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arXiv:1704.04850 · Physical Review A

Scrambling of quantum information in quantum many-body systems

Eiki Iyoda, Takahiro Sagawa

Published Apr 18, 2018• 100 citations
arXivHTMLPDF

Abstract

We systematically investigate scrambling (or delocalizing) processes of quantum information encoded in quantum many-body systems by using numerical exact diagonalization. As a measure of scrambling, we adopt the tripartite mutual information (TMI) that becomes negative when quantum information is delocalized. We clarify that scrambling is an independent property of the integrability of Hamiltonians; TMI can be negative or positive for both integrable and nonintegrable systems. This implies that scrambling is a separate concept from conventional quantum chaos characterized by nonintegrability. Specifically, we argue that there are a few exceptional initial states that do not exhibit scrambling, and show that such exceptional initial states have small effective dimensions. Furthermore, we calculate TMI in the Sachdev-Ye-Kitaev (SYK) model, a fermionic toy model of quantum gravity. We find that disorder does not make scrambling slower but makes it smoother in the SYK model, in contrast to many-body localization in spin chains.

DOI
10.1103/physreva.97.042330

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We systematically investigate scrambling (or delocalizing) processes of quantum information encoded in quantum many-body systems by using numerical exact diagonalization. As a measure of scrambling, we adopt the tripartite mutual information (TMI) that becomes negative when quantum information is delocalized. We clarify that scrambling is an independent property of the integrability of Hamiltonians; TMI can be negative or positive for both integrable and nonintegrable systems. This implies that scrambling is a separate concept from conventional quantum chaos characterized by nonintegrability. Specifically, we argue that there are a few exceptional initial states that do not exhibit scrambling, and show that such exceptional initial states have small effective dimensions. Furthermore, we calculate TMI in the Sachdev-Ye-Kitaev (SYK) model, a fermionic toy model of quantum gravity. We find that disorder does not make scrambling slower but makes it smoother in the SYK model, in contrast to many-body localization in spin chains.
Quantum information scrambling under the dynamics of a closed many-body system is of wide interest. The tripartite mutual information (TMI) can quantify the scrambling via its stable negative value under temporal evolution. Here, we study the quench dynamics of the TMI in a nonintegrable Ising model with different initial states in the regimes of strong and weak thermalization. Our numerical results show that the most efficient scrambling can occur when the inverse temperatures of initial states are near zero and in the regime of strong thermalization, and weak thermalization accompanies slow scrambling. We then present an experimental protocol for observing strong and weak thermalization in a one-dimensional array of superconducting qubits, based on which the relation between scrambling and the degree of thermalization revealed in this work can be directly verified by superconducting quantum simulators. The inevitable decoherence effects in real situations are also analyzed, paving the way for faithful quantum simulations on actual experimental platforms.
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.
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 periodically driven 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 perspective. For general non-integrable models, we use the equilibrium approximation to show that the maximal amount of information is scrambled (as measured by the tripartite mutual information of the time-evolution operator) at late times even when a system conserves energy. In contrast, we explicate how when a system has additional symmetries that lead to degeneracies in the spectrum, the amount of information scrambled must decrease. This general theory is exemplified in case studies of holographic conformal field theories (CFTs) and the Sachdev-Ye-Kitaev (SYK) model. Due to the large Virasoro symmetry in 1+1D CFTs, we argue that, in a sense, these holographic theories are not maximally chaotic, which is explicitly seen by the non-saturation of the second Rényi tripartite mutual information. The roles of particle-hole and U(1) symmetries in the SYK model are milder due to the degeneracies being only two-fold, which we confirm explicitly at both large- and small-N. We reinterpret the operator entanglement in terms of the growth of local operators, connecting our results with the information scrambling described by out-of-time-ordered correlators, identifying the mechanism for suppressed scrambling from the Heisenberg perspective.
Information scrambling plays a pivotal role in comprehending the dynamics of quantum many-body systems. In this work, we investigate information scrambling in a one-dimensional topological system using tripartite mutual information (TMI). We consider the one-dimensional Su-Schrieffer-Heeger model as an example and implement it using a Gaussian stroboscopic simulator, verifying its validity through both dynamical studies and effective Hamiltonian analysis, confirming the presence of topological edge states. We then discuss the dynamical behavior of TMI using this simulator as a prototype. Our results reveal that information scrambling can serve as a probe of the system's topological properties, with its presence or absence distinctly characterizing the topologically nontrivial and trivial phases. Moreover, we introduce the standardized mean TMI as a measure of topological phase transitions, demonstrating its effectiveness in determining the phase diagram. Our findings not only elucidate the interplay between topological properties and quantum information dynamics but also propose methods for detecting topological phase transitions through information dynamics.
Abstract Scrambling of quantum information in both integrable and nonintegrable Floquet spin systems is studied. Our study employs tripartite mutual information (TMI), with negative TMI serving as an indicator of scrambling, where a more negative value suggests a higher degree of scrambling. Both integrable and nonintegrable Floquet systems display scrambling behavior across all periods lying between 0 and π/2\pi /2π/2 π / 2 , except at self-dual point (π/4)(\pi /4)(π/4) ( π / 4 ) . Nonintegrable Floquet systems exhibit more pronounced scrambling compared to integrable ones across all periods. The degree of scrambling increases as we move toward the self-dual point (but not at the self-dual point), regardless of the initial states. TMI demonstrates periodic behavior at the self-dual point, with a period matching the system size in the case of the integrable system while displaying complex patterns in the nonintegrable system. The initial growth of scrambling in both integrable and nonintegrable Floquet systems manifests as a power-law increase for small periods, followed by a sudden jump in scrambling near the self-dual point. Graphical abstract
Many-body states that are invariant under particle relabeling, the permutation symmetric states, occur naturally when the system dynamics is described by symmetric processes or collective spin operators. We derive expressions for the reduced density matrix for arbitrary subsystem decomposition for these states and study properties of permutation symmetric states and their subsystems when the joint system is picked randomly and uniformly. Thus defining an appropriate random matrix ensemble, we find the average linear entropy and von Neumann entropy, which implies that random permutation symmetric states are marginally entangled and as a consequence the tripartite mutual information (TMI) is typically positive, preventing information from being shared globally. Applying these results to the quantum kicked top viewed as a multiqubit system, we find that entanglement, mutual information, and TMI all increase for large subsystems across the Ehrenfest or logarithmic time and saturate at the random state values if there is global chaos. During this time the out-of-time-order correlators evolve exponentially, implying scrambling in phase space. We discuss how positive TMI may coexist with such scrambling.