Quantum information scrambling under many-body dynamics is of fundamental interest. The tripartite mutual information can quantify the scrambling via its negative value. Here, we first study the quench dynamics of tripartite mutual information in a non-integrable Ising model where the strong and weak thermalization are observed with different initial states. We numerically show that the fastest scrambling can occur when the energy density of the chosen initial state possesses the maximum density of states. We then present an experimental protocol for observing weak and strong thermalization in a superconducting qubit array. Based on the protocol, the relation between scrambling and thermalization revealed in this work can be directly verified by superconducting quantum simulations.
Understanding various phenomena in non-equilibrium dynamics of closed quantum many-body systems, such as quantum thermalization, information scrambling, and nonergodic dynamics, is a crucial for modern physics. Using a ladder-type superconducting quantum processor, we perform analog quantum simulations of both the $XX$ ladder and one-dimensional (1D) $XX$ model. By measuring the dynamics of local observables, entanglement entropy and tripartite mutual information, we signal quantum thermalization and information scrambling in the $XX$ ladder. In contrast, we show that the $XX$ chain, as free f…
Out-of-time ordered (OTO) correlation functions describe scrambling of information in correlated quantum matter. They are of particular interest in incoherent quantum systems lacking well defined quasi-particles. Thus far, it is largely elusive how OTO correlators spread in incoherent systems with diffusive transport governed by a few globally conserved quantities. Here, we study the dynamical response of such a system using high-performance matrix-product-operator techniques. Specifically, we consider the non-integrable, one-dimensional Bose-Hubbard model in the incoherent high-temperature re…
We study the quantum thermalization and information scrambling dynamics of an experimentally realizable quantum spin model with homogeneous XX-type all-to-all interactions and random local potentials. We identify the thermalization-localization transition by changing the disorder strength, under a proper all-to-all interaction strength. The scrambling dynamics in the localization phase shows novel behaviors distinct from that of local models. The operator scrambling grows almost equally fast in both phases. In the thermal phase, we show there exhibits fast scrambling without appealing to the s…
Scrambling of quantum information is the process by which information initially stored in the local degrees of freedom of a quantum many-body system spreads over its many-body degrees of freedom, becoming inaccessible to local probes and thus apparently lost. Scrambling and entanglement are key concepts reconciling seemingly unrelated behaviors including thermalization of isolated quantum systems and information loss in black holes, and have revolutionized our understanding of non-equilibrium phenomena. Here, we demonstrate that a family of fidelity out-of-time-order correlators (FOTOCs), rece…
We study quantum information scrambling, specifically the growth of Heisenberg operators, in large disordered spin chains using matrix product operator dynamics to scan across the thermalization-localization quantum phase transition. We observe ballistic operator growth for weak disorder, and a sharp transition to a phase with sub-ballistic operator spreading. The critical disorder strength for the ballistic to sub-ballistic transition is well below the many body localization phase transition, as determined from finite size scaling of energy eigenstate entanglement entropy in small chains. In …
Many proposed quantum mechanical models of black holes include highly nonlocal interactions. The time required for thermalization to occur in such models should reflect the relaxation times associated with classical black holes in general relativity. Moreover, the time required for a particularly strong form of thermalization to occur, sometimes known as scrambling, determines the time scale on which black holes should start to release information. It has been conjectured that black holes scramble in a time logarithmic in their entropy, and that no system in nature can scramble faster. In this…
Collective behavior strongly influences the charging dynamics of quantum batteries (QBs). Here, we study the impact of nonlocal correlations on the energy stored in a system of $N$ QBs. A unitary charging protocol based on a Sachdev-Ye-Kitaev (SYK) quench Hamiltonian is thus introduced and analyzed. SYK models describe strongly interacting systems with nonlocal correlations and fast thermalization properties. Here, we demonstrate that, once charged, the average energy stored in the QB is very stable, realizing an ultraprecise charging protocol. By studying fluctuations of the average energy st…
Quantum scrambling is the dispersal of local information into many-body quantum entanglements and correlations distributed throughout the entire system. This concept underlies the dynamics of thermalization in closed quantum systems, and more recently has emerged as a powerful tool for characterizing chaos in black holes. However, the direct experimental measurement of quantum scrambling is difficult, owing to the exponential complexity of ergodic many-body entangled states. One way to characterize quantum scrambling is to measure an out-of-time-ordered correlation function (OTOC); however, si…
The delocalization or scrambling of quantum information has emerged as a central ingredient in the understanding of thermalization in isolated quantum many-body systems. Recently, significant progress has been made analytically by modeling non-integrable systems as stochastic systems, lacking a Hamiltonian picture, while honest Hamiltonian dynamics are frequently limited to small system sizes due to computational constraints. In this paper, we address this by investigating the role of conservation laws (including energy conservation) in the thermalization process from an information-theoretic …
Out-of-time-ordered correlators (OTOCs) are an effective tool in characterizing black hole chaos, many-body thermalization and quantum dynamics instability. Previous research findings have shown that the OTOCs' exponential growth (EG) marks the limit for quantum systems. However, we report in this letter a periodically-modulated nonlinear Schrödinger system, in which we interestingly find a novel way of information scrambling: super-EG. We show that the quantum OTOCs' growth, which stems from the quantum chaotic dynamics, will increase in a super-exponential way. We also find that in the class…
In ergodic many-body quantum systems, locally encoded quantum information becomes, in the course of time evolution, inaccessible to local measurements. This concept of "scrambling" is currently of intense research interest, entailing a deep understanding of many-body dynamics such as the processes of chaos and thermalization. Here, we present first experimental demonstrations of quantum information scrambling on a 10-qubit trapped-ion quantum simulator representing a tunable long-range interacting spin system, by estimating out-of-time ordered correlators (OTOCs) through randomized measurement…
Tripartite mutual information (TMI) is an efficient observable to quantify the ability of scrambler for unitary time-evolution operator with quenched many-body Hamiltonian. In this paper, we give numerical demonstrations of the TMI in disorder-free (translational invariant) spin models with 3-body and 4-body multiple-spin interactions. The dynamical behavior of the TMI of these models does {\it not} exhibit linear light-cone for sufficiently strong interactions. In early-time evolution, the TMI displays distinct negative increase behavior fitted by a logarithmic-like function. This is in contr…
Quantum simulation elucidates properties of quantum many-body systems by mapping its Hamiltonian to a better-controlled system. Being less stringent than a universal quantum computer, noisy small- and intermediate-scale quantum simulators have successfully demonstrated qualitative behavior such as phase transition, localization and thermalization which are insensitive to imperfections in the engineered Hamiltonian. For more complicated features like quantum information scrambling, higher controllability will be desired to simulate both the forward and the backward time evolutions and to diagno…
In this paper, we look for signatures of quantum revivals in two-dimensional conformal field theories (2d CFTs) on a spatially compact manifold by using operator entanglement. It is believed that thermalization does not occur on spatially compact manifolds as the quantum state returns to its initial state which is a phenomenon known as quantum revival. We find that in CFTs such as the free fermion CFT, the operator mutual information exhibits quantum revival in accordance with the relativistic propagation of quasiparticles while in holographic CFTs, the operator mutual information does not exh…
Quantum information scrambling has attracted much attention amid the effort to reconcile the conflict between quantum-mechanical unitarity and the thermalizaiton-irreversibility in many-body systems. Here we propose an unconventional mechanism to generate quantum information scrambling through a high-complexity mapping from logical to physical degrees-of-freedom that hides the logical information into non-separable many-body-correlations. Corresponding to this mapping, we develop an algorithm to efficiently sample a Slater-determinant wavefunction and compute all physical observables in dynami…
Out-of-time-ordered correlation functions (OTOCs) play a crucial role in the study of thermalization, entanglement, and quantum chaos, as they quantify the scrambling of quantum information due to complex interactions. As a consequence of their out-of-time-ordered nature, OTOCs are difficult to measure experimentally. Here we propose an OTOC measurement protocol that does not rely on the reversal of time evolution and is easy to implement in a range of experimental settings. The protocol accounts for both pure and mixed initial states, and is applicable to systems that interact with environmen…
In a locally interacting many-body system, two isolated qubits, separated by a large distance $r$, become correlated and entangled with each other at a time $t \ge r/v$. This finite speed $v$ of quantum information scrambling limits quantum information processing, thermalization and even equilibrium correlations. Yet most experimental systems contain long range power law interactions -- qubits separated by $r$ have potential energy $V(r)\propto r^{-α}$. Examples include the long range Coulomb interactions in plasma ($α=1$) and dipolar interactions between spins ($α=3$). In one spatial dimensio…
We study dynamics of entanglement in the thermalization process of an isolated quantum many-body system. We propose a simple setup for measuring the propagation speed of entanglement entropy (EE) in numerical simulations and apply it to the integrable/non-integrable spin models in 1D - the transverse Ising (TI) model, the chaotic Ising (CI) model, and the extended chaotic Ising (ECI) model. We find that two distinct time-scales $t^\ast$ and $t_{\rm {diff}}$ arise in the dynamics of EE in the thermalization process: the former represents the time-scale for the saturation of EE and the latter ch…
Thermalization of chaotic quantum many-body systems under unitary time evolution is related to the growth in complexity of initially simple Heisenberg operators. Operator growth is a manifestation of information scrambling and can be diagnosed by out-of-time-order correlators (OTOCs). However, the behavior of OTOCs of local operators in generic chaotic local Hamiltonians remains poorly understood, with some semiclassical and large N models exhibiting exponential growth of OTOCs and a sharp chaos wavefront and other random circuit models showing a diffusively broadened wavefront. In this paper …
We investigate the eigenstate thermalization hypothesis (ETH) for a translationally invariant quantum spin system on the $d$-dimensional cubic lattice under the periodic boundary conditions. It is known that the ETH holds in this model for typical energy eigenstates in the sense that the standard deviation of the expectation values of a local observable in the energy eigenstates within the microcanonical energy shell vanishes in the thermodynamic limit, which is called the weak ETH. Here, it is remarked that the diagonal elements of a local observable in the energy representation shows the lar…
We explore the origin of the arrow of time in an isolated quantum system described by the Schroedinger equation. We provide an explanation from weak values in the configuration space, which are understood as operational properties obtained in the laboratory following a well-defined protocol. We show that quantum systems satisfying the eigenstate thermalization hypothesis can simultaneously provide thermalized ensemble expectation values and nonthermalized weak values of the momentum, both from the same operational probability distribution. The reason why weak values of the momentum may escape …
How violently do two quantum operators disagree? Different fields of physics feature different measures of incompatibility: (i) In quantum information theory, entropic uncertainty relations constrain measurement outcomes. (ii) In condensed matter and high-energy physics, the out-of-time-ordered correlator (OTOC) signals scrambling, the spread of information through many-body entanglement. We unite these measures, proving entropic uncertainty relations for scrambling. The entropies are of distributions over weak and strong measurements' possible outcomes. The weak measurements ensure that the O…
When a non-integrable system evolves out of equilibrium for a long time, local observables are expected to attain stationary expectation values, independent of the details of the initial state. However, intriguing experimental results with ultracold gases have shown no thermalization in non-integrable settings, triggering an intense theoretical effort to decide the question. Here we show that the phenomenology of thermalization in a quantum system is much richer than its classical counterpart. Using a new numerical technique, we identify two distinct thermalization regimes, strong and weak, oc…
Deriving conditions under which a macroscopic system thermalizes directly from the underlying quantum many-body dynamics of its microscopic constituents is a long-standing challenge in theoretical physics. The well-known eigenstate thermalization hypothesis (ETH) is presumed to be a key mechanism, but has defied rigorous verification for generic systems thus far. A weaker variant (weak ETH), by contrast, is provably true for a large variety of systems, including even many integrable models, but its implications with respect to the problem of thermalization are still largely unexplored. Here we…
We first propose and study a quantum toy model of black hole dynamics. The model is unitary, displays quantum thermalization, and the Hamiltonian couples every oscillator with every other, a feature intended to emulate the color sector physics of large-$\mathcal{N}$ matrix models. Considering out of equilibrium initial states, we analytically compute the time evolution of every correlator of the theory and of the entanglement entropies, allowing a proper discussion of global thermalization/scrambling of information through the entire system. Microscopic non-locality causes factorization of red…
We experimentally study the ergodic dynamics of a 1D array of 12 superconducting qubits with a transverse field, and identify the regimes of strong and weak thermalization with different initial states. We observe convergence of the local observable to its thermal expectation value in the strong-thermalizaion regime. For weak thermalization, the dynamics of local observable exhibits an oscillation around the thermal value, which can only be attained by the time average. We also demonstrate that the entanglement entropy and concurrence can characterize the regimes of strong and weak thermalizat…
Scrambling of information in a quantum many-body system, quantified by the out-of-time-ordered correlator (OTOC), is a key manifestation of quantum chaos. A regime of exponential growth in the OTOC, characterized by a Lyapunov exponent, has so far mostly been observed in systems with a high-dimensional local Hilbert space and in weakly-coupled systems. Here, we propose a general criterion for the existence of a well-defined regime of exponential growth of the OTOC in spatially extended systems with local interactions. In such systems, we show that a parametrically long period of exponential gr…
Topological Control of Quantum Chaos Diagnostics: OTOCs, Spectral Statistics, and Information Scrambling in Ising Model
School of Quantum Physics and Matter Institute for Research in Fundamental Sciences (IPM)（量子物理与物质学院，基础科学研究院（IPM））；Ershad Damavand, Institute of Higher Education (EDI)（厄尔沙德达马万德高等教育研究所（EDI））；Department of Mathematics and Computer Science, Amirkabir University of Technology（阿米尔卡比尔理工大学数学与计算机科学系）；Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University（阿尔扎哈拉大学数学科学学院数学系）；Department of Physics, CT.C, Islamic Azad University（伊斯兰阿扎德大学CT.C物理学系）；Institute of Biosocial and Quantum Science and Technologies, CT.C, Islamic Azad University（伊斯兰阿扎德大学CT.C生物社会与量子科学与技术研究所）
By calculating correlation functions for the Lieb-Liniger model based on the algebraic Bethe ansatz method, we conduct a finite-size scaling analysis of the eigenstate thermalization hypothesis (ETH) which is considered to be a possible mechanism of thermalization in isolated quantum systems. We find that the ETH in the weak sense holds in the thermodynamic limit even for an integrable system although it does not hold in the strong sense. Based on the result of the finite-size scaling analysis, we compare the contribution of the weak ETH to thermalization with that of yet another thermalizatio…
Complexity of dynamics is at the core of quantum many-body chaos and exhibits a hierarchical feature: higher-order complexity implies more chaotic dynamics. Conventional ergodicity in thermalization processes is a manifestation of the lowest order complexity, which is represented by the eigenstate thermalization hypothesis (ETH) stating that individual energy eigenstates are thermal. Here, we propose a higher-order generalization of the ETH, named the $ k $-ETH ($ k=1,2,\dots $), to quantify higher-order complexity of quantum many-body dynamics at the level of individual energy eigenstates, wh…
Weakly interacting Fermi gases simulate spin-lattices in energy-space, offering a rich platform for investigating information spreading and spin coherence in a large many-body quantum system. We show that the collective spin vector can be determined as a function of energy from the measured spin density, enabling general energy-space resolved protocols. We measure an out-of-time-order correlation function in this system and observe the energy dependence of the many-body coherence.
