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arXiv:2003.10106 · Journal of the Physical Society of Japan

Entanglement Propagation in Thermalization of an Isolated Quantum System

Ryosuke Yoshii, Shion Yamashika, Shunji Tsuchiya

Published Apr 4, 2022• 6 citations
arXivHTMLPDF

Abstract

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 characterizes spreading of EE over the entire system. Evaluating the propagation speed of entanglement from $t_{\rm diff}$, we find that entanglement propagates ballistically with a constant velocity irrespective of the integrability of the model. The propagation speed of entanglement is found to coincide with the maximum group velocity of quasi-particle excitations in the TI model. We also evaluate the propagation speed of entanglement by mutual information and find the characteristic time-scale $t_{\mathrm{MI}}$. We show that the propagation speeds of entanglement evaluated by $t_{\mathrm{MI}}$ and $t_{\rm {diff}}$ agree well. We discuss the condition for thermalization based on the numerical results and propose that scrambling of the entire system has to take place before saturation of EE for thermalization.

DOI
10.7566/jpsj.91.054601

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Investigating the scrambling of quantum information (also known as quantum chaos) in many-body quantum systems provides valuable insights into the postquench evolution of a system. While various methods are available to understand the chaotic dynamics of these quantum systems, scrambling of mutual information following a quench has recently also emerged as a diagnostic of chaotic dynamics. In this work, we consider a chain of spin-1/21/21/2 particles of a finite length LLL evolving with the mixed-field Ising Hamiltonian and impose open boundary conditions. We simulate the time evolution of entanglement entropy and mutual information following a quench from the N\'eel state in this system using tensor networks and find that the entanglement entropy for nonintegrable systems saturates to a constant value at late times, while it continues to oscillate for integrable systems. We also find that mutual information peaks as a function of distance between intervals decay faster for nonintegrable systems compared to integrable systems, in agreement with the results in literature for XXZ chains. We compare the oscillations in entanglement entropy evolution obtained from simulations in the integrable case with analytic results from the quasiparticle picture and find agreement.
Quantum many-body systems exhibit highly complex interactions that give rise to emergent collective phenomena beyond the behavior of individual particles. One of the most intriguing challenges in modern quantum physics is understanding how isolated quantum systems evolve toward thermal equilibrium despite following fundamentally reversible quantum dynamics. Quantum entanglement has emerged as a central mechanism governing information propagation, thermalization, and the development of statistical equilibrium in isolated many-body systems. Simultaneously, recent advances in quantum information science have enabled researchers to investigate the relationship between entanglement dynamics, information scrambling, and emergent quantum behavior using computational and analytical approaches. This study proposes a Quantum Information Analytics Framework for investigating thermalization processes in isolated many-body quantum systems through the integration of entanglement dynamics, quantum information measures, statistical mechanics, and computational modeling. The proposed framework evaluates system evolution using multiple performance indicators, including entanglement entropy, information propagation, thermal equilibrium, quantum coherence, and computational stability. The findings indicate that quantum entanglement plays a fundamental role in driving thermalization while preserving the underlying unitary evolution of isolated quantum systems. The proposed analytical framework contributes to a deeper understanding of emergent quantum dynamics and provides valuable insights for quantum computing, quantum simulation, condensed matter physics, and next-generation quantum information technologies.
The time evolution of the entanglement entropy in non-equilibrium quantum systems provides crucial information about the structure of the time-dependent state. For quantum quench protocols, by combining a quasiparticle picture for the entanglement spreading with the exact knowledge of the stationary state provided by Bethe ansatz, it is possible to obtain an exact and analytic description of the evolution of the entanglement entropy. Here we discuss the application of these ideas to several integrable models. First we show that for non-interacting systems, both bosonic and fermionic, the exact time-dependence of the entanglement entropy can be derived by elementary techniques and without solving the dynamics. We then provide exact results for interacting spin chains that are carefully tested against numerical simulations. Finally, we apply this method to integrable one-dimensional Bose gases (Lieb-Liniger model) both in the attractive and repulsive regimes. We highlight a peculiar behaviour of the entanglement entropy due to the absence of a maximum velocity of excitations.
We present a general theory of quantum information propagation in chaotic quantum many-body systems. The generic expectation in such systems is that quantum information does not propagate in localized form; instead, it tends to spread out and scramble into a form that is inaccessible to local measurements. To characterize this spreading, we define an information speed via a quench-type experiment and derive a general formula for it as a function of the entanglement density of the initial state. As the entanglement density varies from zero to one, the information speed varies from the entanglement speed to the butterfly speed. We verify that the formula holds both for a quantum chaotic spin chain and in field theories with an AdS/CFT gravity dual. For the second case, we study in detail the dynamics of entanglement in two-sided Vaidya-AdS-Reissner-Nordstrom black branes. We also show that, with an appropriate decoding process, quantum information can be construed as moving at the information speed, and, in the case of AdS/CFT, we show that a locally detectable signal propagates at the information speed in a spatially local variant of the traversable wormhole setup.
The entangling power and operator entanglement entropy are state-independent measures of entanglement. Their growth and saturation is examined in the time-evolution operator of quantum many-body systems that can range from the integrable to the fully chaotic. An analytically solvable integrable model of the kicked transverse-field Ising chain is shown to have ballistic growth of operator von Neumann entanglement entropy and exponentially fast saturation of the linear entropy with time. Surprisingly, a fully chaotic model with longitudinal fields turned on shares the same growth phase, and is consistent with a random matrix model that is also exactly solvable for the linear entropy entanglements. However, an examination of the entangling power shows that its largest value is significantly less than the nearly maximal value attained by the nonintegrable one. The importance of long-range spectral correlations, and not just the nearest-neighbor spacing, is pointed out in determining the growth of entanglement in nonintegrable systems. Finally, an interesting case that displays some features peculiar to both integrable and nonintegrable systems is briefly discussed.
Information scrambling is a hallmark of quantum chaos and thermalization in isolated quantum many-body systems. We investigate scrambling dynamics in the all-to-all interacting spin Sachdev-Ye-Kitaev (SYK)-qqq model using both pure- and mixed-state entanglement measures. We show that von-Neumann and Rényi entropies exhibit rapid growth followed by saturation near Haar-random values, signaling efficient scrambling. The scrambling rate reveals a nontrivial dependence on the interaction order, system size, and Hamiltonian scaling. We further employ mixed-state entanglement as a powerful probe of information scrambling. We numerically find a universal relation between the Rényi-1/2 mutual information and entanglement negativity for minimal interaction order in the early growth regime. Furthermore, entanglement negativity displays a Page-curve-like behavior under unequal subsystem partitioning, characterized by the birth, spread, and eventual death of quantum correlations. Our results provide a generic description of information scrambling using entanglement dynamics in all-to-all interacting spin systems with multi-body interactions.