Disorder’s Ripple: How Scrambled Quantum Information Spreads

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Author: Denis Avetisyan

New research reveals how strongly disordered quantum systems spread information differently than their more ordered counterparts, impacting the speed and nature of quantum chaos.

The study of the Heisenberg XXZ model in strongly disordered conditions-specifically with a disorder parameter of $h=14$ and averaged over 5000 realizations at a system size of $N=13$-reveals that out-of-time-order commutators, $C_x(r,t)$, exhibit light-cone-like behavior with characteristic timescales and spatial decay rates determined by thresholds $\theta \in \{0.25, 0.5, 1\}$, and fitting to functions of the form $t_\theta \propto e^{\beta r}$ and $t_\theta \propto r^\beta$.
The study of the Heisenberg XXZ model in strongly disordered conditions-specifically with a disorder parameter of $h=14$ and averaged over 5000 realizations at a system size of $N=13$-reveals that out-of-time-order commutators, $C_x(r,t)$, exhibit light-cone-like behavior with characteristic timescales and spatial decay rates determined by thresholds $\theta \in \{0.25, 0.5, 1\}$, and fitting to functions of the form $t_\theta \propto e^{\beta r}$ and $t_\theta \propto r^\beta$.

This study demonstrates algebraic light cones for operator growth in strongly disordered Rydberg spin systems and proposes a method to experimentally measure out-of-time-order correlators in Rydberg atom arrays.

While many-body systems with short-range interactions are well-understood, the dynamics of their long-range counterparts remain largely unexplored. This work, ‘Quantum information scrambling in strongly disordered Rydberg spin systems’, investigates information propagation-specifically, operator growth-in strongly disordered systems featuring power-law interactions. We demonstrate that such systems exhibit algebraic light cones for operator spreading, a marked deviation from the logarithmic behavior observed in nearest-neighbor models, and propose a feasible experimental protocol utilizing Rydberg atom arrays to directly measure these dynamics via out-of-time-order correlators. Could this understanding of non-logarithmic scrambling unlock novel approaches to quantum information processing and many-body localization?

The Illusion of Predictability: Information’s Stubborn Path

The propagation of information isn’t limited to digital networks; it’s a fundamental process occurring across diverse physical systems, from the spread of forest fires and the collective motion of bird flocks to the dynamics of spin glasses and neural networks. However, quantifying how information travels within these systems presents a significant challenge. Determining the speed at which a signal diminishes with distance, or the ultimate reach of that signal before it’s lost to noise or system limitations, requires sophisticated analytical techniques and often, extensive computational modeling. The inherent complexity arises from the interplay of numerous factors – the system’s geometry, the nature of the interactions between its components, and the presence of inherent randomness or disorder. Simply tracking a signal’s decay isn’t enough; understanding the pathways it takes, the bottlenecks it encounters, and the way information is encoded and transformed along the way demands a nuanced approach to measurement and analysis, highlighting the frontiers of research in this interdisciplinary field.

Conventional understandings of how information travels through complex systems frequently rely on the premise that interactions between components diminish with distance, often modeled using power-law decay – a relationship where influence decreases proportionally to a power of the separation. While mathematically convenient and broadly applicable, this simplification doesn’t always reflect reality; many natural and engineered systems exhibit more nuanced interaction profiles. Researchers are discovering that the assumption of simple distance-based decay overlooks critical factors like network topology, heterogeneity in component strengths, and the presence of long-range connections. Consequently, predictions based solely on power-law interactions can significantly misrepresent the true extent and speed of information propagation, necessitating more sophisticated models that account for these complexities and offer a more accurate portrayal of information flow within these systems.

When disorder-randomness in structure or properties-becomes prominent within a system, the conventional understanding of information spread breaks down. Traditional models, relying on predictable interactions that diminish with distance – often described by a power law, $f(r) \propto r^{-\alpha}$ – fail to account for the complex pathways created by these irregularities. Research indicates that strong disorder doesn’t simply slow information propagation; it fundamentally alters the mechanism itself. Instead of a smooth, predictable diffusion, information navigates through localized ‘hotspots’ and unexpected connections, creating a fragmented and potentially much more efficient, though less predictable, transfer. This suggests the existence of emergent phenomena, where the system’s overall behavior isn’t simply the sum of its parts, and that disorder can, paradoxically, enhance information transfer in specific contexts.

Analysis of out-of-time-order commutators across 5000 disorder realizations reveals that the averaged behavior (red curves) closely matches the analytical solution for power-law Ising interactions (dashed white curve), demonstrating consistent system dynamics for nearest-neighbor and dipolar interactions at r=3.
Analysis of out-of-time-order commutators across 5000 disorder realizations reveals that the averaged behavior (red curves) closely matches the analytical solution for power-law Ising interactions (dashed white curve), demonstrating consistent system dynamics for nearest-neighbor and dipolar interactions at r=3.

The XXZ Model: A Convenient Illusion of Control

The XXZ Heisenberg model is a quantum mechanical model used in condensed matter physics to describe the interactions between spins. It represents a chain of spins-$1/2$ particles, where interactions occur between neighboring spins in three spatial directions: $x$, $y$, and $z$. The Hamiltonian for the model includes terms representing these interactions, as well as an anisotropy parameter, $\Delta$, which controls the relative strength of interactions along the $z$-axis compared to the $x$ and $y$ axes. This model is particularly useful for studying many-body localization and thermalization, as the interplay between interactions and disorder can lead to complex dynamical behavior. Its application to information scrambling arises from the model’s ability to simulate the propagation of quantum information through a many-body system, with the rate of information spread being determined by the strength of the interactions and the degree of disorder.

The introduction of disorder into the XXZ Heisenberg model simulates realistic imperfections found in quantum systems and allows for the study of many-body localization (MBL). In MBL, the presence of disorder inhibits the delocalization of quantum information, fundamentally altering the dynamics of the system. Specifically, disorder creates a localized phase where interactions are insufficient to allow information to propagate ballistically; instead, information diffusion becomes sub-diffusive or even completely arrested. Analyzing the XXZ model with disorder provides a controlled environment to investigate the mechanisms governing this transition from ballistic to localized behavior and to quantify how the strength and type of disorder influence the rate of quantum information propagation, measured by the growth of operators acting on local degrees of freedom.

Operator growth, within the context of the XXZ Heisenberg model at a disorder strength of $h = 14$, serves as a quantifiable metric for information propagation. Specifically, the rate at which the initially localized operator spreads throughout the system directly correlates to the speed at which quantum information is scrambled and disseminated. Measurements of this operator growth, typically assessed via the out-of-time-ordered correlation function, demonstrate a linear increase with time, the slope of which defines the operator’s growth rate – a value representing the effective speed of information scrambling. This methodology allows for a precise determination of how quickly information escapes its initial locality due to the interactions and disorder present within the model.

Operator growth, measured via out-of-time-order commutators for the Heisenberg XXZ model with van der Waals interactions at strong disorder, exhibits exponential growth with distance and time, with the rate of growth differing between nearest-neighbor and van der Waals interactions, as demonstrated by analysis of 5000 disorder realizations at system size N=13.
Operator growth, measured via out-of-time-order commutators for the Heisenberg XXZ model with van der Waals interactions at strong disorder, exhibits exponential growth with distance and time, with the rate of growth differing between nearest-neighbor and van der Waals interactions, as demonstrated by analysis of 5000 disorder realizations at system size N=13.

Light Cones: Mapping the Boundaries of What We Think We Know

Operator growth, which quantifies the spreading of quantum information, is fundamentally connected to the geometry of light cones. A light cone defines the maximal region in spacetime that can be influenced by, or receive information from, a given event or operator. The rate at which operators grow is directly constrained by the boundaries established by the light cone; information cannot propagate faster than the speed of light, and the light cone visually represents this limitation. Specifically, the dimensions of the accessible region – defined by the light cone – determine the number of operators that can effectively interact and contribute to the system’s dynamics at a given time. Therefore, alterations to the light cone’s shape directly impact the rate of operator growth and the system’s ability to process information.

In systems governed by simple, short-range interactions, the geometry of light cones – representing the propagation of information – scales algebraically with both time and spatial distance. Specifically, the spread of the light cone, defining the furthest point causally connected to an initial event, increases proportionally to $t^z$ in time and $x^z$ in space, where $z$ is a dynamic exponent. This algebraic behavior arises from the predictable nature of interactions and the resulting ballistic propagation of information. The exponent $z$ is determined by the dimensionality of the system and the nature of the interactions, providing a quantifiable measure of how quickly information can disseminate throughout the system without the influence of disorder.

In systems exhibiting strong disorder, the geometry of light cones deviates from the algebraic dependence observed in simpler systems. Specifically, the shape of these light cones transitions to one characterized by an exponent of $1.478 \pm 0.041$. This altered light cone shape is a key indicator of many-body localization (MBL), a phase of matter where localization prevents thermalization and leads to a breakdown of diffusion. The precise value of this exponent provides quantitative evidence supporting the presence of MBL and allows for characterization of the localized phase.

A modified measurement sequence effectively cancels out unwanted fields and implements the XXZ model with random disorder, as demonstrated by accurate out-of-time-order commutator calculations (black dashed line) matching Floquet propagation results across varying disorder strengths and cycle times.
A modified measurement sequence effectively cancels out unwanted fields and implements the XXZ model with random disorder, as demonstrated by accurate out-of-time-order commutator calculations (black dashed line) matching Floquet propagation results across varying disorder strengths and cycle times.

Rydberg Arrays: A Playground, Not a Solution

Rydberg atom arrays provide a configurable platform for quantum simulation due to the strong, controllable interactions between highly excited, or Rydberg, atoms. These arrays function as a scalable system where individual atoms serve as qubits, and interactions are mediated by dipole-dipole interactions with a strength dependent on the interatomic distance. This allows for the creation of arbitrary geometries and connectivity, enabling the simulation of diverse quantum systems, including those exhibiting Many-Body Localization (MBL). MBL is a phase of matter where localization prevents thermalization, and Rydberg arrays can be programmed to realize the disordered interactions and connectivity required to observe and study MBL phenomena. The ability to individually address and control each atom with lasers further enhances the system’s versatility for simulating complex Hamiltonians and exploring quantum dynamics.

Floquet engineering, applied to Rydberg atom arrays, utilizes time-periodic driving fields to modulate the interactions between atoms. This technique effectively creates an artificial, time-dependent Hamiltonian, allowing for the simulation of static Hamiltonians that would otherwise be inaccessible or difficult to realize. By carefully controlling the frequency and amplitude of the driving field, researchers can tune the strength and nature of interactions, such as the $V_{ij}$ between atoms $i$ and $j$, and introduce novel many-body phenomena. This precise control extends to engineering effective interactions beyond the native dipole-dipole interactions of the Rydberg atoms, enabling the investigation of a broader range of quantum systems and models.

Implementation of state transfer time-reversal symmetry within Rydberg atom arrays is achievable with a cycle time of $t_c = 0.1 J^{-1}$. This precise temporal control allows for the investigation of quantum information dynamics by effectively reversing the state transfer process. By manipulating the system with this cycle time, researchers can probe the effects of time-reversal symmetry on quantum states and explore phenomena related to quantum coherence and decoherence. The defined $t_c$ parameter facilitates controlled experiments aimed at understanding and manipulating the evolution of quantum information within the array.

The pursuit of perfectly localized many-body systems, as explored in this work, feels remarkably like chasing a stable sort order in a perpetually shifting dataset. This paper meticulously details how strong disorder alters operator growth – algebraic light cones instead of the expected logarithmic spread. It’s a beautifully complex demonstration of how reality consistently complicates elegant theories. One suspects that even with Floquet engineering and Rydberg atom arrays, production systems will find a way to introduce unforeseen interactions. As John Bell famously stated, ‘No phenomenon is a genuine phenomenon until it is a reproducible phenomenon.’ Reproducibility, of course, is the bane of any experimentalist attempting to tame chaos, and the Rydberg systems are no exception. The authors propose measuring out-of-time-order correlators; one imagines the debugging sessions will be
 extensive. It’s not about building something that can’t fail, but rather about understanding how it fails – and documenting it thoroughly for those digital archaeologists.

What’s Next?

The observation of algebraic light cones in these strongly disordered Rydberg systems is
 predictable. It shifts the goalposts, naturally. The logarithmic scaling of cleaner systems was always a convenient idealization, a theoretical scaffolding destined to crumble under the weight of real-world imperfections. The algebraic behavior simply reveals that ‘many-body localization’ isn’t a binary switch, but a gradient, and gradients are messy. The crucial test, of course, will be determining how robust these light cones are against more disorder-because adding disorder to a system already touted for its resistance to thermalization feels
circular.

The proposed experimental protocol, measuring out-of-time-order correlators, is a familiar refrain. Every new platform promises a direct probe of quantum chaos, and the data invariably requires increasingly elaborate post-processing to extract meaningful signals. One anticipates a blossoming of numerical techniques designed to ‘correct’ for the inevitable imperfections in the array, a process that will likely obscure the fundamental physics it seeks to reveal. If all the calibrations succeed, it’s probably because the tests don’t actually probe the relevant regime.

The field now faces the inevitable question: how much control is enough control? Each layer of ‘Floquet engineering’ adds complexity, and complexity is a tax on understanding. The pursuit of ever-more-exotic Hamiltonians may yield technically impressive demonstrations, but the true challenge lies in identifying the minimal ingredients required for genuinely novel quantum behavior. The next decade will likely be defined not by breakthroughs, but by the painstaking process of disentangling signal from noise-a task that, frankly, has been keeping physicists employed for quite some time.

Original article: https://arxiv.org/pdf/2512.19856.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-24 19:48