molecules are known to undergo a transition from regular dynamics at low energy to facile energy flow at sufficiently high energy. Molecules therefore represent ideal quantum systems to study scrambling in many-body systems of moderate size (here 6 to 36 degrees of freedoms). By computing quantum OTOCs and their classical counterparts we quantify how information becomes ``scrambled'' quantum mechanically in molecular systems. Between early ``ballistic'' dynamics, and late ``saturation'' of the OTOC when the full density of states is explored, there is indeed a regime where a quantum Lyapunov coefficient can be defined for all molecules in this study. Comparison with experimental rate data shows that slow scrambling as measured by the OTOC can reach the timescale of molecular reaction dynamics. Even for the smallest molecules we discuss, the Maldacena bound remains satisfied by regularized OTOCs, but not by unregularized OTOCs, highlighting that the former are more useful for discussing information scrambling in this type of moderate-size quantum system.
Abstract Out-of-time-order correlators (OTOCs) have emerged as the preeminent theoretical and experimental probe of quantum information scrambling — the delocalization of quantum information across many-body degrees of freedom. Here we present a comprehensive computational study of OTOCs across three paradigmatic models: the Sachdev-Ye-Kitaev (SYK) model, the non-integrable transverse-field Ising chain, and random unitary circuits. Through exact diagonalization and large-N analytics, we extract Lyapunov exponents, scrambling times, and butterfly velocities, demonstrating that all chaotic systems studied approach, but do not saturate, the Maldacena-Shenker-Stanford (MSS) bound λ_L ≤ 2π/β. Crucially, we introduce the Spectral Out-of-Time-Order Correlator (SOTOC), a novel frequency-domain diagnostic that decomposes information scrambling into its constituent spectral modes. The SOTOC achieves a 32% improvement in Lyapunov exponent extraction precision over conventional log-linear fitting and maintains robust performance at noise levels σ > 0.30, where traditional methods fail. Our phase diagram reveals a sharp transition in scrambling dynamics at h/J ≈ 1.0 in the TFIM, coinciding with the onset of quantum chaos as measured by level-spacing statistics. These results establish SOTOC as a practical and theoretically grounded tool for probing scrambling in near-term quantum devices and holographic systems.
Scrambling, the delocalization of initially localized quantum information, is commonly characterized by the out-of-time ordered correlator (OTOC). Employing the OTOC-Renyi-2 entropy theorem we derive a quantum speed limit for the OTOC, which sets an lower bound for the rate with which information can be scrambled. This bound becomes particularly tractable by describing the scrambling of information in a closed quantum system as an effective decoherence process of an open system interacting with an environment. We prove that decay of the OTOC can be bounded by the strength of the system-environment coupling and two-point environmental correlation functions. We validate our analytic bound numerically using the non-integrable transverse field Ising model. Our results provide a universal and model-agnostic quantitative framework for understanding the dynamical limits of information spreading across quantum many-body physics, condensed matter, and engineered quantum platforms.
Out-of-time-order correlators (OTOCs) are central probes of quantum scrambling, and their generalizations have recently become key primitives for both benchmarking quantum advantage and learning the structure of Hamiltonians. Yet their behavior has lacked a unified algorithmic interpretation. We show that higher-order OTOCs naturally fit within the framework of quantum signal processing (QSP): each OTOC(k)\mathrm{OTOC}^{(k)}OTOC(k) measures the 2k2k2k-th Fourier component of the phase distribution associated with the singular values of a spatially resolved truncated propagator. This explains the contrasting sensitivities of time-ordered correlators (TOCs) and higher-order OTOCs to causal-cone structure and to chaotic, integrable, or localized dynamics. Based on this understanding, we further generalize higher-order OTOCs by polynomial transformation of the singular values of the spatially resolved truncated propagator. The resultant signal allows us to construct frequency-selective filters, which we call \emph{OTOC spectroscopy}. This extends conventional OTOCs into a mode-resolved tool for probing scrambling and spectral structure of quantum many-body dynamics.