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arXiv:2205.03008 · Physical review. B./Physical review. B

Quantum information spreading in random spin chains with topological order

Takahiro Orito, Yoshihito Kuno, Ikuo Ichinose

Published Sep 21, 2022• 11 citations
arXivHTMLPDF

Abstract

Quantum information spreading and scrambling in many-body systems attract interest these days. Tripartite mutual information (TMI) based on operator-based entanglement entropy (EE) is an efficient tool for measuring them. In this paper, we study random spin chains that exhibit phase transitions accompanying nontrivial changes in topological properties. In their phase diagrams, there are two types of many-body localized (MBL) states and one thermalized regime intervening these two MBL states. Quench dynamics of the EE and TMI display interesting behaviors, providing an essential perspective concerning the encoding of quantum information. In particular, one of the models is self-dual, but information spreading measured by the TMI does not respect this self-duality. We investigate this phenomenon from the viewpoint of spatial structure of the stabilizers. In general, we find that knowledge of the phase diagram corresponding to the qubit system is useful for understanding the nature of quantum information spreading in that system. The connection between the present paper and random circuit of projective measurements and also topological Majorana quantum memory is remarked upon.

DOI
10.1103/physrevb.106.104204

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Quantum information spreading and scrambling in many-body systems attract interest these days. Tripartite mutual information (TMI) based on operator-based entanglement entropy (EE) is an efficient tool for measuring them. In this paper, we study random spin chains that exhibit phase transitions accompanying nontrivial changes in topological properties. In their phase diagrams, there are two types of many-body localized (MBL) states and one thermalized regime intervening these two MBL states. Quench dynamics of the EE and TMI display interesting behaviors, providing an essential perspective concerning the encoding of quantum information. In particular, one of the models is self-dual, but information spreading measured by the TMI does not respect this self-duality. We investigate this phenomenon from the viewpoint of spatial structure of the stabilizers. In general, we find that knowledge of the phase diagram corresponding to the qubit system is useful for understanding the nature of quantum information spreading in that system. The connection between the present paper and random circuit of projective measurements and also topological Majorana quantum memory is remarked upon.
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.
Tripartite mutual information (TMI) is an efficient observable to quantify the ability of a scrambler for the unitary time-evolution operator with a quenched many-body Hamiltonian. In this paper we give numerical demonstrations of the TMI in disorder-free (translationally invariant) spin models with three-body and four-body multiple-spin interactions. The dynamical behavior of the TMI of these models does not exhibit a linear light cone for sufficiently strong interactions. In early-time evolution, the TMI displays a distinct negative increase fitted by a logarithmiclike function. This is in contrast to the conventional linear light-cone behavior present in the XXZXXZXXZ model and its near-integrable vicinity. The late-time evolution of the TMI in finite-size systems is also numerically investigated. The multiple-spin interactions make the system nearly integrable and weakly suppress the spread of information and scrambling. The observation of the late-time value of the TMI indicates that the scrambling nature of the system changes by interactions and this change can be characterized by a phase-transition-like behavior of the TMI, reflecting the system's integrability and violating the eigenstate thermalization hypothesis.
Chaotic transverse-field Ising model with measurements exhibits interesting purification dynamics. An ensemble of nonunitary dynamics of a chaotic many-body system with measurements exhibits a purification phase transition. We numerically find that the law of the increase dynamics of the purity changes by projective measurements in the model. In order to study this behavior in detail, we construct the formalism of the tripartite mutual information (TMI) for a nonunitary time evolution operator by using the state-channel map. The numerical result of the saturation value of the TMI indicates the existence of a measurement-induced phase transition. This implies the existence of two distinct phases, a mixed phase and purified phase. Furthermore, the real-space spread of the TMI is investigated to explore spatial patterns of information spreading. Even in the purified phase, the spatial pattern of the light-cone spread of quantum information is not deformed, but its density of information propagation is reduced on average by the projective measurements. We also find that this spatial pattern of the TMI distinguishes the chaotic and integrable regimes of the system.
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.
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.