⚛️ high-energy theory

Wigner negativity in Krylov space and emergent semiclassicality

This paper proposes that the Krylov basis provides a semiclassical representation for complex many-body systems by demonstrating that Wigner negativity, a measure of classical simulation complexity, remains constant or grows slowly in various solvable models, thereby indicating emergent semiclassicality in Krylov space.

Original authors: Vijay Balasubramanian, Pawel Caputa, Onkar Parrikar, Vivek Singh

Published 2026-07-03

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CC BY 4.0

Original authors: Vijay Balasubramanian, Pawel Caputa, Onkar Parrikar, Vivek Singh

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

Imagine you are trying to describe the motion of a chaotic, complex system—like a swarm of billions of bees or a turbulent ocean. In the world of quantum physics, describing such a system is usually a nightmare. The math requires tracking an exponentially huge number of possibilities, making it impossible for even the fastest supercomputers to simulate. It's like trying to write down the exact position and speed of every single grain of sand on a beach simultaneously.

However, this paper suggests there is a special "secret language" or a specific way of looking at the system where the chaos suddenly looks simple and almost classical (like everyday objects). The authors call this the Krylov basis.

Here is a breakdown of their findings using simple analogies:

1. The Problem: The "Messy Room" vs. The "Organized Shelf"

Usually, when a quantum system evolves over time, its state spreads out like a drop of ink in water, becoming a messy, complex superposition of millions of different states. If you try to describe this using standard methods (the "computational basis"), you need to track an impossible amount of information.

The Krylov basis is like a magical, custom-built shelf.

The Setup: You start with a specific initial state (like a single book on the shelf).
The Magic: As time passes, the system doesn't spread out randomly into the whole room. Instead, it moves in a very orderly, local way along this specific shelf. It hops from one spot to the next, like a person walking down a hallway.
The Result: Even though the system is huge and complex, in this specific "Krylov hallway," the state stays relatively simple and confined.

2. The Test: The "Wigner Negativity" Meter

How do we know if the system is behaving like a simple, classical object (like a ball rolling) or a weird, complex quantum object (like a ghost)? The authors use a tool called Wigner negativity.

The Analogy: Think of Wigner negativity as a "Quantum Weirdness Meter."
- If the meter reads zero, the system behaves like a normal, classical object. You could simulate it easily on a regular computer.
- If the meter reads high, the system is deeply quantum and "weird." Simulating it would require a supercomputer.
The Goal: The authors wanted to see if this "Weirdness Meter" stays low when they watch the system evolve in their special "Krylov hallway."

3. The Experiments: Three Different Worlds

The authors tested this idea in three different theoretical worlds to see if the "Krylov hallway" always keeps the system simple.

A. The 2D Conformal Field Theory (The Infinite Line)

The Scenario: They looked at a quantum field on an infinite line, starting with a vacuum (empty space) and then poking it with a particle.
The Finding: Even though the "spread" of the system (how far it walks down the hallway) grows very fast, the Weirdness Meter stays low and constant.
The Takeaway: The system behaves almost entirely like a classical object in this view. The complexity is there, but it's a "classical complexity" (like a long line of dominoes falling), not a "quantum weirdness."

B. Random Matrix Theory (The Chaotic Dice)

The Scenario: They used a model where the system is essentially a giant, random jumble of numbers (like rolling a trillion dice). This is usually the definition of maximum chaos.
The Finding: In this chaotic world, the Weirdness Meter does start to rise, but it grows very slowly (like the square root of time). It doesn't explode to infinity.
The Takeaway: Even in a completely random, chaotic system, if you look at it through the Krylov lens, it still retains a "semi-classical" nature for a long time. It's not fully quantum weird yet.

C. The Double-Scaled SYK Model (The Black Hole Toy)

The Scenario: This is a famous model used to study black holes and gravity.
The Finding: They found two distinct phases:
1. Early Times: The system is perfectly classical (Weirdness Meter is flat).
2. Late Times: The system slowly starts to become more quantum (Weirdness Meter grows slowly), but it never immediately becomes totally chaotic.
The Takeaway: This mirrors how gravity works. For a while, a black hole looks like a smooth, classical object. Only after a very long time does it start to show its deep quantum nature.

4. The Big Picture: Gravity as a "Low-Complexity" View

The authors connect this to the famous AdS/CFT correspondence (a theory linking quantum mechanics to gravity).

The Idea: Gravity (like General Relativity) is often seen as a "low-complexity" description of a much more complex quantum world.
The Conclusion: The paper argues that the Krylov basis is exactly the "lens" that gravity uses. Gravity isn't just a random approximation; it is the specific way of looking at the quantum system where the "Weirdness Meter" stays low, allowing us to describe the universe with simple, classical laws (like Einstein's equations) instead of impossible quantum math.

Summary

In short, this paper claims that for many complex quantum systems, there exists a special way of organizing the information (the Krylov basis) where the system looks surprisingly simple and classical. Even when the system is spreading out and getting "complex," it doesn't necessarily get "quantum weird." This suggests that the classical world we see (and the laws of gravity) might just be the result of looking at the universe through this specific, efficient lens.

Technical Summary: Wigner Negativity in Krylov Space and Emergent Semiclassicality

Problem Statement
Quantum dynamics in large- $N$ , complex many-body systems typically involve states that spread over an exponentially large Hilbert space, rendering classical simulation intractable. While the Krylov basis is known to minimize the spread of a wavefunction (spread complexity) and minimize Wigner negativity at early times, it remains an open question whether this basis provides a robust, low-complexity, semiclassical effective description for general chaotic systems at finite but large times. Specifically, the authors investigate whether time evolution in the Krylov basis remains "semiclassical"—defined here as having Wigner negativity that does not diverge with the system size $D$ (or $N$ )—even when the spread complexity grows rapidly.

Methodology
The authors employ Wigner negativity, a measure of non-stabilizerness and classical simulation complexity, as a probe of semiclassicality within the Krylov basis. They define a Krylov-Wigner function for infinite-dimensional Hilbert spaces by taking the limit of discrete Wigner functions on truncated Krylov chains. The study focuses on three distinct solvable models in the large- $N$ (or large-dimension) limit:

2d Conformal Field Theories (CFTs): Analyzing local operator quenches on the vacuum and thermofield double (TFD) states. The Krylov dynamics are mapped to the $SL(2, \mathbb{R})$ model, allowing for exact analytical solutions for Lanczos coefficients and wavefunctions.
Random Matrix Theory (RMT): Considering the Gaussian Unitary Ensemble (GUE) with a maximally entangled initial state. The authors utilize the known asymptotic behavior of Lanczos coefficients (constant $b_n$ ) to derive an effective Hamiltonian and employ stationary phase approximations to analyze the Wigner function at late times.
Double-Scaled SYK (DSSYK) Model: Studying the time evolution of the TFD state. This model interpolates between the $SL(2, \mathbb{R})$ (Heisenberg-Weyl) regime at early times and the GUE regime at late times, allowing for a unified analysis of the transition between different dynamical phases.

Key Contributions and Results

2d CFTs: For operator dimensions $\Delta > 1/2$ , the authors find that while the spread complexity grows rapidly (quadratically in the vacuum case and exponentially in the TFD case), the Wigner negativity saturates to an $O(1)$ constant at late times. This indicates that despite the rapid spreading of the wavefunction in the Krylov chain, the dynamics remain approximately classical. Analytical results in a scaling limit confirm that the negativity saturates because the wavefunction becomes squeezed in the conjugate momentum basis. For $\Delta \leq 1/2$ , negativity grows, signaling a breakdown of this semiclassical behavior.
Random Matrix Theory (RMT): In the large- $D$ limit, the authors demonstrate that Wigner negativity in the Krylov basis grows as $t^{1/2}$ at large but $O(1)$ times. This growth is derived using a stationary phase approximation of the Bessel-function-based wavefunctions. Crucially, this growth is independent of the Hilbert space dimension $D$ , implying that for any fixed time scale not scaling with system size, the dynamics remain semiclassical. The negativity only becomes exponentially large (saturating to a typical state value) at times $t \sim D$ .
Double-Scaled SYK (DSSYK): The model exhibits a two-phase behavior. At early times (corresponding to the $SL(2, \mathbb{R})$ or Heisenberg-Weyl limit), the negativity remains constant, indicating an approximately classical phase. At late times (approaching the GUE limit), the negativity transitions to a $t^{1/2}$ growth, mirroring the RMT results. This confirms that the Krylov basis captures a semiclassical description that evolves into a power-law growing but still dimension-independent regime before the exponential saturation time.

Significance and Claims
The paper argues that the Krylov basis provides a universal, low-complexity effective description for the dynamics of large, chaotic quantum systems at $O(1)$ times. The central claim is that emergent semiclassicality can be operationally defined by the behavior of Wigner negativity: if the negativity remains $O(1)$ (or grows slowly without scaling with system size), the system admits a semiclassical description in that basis.

The authors suggest that this framework offers a general mechanism for understanding how gravity emerges in the AdS/CFT correspondence. They posit that bulk gravity acts as a low-complexity effective theory precisely because it corresponds to the Krylov basis description of boundary dynamics, where the wavefunction remains confined to a manageable subspace and retains semiclassical properties (low Wigner negativity) even when the boundary system is highly chaotic. The work establishes that the growth of spread complexity does not necessarily imply a loss of classicality; rather, the Krylov basis can simultaneously support rapid spreading and low simulation complexity (semiclassicality) for a significant duration.

The paper concludes that for all examined models, the Krylov basis minimizes the "quantumness" of the dynamics as measured by Wigner negativity, providing a robust language for quantifying the emergence of semiclassical spacetime from complex quantum systems.