⚛️ quantum physics

When level repulsion fails: non-normality and chaos in open quantum systems

This paper demonstrates that level repulsion statistics are an unreliable diagnostic for quantum chaos in open quantum systems because the strong non-normality of Lindbladians decouples spectral statistics from physical dynamics, allowing chaotic signatures to be absent even when level repulsion is present.

Original authors: Caio B. Naves, Thomas Klein Kvorning, Jonas Larson

Published 2026-04-02

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CC BY 4.0

Original authors: Caio B. Naves, Thomas Klein Kvorning, Jonas Larson

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

The Big Idea: A Broken Compass for Chaos

Imagine you are a detective trying to figure out if a machine is "chaotic" (wild, unpredictable, and sensitive to tiny changes) or "regular" (calm, predictable, and orderly).

For a long time, scientists had a trusted tool for this job: Level Repulsion.

The Theory: If you look at the "notes" (energy levels) a machine plays, a chaotic machine will have notes that refuse to be the same distance apart. They push each other away, like people in a crowded elevator trying not to touch. A regular machine, however, might have notes that are randomly spaced, like raindrops hitting a roof.
The Rule: In closed systems (like a perfect box with no leaks), this rule works perfectly. If the notes push each other away, the system is chaotic.

This paper says: "Stop! That rule doesn't work for open systems."

Open systems are like machines with leaks, pumps, or friction (think of a cup of coffee cooling down, or a quantum computer interacting with its environment). The authors show that in these systems, the "notes" can look like they are pushing each other away (mimicking chaos) even when the machine is actually perfectly calm and predictable.

It's like looking at a calm lake and seeing ripples that look like a storm, but the water is actually still. The tool (Level Repulsion) is giving you a false alarm.

The Two Main Characters: The "Drunk" and the "Mirror"

To understand why this happens, we need to understand two concepts the paper introduces: Non-Normality and the Skin Effect.

1. The "Drunk" Matrix (Non-Normality)

In the world of math, some machines (matrices) are "Normal." They are like a well-behaved mirror: if you poke them slightly, they reflect a tiny change.

Normal Matrix: You push the mirror a little bit; the reflection moves a little bit.
Non-Normal Matrix: This is like a drunk tightrope walker. If you push them even a tiny bit, they might fall off the rope or spin wildly.

In open quantum systems, the math describing the system is often "Non-Normal." This means the system is incredibly sensitive to tiny errors. Even a microscopic mistake (like a tiny rounding error in a computer calculation) can make the system's "notes" jump around wildly.

2. The "Skin Effect" (The Crowd at the Door)

The paper also talks about the Non-Hermitian Skin Effect. Imagine a long hallway with people (eigenstates) standing in it.

In a normal system, people are spread out evenly.
In these "open" systems, the rules of the hallway force almost everyone to crowd up against one specific wall (the boundary).

When you try to measure the "notes" of this system, the computer sees this massive crowd at the wall. Because the crowd is so dense and jumbled, the computer's math gets confused. It starts seeing "pushing away" (level repulsion) just because the crowd is squished together, not because the system is actually chaotic.

The Experiment: Two Simple Machines

The authors tested this with two simple, boring machines that are definitely not chaotic:

A Driven Harmonic Oscillator: Think of a swing being pushed by a fan. It's a simple, predictable motion.
A Tight-Binding Model: Think of a row of buckets where water leaks from one to the next. Also very predictable.

What happened?
When they ran these machines on a computer:

The Math: The computer calculated the "notes" and found they were pushing each other away perfectly. By the old rules, the computer screamed, "CHAOS! CHAOS!"
The Reality: The authors watched the actual movement of the system. It was smooth, calm, and predictable. No chaos at all.

Why the lie?
The computer had to cut off the infinite size of the universe to fit it on a screen (a process called "truncation"). This cut-off created a "wall." Because the system was "Non-Normal" (the drunk tightrope walker), the presence of this wall made the math go haywire. The computer saw the "crowd at the wall" and misinterpreted it as chaos.

The Takeaway: Don't Trust the "Notes" Alone

The paper concludes with a warning for scientists:

"Just because the notes look chaotic, doesn't mean the music is chaotic."

In open quantum systems (which include almost all real-world quantum devices), the "level statistics" (the spacing of the notes) are a broken compass. They can be tricked by:

Boundary conditions: Where you draw the line of your simulation.
Non-normality: How sensitive the system is to tiny errors.
Computer errors: Even the tiniest rounding error in a calculation can create a fake "chaotic" pattern.

The New Rule:
If you want to know if an open quantum system is truly chaotic, you cannot just look at the spectrum (the notes). You have to watch the movie (the actual dynamics over time). You need to see if the system actually behaves wildly, not just if its math looks messy.

Summary Analogy

Imagine you are trying to tell if a party is wild (chaotic) or quiet (regular).

The Old Way: You look at the dance floor from a distance. If people are bumping into each other, you assume it's a wild party.
The Problem: In this specific type of party (Open Systems), the dance floor is so slippery (Non-Normal) that even if everyone is standing still, a tiny breeze (computer error) makes them stumble and bump into each other.
The Result: You see them bumping and think, "Wow, what a wild party!" but in reality, everyone was just standing still.
The Solution: Don't just look at the bumps. Watch how they move over time. If they aren't actually dancing wildly, it's not a wild party, no matter how much they bump.

1. Problem Statement

The paper addresses a critical flaw in the application of spectral level statistics as a diagnostic tool for quantum chaos in open quantum systems.

Context: In closed Hamiltonian systems, the statistical distribution of energy level spacings (e.g., Wigner-Dyson statistics) is a robust indicator of quantum chaos, distinguishing it from integrable (Poissonian) systems. This logic was extended to open systems via the Grobe–Haake–Sommers (GHS) conjecture, which posits that open quantum systems with chaotic classical counterparts should exhibit level repulsion in their Lindbladian spectrum (governed by non-Hermitian random matrix theory, specifically the Ginibre ensemble).
The Issue: The authors argue that this analogy fails because Hamiltonian and Lindbladian operators have fundamentally different mathematical properties. Hamiltonians are normal (commute with their adjoints), making their spectra stable against perturbations. In contrast, Lindbladians are typically non-normal, leading to extreme spectral instability.
Core Question: Can level repulsion in the Lindbladian spectrum reliably distinguish between chaotic and regular dynamics in open quantum systems, or is it an artifact of non-normality and numerical boundaries?

2. Methodology

The authors employ a combination of analytical theory, numerical simulation, and perturbation analysis to dissect the relationship between spectral statistics and dynamical behavior.

Model Systems: Two exactly solvable, integrable models were chosen to test the GHS conjecture:
1. Driven Quantum Harmonic Oscillator: Coupled to a thermal bath with incoherent gain and loss.
2. Open Tight-Binding Model: A 1D lattice with incoherent tunneling.
- Note: Both models are analytically known to be integrable (non-chaotic) and possess regular dynamics.
Numerical Diagonalization: The authors numerically diagonalized the Lindbladian superoperators for these models under two conditions:
- Truncation: Imposing a cutoff $N_{tr}$ in the Fock space for the oscillator.
- Boundary Conditions: Comparing Periodic Boundary Conditions (PBC) with Open Boundary Conditions (OBC) for the tight-binding model.
Statistical Analysis:
- Calculated the level spacing distribution $P(s)$ and the complex spacing ratio $z_n$ (radial and angular components).
- Compared results against the Ginibre Unitary Ensemble (GinUE) (chaotic) and 2D Poisson (regular) distributions.
Dynamical Probes: To verify the absence of chaos, the authors computed the Uhlmann fidelity between the numerically truncated/open-boundary evolution and the exact analytical (or PBC) evolution. They also analyzed the eigenvalue condition numbers to quantify non-normality.
Theoretical Framework:
- Applied non-normal matrix theory, specifically focusing on the condition number of the eigenvector matrix ( $\kappa$ ).
- Investigated the $\epsilon$ -pseudospectrum to understand spectral sensitivity.
- Linked the observed phenomena to the non-Hermitian skin effect (accumulation of eigenvectors at boundaries).

3. Key Contributions

Refutation of the GHS Conjecture for Generic Open Systems: The paper demonstrates that the GHS conjecture does not hold generally. Level repulsion in the Lindbladian spectrum can arise in systems with completely regular (integrable) dynamics solely due to strong non-normality.
Identification of Non-Normality as the Driver: The authors establish that the "chaotic" spectral signatures are not dynamical but structural. They arise from the extreme spectral instability of non-normal operators, where infinitesimal perturbations (like numerical round-off errors or boundary truncations) are exponentially amplified.
Decoupling of Spectral Statistics and Dynamics: The study proves that one can tune level statistics to appear chaotic (Ginibre-like) without any corresponding chaotic behavior in the time evolution (fidelity remains near unity, no operator growth).
Mechanism of Failure: The failure is linked to the non-Hermitian skin effect in Liouville space. Boundaries (physical or numerical) induce a drift that localizes eigenvectors, causing the condition number to grow exponentially with system size. This makes the spectrum effectively random, mimicking chaos.

4. Key Results

Spectral Statistics vs. Dynamics:
- For both the driven oscillator and the tight-binding model, numerical diagonalization yielded spectra with cubic level repulsion ( $P(s) \sim s^3$ ) and a "bitten-donut" complex spacing ratio, characteristic of the Ginibre ensemble (chaos).
- However, the Uhlmann fidelity between the "chaotic-looking" spectrum's dynamics and the exact integrable dynamics remained close to 1.0 for arbitrarily long times (as system size increased). This confirms the dynamics are regular, not chaotic.
Role of Boundaries and Truncation:
- The "chaotic" spectral features appeared only when boundaries were introduced (via truncation $N_{tr}$ or OBC).
- Analytical spectra (PBC or infinite limit) showed no level repulsion.
- The discrepancy is caused by numerical round-off errors acting as perturbations on a highly non-normal matrix.
Condition Number Scaling:
- The condition number of bulk eigenvalues grows exponentially with system size ( $N_{tr}$ or $L$ ), while the steady-state eigenvalue remains stable.
- This exponential growth implies that the $\epsilon$ -pseudospectrum covers a vast region of the complex plane. Consequently, even machine-precision errors shift eigenvalues enough to create artificial level repulsion.
Breakdown of Poisson Statistics:
- Unlike Hamiltonian systems where Poisson statistics are a robust signature of integrability, in non-normal Lindbladians, Poisson statistics are fragile. Any slight perturbation (including numerical noise) will destroy the Poisson structure and force the spectrum toward random-matrix statistics.

5. Significance and Implications

Invalidation of a Standard Diagnostic: The paper fundamentally challenges the use of Lindbladian level statistics as a universal diagnostic for quantum chaos. Researchers cannot rely on spectral repulsion to claim a system is chaotic; it may simply be a non-normal system with boundaries.
Need for Alternative Diagnostics: The authors advocate for dynamical probes (e.g., Out-of-Time-Ordered Correlators (OTOCs), operator growth, or fidelity decay) that are robust against the spectral instabilities caused by non-normality.
Implications for Numerical Simulations:
- Standard numerical methods (Krylov/Arnoldi, Matrix Product Operators) that assume small errors lead to small spectral deviations are invalid for highly non-normal open systems.
- Apparent spectral gaps and decay rates extracted from such simulations may be artifacts of truncation and non-normality rather than physical properties.
Broader Physical Context: The findings extend to the non-Hermitian skin effect, suggesting that boundary-induced localization in Liouville space is the root cause of the spectral instability. This has implications for quantum sensing, error correction thresholds, and the simulation of dissipative phase transitions.

Conclusion: The paper concludes that in open quantum systems, spectral statistics are not a reliable indicator of chaos due to the inherent non-normality of Lindbladian generators. The "chaos" observed in spectra is often a mathematical artifact of boundary conditions and numerical precision, not a reflection of the system's dynamical complexity.