Developing quantum technologies requires the control and understanding of the nonequilibrium dynamics of quantum information in many-body systems. Local information propagates in the system by creating complex correlations known as information scrambling, as this process prevents extracting the information from local measurements. In this paper, we develop a model adapted from solid-state NMR methods, to quantify the information scrambling. The scrambling is measured via time-reversal Loschmidt echo (LE) and multiple quantum coherences experiments that intrinsically contain imperfections. Considering these imperfections, we derive expressions for out-of-time order correlators (OTOCs) to quantify the observable information scrambling based on measuring the number of active spins where the information was spread. Based on the OTOC expressions, decoherence effects arise naturally by the effects of the nonreverted terms in the LE experiment. Decoherence induces localization of the measurable degree of information scrambling. These effects define a localization cluster size for the observable number of active spins that determines a dynamical equilibrium. We contrast the model's predictions with quantum simulations performed with solid-state NMR experiments, that measure the information scrambling with time-reversal echoes with controlled imperfections. An excellent quantitative agreement is found with the dynamics of quantum information scrambling and its localization effects determined from the experimental data. The presented model and derived OTOCs set tools for quantifying the quantum information dynamics of large quantum systems (more than ${10}^{4}$ spins) consistent with experimental implementations that intrinsically contain imperfections.
Developing quantum technologies requires the control and understanding of the nonequilibrium dynamics of quantum information in many-body systems. Local information propagates in the system by creating complex correlations known as information scrambling, as this process prevents extracting the information from local measurements. In this paper, we develop a model adapted from solid-state NMR methods, to quantify the information scrambling. The scrambling is measured via time-reversal Loschmidt echo (LE) and multiple quantum coherences experiments that intrinsically contain imperfections. Considering these imperfections, we derive expressions for out-of-time order correlators (OTOCs) to quantify the observable information scrambling based on measuring the number of active spins where the information was spread. Based on the OTOC expressions, decoherence effects arise naturally by the effects of the nonreverted terms in the LE experiment. Decoherence induces localization of the measurable degree of information scrambling. These effects define a localization cluster size for the observable number of active spins that determines a dynamical equilibrium. We contrast the model's predictions with quantum simulations performed with solid-state NMR experiments, that measure the information scrambling with time-reversal echoes with controlled imperfections. An excellent quantitative agreement is found with the dynamics of quantum information scrambling and its localization effects determined from the experimental data. The presented model and derived OTOCs set tools for quantifying the quantum information dynamics of large quantum systems (more than 104 spins) consistent with experimental implementations that intrinsically contain imperfections.
Out-of-time-order correlators (OTOCs) serve as a proxy for quantum information scrambling, which refers to the process where information stored locally disperses across the many-body degrees of freedom in a quantum system, rendering it inaccessible to local probes. Most experimental implementations of OTOCs to probe information scrambling rely on indirect measurements based on global observables, using techniques such as Loschmidt echoes andmultiple quantum coherences, via time-reversal evolutions. In this article, we establish a direct connection between OTOCs with global and local observables in the context of NMR experiments, where the observable is the total magnetization of the system. We conduct a numerical analysis to quantify the differences in the evolution of both magnitudes, evaluating the excitation dynamics in spin ring systems with 8 to 16 spins, using a many-body Hamiltonian and long-range interactions. Our analysis decomposes the global echo into a sum of local echoes and cross-contributions, leading to local and global OTOCs. The results indicate that, after an initial transient period, local OTOCs determine the global ones. We observe that the difference between the average of local OTOCs and the global one, as well as their fluctuations, becomes negligible as the system size increases. Thus, for large homogeneous systems, global and local OTOCs become equivalent. This behavior aligns with that observed in highly interacting or chaotic systems in several experiments.
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.
Quantum information scrambling, typically explored in closed quantum systems, describes the spread of initially localized information throughout a system and can be quantified by measures such as the Loschmidt echo (LE) and out-of-time-order correlator (OTOC). In this paper, we explore information scrambling in the presence of dissipation by generalizing the concepts of LE and OTOC to open quantum systems governed by Lindblad dynamics. We investigate the universal dynamics of the generalized LE across regimes of weak and strong dissipation. In the weak dissipation regime, we identify a universal structure, while in the strong dissipation regime, we observe a distinctive two-local-minima structure, which we interpret through an analysis of the Lindblad spectrum. Furthermore, we establish connections between the thermal averages of LE and OTOC and prove a general relation between OTOC and Rényi entropy in open systems. Finally, we propose an experimental protocol for measuring OTOC in open systems. These findings provide deeper insights into information scrambling under dissipation and pave the way for experimental studies in open quantum systems.
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.
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
Abstract How does quantum chaos lead to rapid scrambling of information as well as errors across a system when one introduces perturbations in the dynamics? What are its consequences for the reliability of quantum simulations and quantum information processing? We employ continuous measurement quantum tomography as a paradigm to study these questions. The measurement record is generated as a sequence of expectation values of a Hermitian observable evolving under repeated application of the Floquet map of the quantum kicked top. Interestingly, we find that the reconstruction fidelity initially increases regardless of the degree of chaos or the strength of perturbations in the dynamics. For random states, when the measurement record is obtained from a random initial observable, the subsequent drop in the fidelity obtained is inversely correlated to the degree of chaos in the dynamics. More importantly, this also gives us an operational interpretation of Loschmidt echo for operators by connecting it to the performance of quantum tomography. We define a quantity to capture the scrambling of errors, an out-of-time-ordered correlator (OTOC) between two operators under perturbed and unperturbed system dynamics that serves as a signature of chaos and quantifies the spread of errors. Our results demonstrate not only a fundamental link between Loschmidt echo and scrambling of errors, as captured by OTOCs, but that such a link can have operational consequences in quantum information processing.