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Complex Operator Growth in Dissipative Quantum Systems

This paper extends the universal operator-growth hypothesis to dissipative quantum systems by demonstrating that while coupling to a Markovian environment introduces an imaginary growth rate proportional to dissipation strength, the real complexity-growth rate remains protected to second order, preserving the bound on the Lyapunov exponent without suppressing operator growth.

Original authors: Hikaru Wakaura, Taiki Tanimae

Published 2026-07-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hikaru Wakaura, Taiki Tanimae

Original paper licensed under CC BY 4.0 (https://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 Invisible Dance of Chaos and Noise

Imagine you are watching a drop of ink fall into a glass of water. At first, it's a neat, simple circle. But as time passes, it swirls, stretches, and spreads out until it becomes a uniform, messy gray. In the world of quantum physics, this spreading isn't just about ink; it's about information. When a quantum system is "chaotic," a tiny, simple piece of information (like the spin of a single atom) doesn't just stay put. It gets scrambled, spreading its influence across the entire system in a complex dance. Scientists call this "operator growth." It's the engine behind why quantum computers are so hard to simulate and why the universe eventually settles into a state of thermal equilibrium.

For a long time, physicists studied this dance in a perfect, sealed room where nothing could get in or out—a "closed system." They found a universal rule: the complexity of the dance grows at a steady, predictable speed, like a car accelerating on a straight highway. This speed sets a hard limit on how fast chaos can spread, a rule known as the "chaos bound." But the real world isn't a sealed room. Our quantum systems are always leaking a little bit of energy or information into their surroundings, a process called "dissipation." It's like trying to watch that ink drop spread while someone is constantly blowing air across the glass. The big question was: Does this wind slow down the dance, speed it up, or change the rhythm entirely? This is the puzzle that Hikaru Wakaura and Taiki Tanimae set out to solve.

The Paper's Discovery: A Dance in a Windy Room

In their new research, Wakaura and Tanimae explore what happens to this quantum dance when the system is open and interacting with a noisy environment. They discovered that the simple, straight-line acceleration of chaos doesn't just slow down or stop; it actually gets a strange, invisible twist. Instead of just moving forward, the growth rate of the complexity spins off into a new dimension, becoming a "complex" number with both a real part (the speed we can see) and an imaginary part (a hidden, rotational speed).

Think of the growth rate as a car driving down a road. In a closed system, the car drives in a perfectly straight line. When you add dissipation (the wind), the paper finds that the car doesn't just slow down. Instead, it starts driving in a slightly tilted circle. The "real" part of the speed, which determines how fast the complexity actually grows, barely changes at all. In fact, the authors show that for weak noise, the growth speed is protected; it only gets a tiny, quadratic nudge, meaning the chaos keeps growing almost as fast as it did in the perfect, sealed room.

However, the "imaginary" part of the speed is the real star of the show. This isn't a mathematical trick; it's a physical measurement of how the system is losing information to its environment. The paper proves that this imaginary speed is directly tied to how much the "messy" operator (the spreading ink) is interacting with the noise. If the noise is like a fan blowing on the ink, the imaginary rate measures exactly how fast the ink is getting pushed around by that fan. The authors found a precise formula for this: the imaginary speed is exactly twice the strength of the noise multiplied by a specific "weight" of the operator. It's like having a speedometer that doesn't just tell you how fast you're going, but also tells you exactly how hard the wind is hitting your car.

The researchers tested these ideas using two very different models: a chain of interacting spins (like a row of tiny magnets) and a famous model called the SYK model (a theoretical playground for quantum chaos). In the spin chain, they saw that as the noise increased, the imaginary speed grew in a perfectly straight line, just as their theory predicted. In the SYK model, they found a neat, closed-form equation where the imaginary speed depends only on how far apart the particles interact, not on how strongly they interact. This suggests that the "wind" of dissipation doesn't just dampen the system; it reveals a new, hidden layer of structure that was always there but invisible in a closed room.

Perhaps most importantly, the paper argues that the famous "chaos bound"—the limit on how fast information can scramble—still holds true in this noisy world. The authors suggest that the limit is now set by the "real" part of the speed (the straight-line component), which remains almost unchanged even with the noise. They propose that the wind doesn't stop the car from accelerating; it just makes the car's path curve slightly. This means that even in a messy, open world, the fundamental speed limit of quantum chaos is surprisingly robust.

The study also clarifies what doesn't happen. A common intuition might be that adding noise (dissipation) should act like a brake, slowing down the spreading of information. The paper explicitly rules this out. While the peak height of the complexity might be lower because the system is losing energy, the rate at which it grows initially actually speeds up slightly or stays the same. The noise doesn't suppress the growth; it just rotates the direction of that growth into the complex plane.

In summary, Wakaura and Tanimae have shown that when quantum systems leak information, the growth of chaos doesn't break; it evolves. It gains a new, imaginary component that acts as a precise ruler for how the system interacts with its environment, while the core speed of the chaos remains protected. It's a reminder that even in a noisy, imperfect world, the fundamental laws of quantum chaos are resilient, just wearing a slightly different, more colorful hat.

Technical Summary: Complex Operator Growth in Dissipative Quantum Systems

Problem Statement
The Universal Operator-Growth Hypothesis (OGH) posits that in closed chaotic quantum systems, the Lanczos coefficients ( $b_n$ ) governing operator growth in the Krylov basis grow linearly ( $b_n \simeq \alpha n$ ). This rate $\alpha$ bounds the Lyapunov exponent ( $\lambda_L \leq 2\alpha$ ) and controls the exponential growth of Krylov complexity. However, real physical systems are open and subject to Markovian dissipation. In such systems, the generator of dynamics is the Lindbladian, which is non-Hermitian. Consequently, the standard symmetric Lanczos algorithm fails, necessitating the use of the Arnoldi iteration or bi-Lanczos schemes. While previous studies have computed coefficients numerically for weakly dissipative systems (e.g., large- $q$ SYK), a physical and analytic understanding of the resulting complex coefficients—specifically the diagonal terms—has been lacking. The central question is how the operator-growth structure is modified when the generator becomes non-Hermitian, and whether the chaos bound survives.

Methodology
The authors analyze operator growth at infinite temperature using the Wightman inner product $(A|B) = \text{Tr}(A^\dagger B)/\dim$ . The dynamics are generated by the adjoint Lindbladian $G = L_0 + D$ , where $L_0 = i[H, \cdot]$ is the anti-Hermitian Liouvillian and $D$ is the dissipator.

Recursion Scheme: The authors apply the Arnoldi iteration with full re-orthogonalization to the vectorized Lindbladian acting on an initial operator $O_0$ . This generates an upper-Hessenberg matrix $h$ with real sub-diagonal elements $b_n$ and complex diagonal elements $a_n$ .
Complex Rate Definition: The asymptotic growth is characterized by a complex rate $\alpha_C = \alpha_R + i\alpha_I$ , defined by the limits $b_n \to \alpha_R n$ and $-\text{Re}(a_n) \to \alpha_I n$ .
Models Studied: The analysis is performed on:
- A chaotic mixed-field Ising chain with bulk dephasing ( $L_i = \sqrt{\gamma} Z_i$ ).
- The large- $q$ Sachdev-Ye-Kitaev (SYK) model with Majorana dephasing.
- A Bose-Hubbard trimer (continuous-variable system) to test robustness.
Analytic Tools: The authors utilize exact identities derived from the generator's symmetry properties ( $G^\dagger(\gamma) = -G(-\gamma)$ ) and moment bounds derived from generalized Lieb-Robinson bounds for local Markovian generators.

Key Contributions and Results

1. Exact Identity for the Imaginary Rate (Pillar 1)
The authors derive an exact identity for the real part of the diagonal Arnoldi coefficient:
$\text{Re}(a_n) = -2\gamma \langle \hat{G} \rangle_n$
where $\hat{G}$ is a "grading" operator that counts the weight of the operator in the dissipation channels (e.g., transverse weight for dephasing spins).

Physical Interpretation: The imaginary rate $\alpha_I = 2\gamma \sigma_G$ (where $\sigma_G$ is the slope of the grading weight along the Krylov chain) measures the rate at which the growing operator accumulates weight in the environment's coupling channels.
Symmetry and Parity: The generator symmetry $G^\dagger(\gamma) = -G(-\gamma)$ $G^{†} (γ) = - G (- γ)$ implies that the sub-diagonal coefficients $b_n$ $b_{n}$ are even functions of the dissipation strength $\gamma$ $γ$ , while $\text{Re}(a_n)$ $Re (a_{n})$ is odd. Consequently:
- $\alpha_R(\gamma) = \alpha_R(0) + O(\gamma^2)$ : The complexity growth rate is renormalized only quadratically.
- $\alpha_I(\gamma) = 2\kappa_0 \gamma + O(\gamma^3)$ : The imaginary rate turns on linearly, determined entirely by the closed-system transverse-weight slope $\kappa_0$ .
Large- $q$ SYK: For the SYK model with Majorana dephasing, the authors obtain a closed-form expression:
$\alpha_C = J - 2i(q-2)\gamma$
Here, the imaginary part is fixed solely by the interaction range $q$ and is independent of the coupling $J$ .

2. Dissipative Chaos Bound (Pillar 2)
The authors argue that the OGH bound extends to open systems. By establishing that the generator $G$ is a sum of local, bounded superoperators, they derive a bound on the norm of the $n$ -th power of the generator: $\|G^n O\| \leq (cw)^n n! \|O\|$ .

Analyticity Strip: This implies the autocorrelation function $C(t)$ is analytic in a strip $|\text{Im } t| < \tau(\gamma)$ , where the width is narrowed by at most $O(\gamma)$ due to dissipation.
The Bound: Assuming the regularity condition that factorial moment bounds imply linear growth of Lanczos coefficients (the same assumption underlying the closed OGH), the authors propose:
$\lambda_L^{\text{open}} \leq 2\alpha_R(\gamma) \leq 2\alpha_R(0) + O(\gamma)$
This suggests that dissipation does not suppress the rate of operator growth; rather, it rotates the rate into the complex plane.

3. Distinction Between Rate and Peak Complexity
The authors distinguish between the asymptotic growth rate and the finite-time peak of Krylov complexity. While $\alpha_R$ (the rate) increases or remains stable with weak dissipation, the peak Krylov complexity decreases monotonically with $\gamma$ . This is because the depth-dependent damping term $\text{Re}(a_n) \propto -n$ removes amplitude at large Krylov depths, suppressing the height of the complexity curve even as the initial slope increases.

Significance and Claims
The paper claims that the imaginary rate $\alpha_I$ is a new, environment-resolved observable that provides an exact, parity-odd measure of dissipative operator spreading.

Universality vs. Non-Universality: While the magnitude of the imaginary rate ( $\sigma_G$ ) is non-universal (model-dependent), the structural features—linear onset in $\gamma$ , the exact identity linking it to the grading operator, and the parity symmetry—are universal.
Robustness of Growth: A central conclusion is that dissipation does not slow down operator growth in the sense of reducing the complexity growth rate $\alpha_R$ . Instead, the growth rate undergoes a "rigid rotation" into the complex plane.
Experimental Accessibility: The authors note that the diagonal coefficients $a_n$ (and thus $\alpha_I$ ) are encoded in the component of the autocorrelation function $C(t)$ that is not even in time, making them accessible via standard recursion methods used to extract Lanczos coefficients.

The authors remain modest regarding the chaos bound, explicitly stating that Equation (8) is a conjecture "at the same footing as the OGH itself," relying on the unproven regularity assumption that factorial moment bounds imply linear coefficient growth. They do not claim to prove the bound but rather to extend the OGH framework to open systems with the same logical constraints.

The Invisible Dance of Chaos and Noise

The Paper's Discovery: A Dance in a Windy Room

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