⚛️ quantum physics

How to distinguish chaos from integrability using OTOC

This paper demonstrates that while OTOC decay alone cannot distinguish chaotic from integrable systems, the late-time saturation value of operator purity provides a robust distinction through its distinct scaling with system size, a result derived analytically using the Mazur bound and supported by numerical simulations of one-dimensional spin chains.

Original authors: Alexey Milekhin, Anders Wallberg

Published 2026-10-06

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CC0 1.0

Original authors: Alexey Milekhin, Anders Wallberg

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, the line between order and chaos is often harder to draw than in our everyday experience. In classical physics, chaos is easy to spot: a small change in the starting position of a billiard ball leads to a wildly different path. Quantum systems are more subtle. They are governed by rules that allow for perfect predictability in theory, yet in practice, many of them behave as if they are completely random. Physicists have long sought a reliable way to tell the difference between a system that is truly chaotic and one that is "integrable," a technical term for a system that follows hidden, rigid rules and preserves a vast number of specific quantities as it evolves. For years, scientists have looked at how information spreads through these systems, hoping to find a signature that separates the two. One popular tool for this is a measurement called the out-of-time-ordered correlator, which tracks how a piece of information, once localized, gets scrambled and spread out across the entire system. While this scrambling happens in both chaotic and integrable systems, the way it settles down at the very end of the process might hold the key to distinguishing them.

A team of researchers has now proposed a new way to make this distinction, focusing not on how fast information scrambles, but on how much of it remains trapped in a specific region after a long time has passed. They studied a quantity known as operator purity, which essentially measures how "mixed up" a quantum system has become. Imagine a system where you can track a specific piece of information as it evolves. In a chaotic system, this information spreads out so thoroughly that it becomes indistinguishable from random noise, and the system settles into a state where the remaining order is minimal and depends only on the size of the region you are looking at relative to the whole. However, in an integrable system, the presence of those hidden, rigid rules prevents the information from spreading completely. The researchers found that in these ordered systems, a significant amount of information remains localized, and the amount of this leftover order grows steadily as the system gets larger.

To uncover this difference, the authors used a mathematical tool called the Mazur bound. This tool allows physicists to estimate the minimum amount of correlation that must remain in a system if it possesses conserved quantities—things that do not change over time. In a chaotic system, the only thing that is strictly conserved is the total energy. In an integrable system, however, there are many other conserved quantities, and crucially, these are often "local," meaning they are tied to specific, small parts of the system rather than the whole. The researchers realized that because these local rules exist, they act as a barrier that stops information from fully scrambling. By applying this logic, they derived a prediction: if you measure the leftover order in a chaotic system, the value will stay constant regardless of how big the system is, as long as the ratio of the part you are watching to the whole remains the same. But in an integrable system, that value will grow linearly with the size of the part you are watching.

The team tested this idea using computer simulations of one-dimensional chains of quantum spins, which are simple models of magnetic materials. They simulated two types of systems: one that was known to be chaotic and another that was known to be integrable. In the chaotic case, the simulations confirmed their prediction: the measure of leftover order settled to a steady value that depended only on the geometry of the setup. In the integrable case, the measure did not settle to a constant; instead, it grew larger as the system size increased, exactly as the theory of local conserved charges suggested. The results were clear and robust, showing that the way a system retains information at late times is a reliable fingerprint of whether it is chaotic or ordered.

This finding is significant because it offers a practical way to distinguish between these two fundamental types of quantum behavior without needing to observe the system for impossibly long periods. Previous methods often required watching the system for times that grow exponentially with its size, making them impossible to use for large systems. The new approach suggests that the distinction can be seen much earlier, at times that grow only as a power of the system size, which is much more manageable. The researchers also noted that the fluctuations in the data were much larger in the integrable systems than in the chaotic ones, providing another clue that the two behave differently. While the study relied on simulations of specific models, the underlying logic is based on general principles of quantum mechanics and conservation laws, suggesting that this behavior should hold true for a wide range of physical systems.

The work also sheds light on why the "locality" of the conserved rules matters so much. In a chaotic system, the only rules are global, like the total energy, which do not prevent information from spreading out. In an integrable system, the rules are local, acting like a series of small locks that keep information from moving freely between different parts of the system. The researchers showed that these local locks are what cause the leftover order to grow with the size of the system. This insight helps explain why some quantum systems are so good at preserving information while others scramble it instantly. It also connects to broader questions in physics, from how black holes might process information to how quantum computers might maintain their data. By focusing on the final state of the system rather than the journey there, the researchers have provided a new, clear lens through which to view the complex dance of quantum information, revealing that the difference between chaos and order is written in the amount of memory a system keeps.

Technical Summary: Distinguishing Chaos from Integrability Using OTOC

Problem Statement
While the decay of out-of-time-ordered correlators (OTOCs) to a small residual value is a generic signature of information scrambling in interacting many-body systems, this behavior alone fails to distinguish between chaotic and integrable theories. Both classes of systems exhibit OTOC decay, rendering the early-time Lyapunov regime (specific to systems with large semiclassical parameters) and the generic late-time decay insufficient for operational differentiation. The central challenge is to identify a physical observable that robustly distinguishes chaotic systems from integrable ones based on the locality of their conserved charges.

Methodology
The authors propose analyzing the late-time saturation value of operator purity (equivalent to the second Rényi entropy of the Choi–Jamiolkowski state associated with the time-evolution operator), denoted as $\text{Tr}\rho_{A_1 A_2}^2$ . This quantity is related to the OTOC averaged over all operators in a subsystem.

The core analytical tool employed is the Mazur bound, which provides a lower bound on the infinite-time average of an autocorrelation function in terms of the system's conserved charges $\{Q_k\}$ . The authors apply this bound to the autocorrelation of the SWAP operator ( $\text{SWAP}_A$ ) in a doubled system (two copies of the original system). The operator purity is expressed as:
$\text{Tr}\rho_{A_1 A_2}^2 = \frac{1}{d^2} \langle \text{SWAP}_A(0) \text{SWAP}_A(t) \rangle$
where $d$ is the dimension of the full Hilbert space and $d_A$ is the dimension of the subsystem $A$ .

The analysis relies on the following steps:

Decomposition of Charges: The Mazur bound is constructed using the set of conserved charges of the doubled system, specifically $Q_k \otimes Q_k$ , where $Q_k$ are the quasi-local conserved charges of the original Hamiltonian.
Locality Analysis: The authors explicitly calculate the overlap between the SWAP operator and these conserved charges. They demonstrate that for local or quasi-local charges (where the charge density is supported on a finite number of sites), the partial trace $\text{Tr}_A(Q_k)$ is non-zero only if the charge fits entirely within the subsystem $A$ .
Scaling Derivation: By summing the contributions of these charges to the Mazur bound, the authors derive the scaling behavior of the residual operator purity with respect to the subsystem size $L_A$ and the total system size $L$ .

Key Contributions and Results

Distinct Scaling Laws:
The paper establishes that the late-time saturation value of the operator purity scales differently for chaotic and integrable systems:
- Chaotic Systems: Assuming the Hamiltonian $H$ is the only relevant conserved charge (and potentially its powers), the Mazur bound yields a residual value that depends only on the ratio $L_A/L$ . Specifically, $d_A^2 \times \text{Tr}\rho_{A_1 A_2}^2 \geq C(L_A/L)$ . The value remains constant as $L_A$ increases at a fixed ratio.
- Integrable Systems: In the presence of an extensive number of quasi-local conserved charges $Q_k$ (e.g., in the Heisenberg XYZ chain), the bound scales linearly with the subsystem size $L_A$ at a fixed ratio $L_A/L$ . The derived lower bound is:
  $d_A^2 \times \text{Tr}\rho_{A_1 A_2}^2 \geq L_A \times I(L_A/L)$
  This linear growth arises because the number of quasi-local charges that fit entirely within the subsystem $A$ grows linearly with $L_A$ .
Role of Charge Locality:
The analysis clarifies why the locality of conserved charges is critical. Projectors onto energy eigenstates (which are non-local) contribute an exponentially small term ( $\sim d_A^2/d^2$ ) to the bound. In contrast, local integrable charges contribute terms of order $(L_A/L)^2$ , leading to the observed linear growth in the integrated purity.
Numerical Verification:
The authors support their analytical findings with exact diagonalization simulations of one-dimensional spin chains:
- Models: The XXX and XYZ spin chains (integrable) and the mixed-field Ising model (chaotic when integrability is broken).
- Findings: The numerical data confirms the theoretical predictions. In chaotic cases, the operator purity saturates to a constant value dependent on $L_A/L$ . In integrable cases, the quantity grows significantly with $L_A$ , consistent with the derived linear scaling (and in some cases, exhibiting even faster growth due to additional conserved charges in the doubled system).
Saturation Time:
The paper argues that while the Mazur bound applies to the infinite-time average, the saturation of OTOCs and operator purity occurs at the Thouless time (polynomial in system size, e.g., $L^2/D$ for diffusive systems), rather than the exponentially long Heisenberg time. This makes the distinction between chaos and integrability practically observable.

Significance and Claims
The paper claims to provide a robust, operational method to distinguish chaotic from integrable quantum systems using the late-time behavior of operator purity. Unlike previous diagnostics that rely on spectral statistics (requiring exponentially long observation times) or early-time Lyapunov exponents (requiring large- $N$ limits), this approach relies on the scaling of the residual value with system size.

The authors emphasize that their approach makes explicit the connection between the locality of conserved charges and the obstruction of information scrambling. They note that while the Mazur bound provides a lower limit, their numerical results suggest the actual values in integrable systems may exceed these bounds, potentially due to the complex structure of conserved charges in the doubled system (products of charges). The work suggests that operator purity is a sensitive probe of the underlying conservation laws, distinguishing integrable systems (with extensive quasi-local charges) from chaotic ones (where only global charges like energy exist).

The paper concludes by noting that these results may have broader implications for understanding quantum advantage and the black hole information paradox, given the relevance of operator entropy in those contexts, though the primary focus remains on the chaos-versus-integrability distinction.

Technical Summary: Distinguishing Chaos from Integrability Using OTOC

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