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Complexity study of the Hartle-Hawking state in JT gravity

This paper investigates the growth of quantum complexity for the Hartle-Hawking state in Jackiw-Teitelboim gravity using the Wigner function negativity in the length basis, revealing that the state remains a minimum-uncertainty Gaussian with positive Wigner function at early times before saturating at late times, thereby demonstrating that the length basis provides an ideal semi-classical framework for describing chaotic quantum dynamics without exhibiting the spectral ramp seen in other measures.

Original authors: Ritam Basu

Published 2026-10-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ritam Basu

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

In the deepest corners of modern physics, researchers are trying to understand how the strange, fuzzy world of quantum mechanics gives rise to the solid, predictable reality we experience every day. A central puzzle in this quest is figuring out how to tell when a quantum system is behaving in a way that is truly "classical" versus when it is exhibiting the wild, complex behavior that makes it impossible to simulate on a standard computer. To do this, physicists use a mathematical tool called the Wigner function, which acts like a map of a quantum system's possible states. If this map shows only positive numbers, the system behaves like a classical object, such as a ball rolling down a hill. However, if the map dips into negative values, it signals that the system is deeply quantum and incredibly difficult to describe using classical logic. The amount of this "negativity" serves as a precise measure of the system's complexity.

This question becomes particularly urgent when studying gravity and black holes. In recent years, a specific theory of gravity in two dimensions, known as Jackiw–Teitelboim gravity, has emerged as a powerful laboratory for testing these ideas. This theory is linked to a quantum system that is chaotic and complex, yet simple enough to be analyzed mathematically. Physicists are eager to know if this gravitational system can be described by a simple, classical picture, or if its complexity grows so fast that it defies such descriptions. The answer could reveal whether gravity itself is a form of efficient classical computation, a concept that would reshape our understanding of the universe.

In a new study, a researcher at the Tata Institute of Fundamental Research in India has taken a close look at a specific quantum state in this gravitational theory, known as the Hartle–Hawking state. This state represents a universe that is smooth and symmetric at its beginning, much like a calm, flat surface before a storm. The researcher asked a simple but profound question: as time passes, does this state remain simple and classical, or does it quickly become a tangled, complex mess that requires a supercomputer to simulate? To find out, the team tracked the "negativity" of the Wigner function for this state as it evolved over time.

The results were surprising and highly specific. The study found that for an incredibly long period of time—long enough to be considered "sub-exponential" in the language of physics—the state remains remarkably simple. At the very beginning, the state is a perfect, smooth wave packet sitting at rest. As time moves forward, it reflects off a theoretical barrier and then travels freely. During this entire journey, the map of its states stays positive. The "negativity," which measures the quantum complexity, does not grow. It stays at its minimum possible value, effectively frozen. This means that for a vast stretch of time, the chaotic quantum system behaves exactly like a classical object. It does not become difficult to simulate, nor does it develop the complex, tangled structure that usually signals a breakdown of classical descriptions.

This finding stands in sharp contrast to what happens with other measures of complexity. In many chaotic systems, complexity grows steadily and linearly over time, creating a "ramp" that signals the system is becoming harder and harder to understand. The study explicitly shows that this linear growth does not happen here. The negativity remains flat and constant. The researchers explain that this is because the system, in this specific setup, evolves in a way that preserves its simple, Gaussian shape. It is only when the system is allowed to evolve for an exponentially long time—far beyond the scales usually considered in these models—that the complexity finally begins to rise. At that point, the system reaches a "plateau" where the negativity becomes enormous, indicating that the state has become fully quantum and chaotic. However, this final explosion of complexity is driven by the discrete, grainy nature of the system's energy levels, a feature that only appears after an immense amount of time has passed.

The paper also clarifies why this result is so important for the broader field. It demonstrates that the choice of how we measure the system matters deeply. When the researchers looked at the system using a specific set of coordinates related to the length of a wormhole in the gravitational theory, the system appeared simple and classical for a very long time. This suggests that gravity might provide a natural, efficient way to describe complex quantum dynamics, acting as a "classical computer" that keeps the complexity in check for a long duration. The study rules out the idea that this system immediately becomes a chaotic mess; instead, it shows a period of stability where the quantum world mimics the classical one.

Ultimately, the research paints a picture of a universe that is not instantly chaotic. It reveals a phase where the Hartle–Hawking state, despite being part of a chaotic system, remains a calm, classical wave. The complexity is not absent, but it is held in reserve, waiting for a time scale so vast that it is almost unimaginable. Until that moment arrives, the system behaves with a simplicity that allows physicists to understand it without needing the most powerful computers imaginable. This work provides a concrete example of how gravity might act as a bridge between the quantum and classical worlds, keeping the chaos at bay for a surprisingly long time.

Technical Summary: Complexity Study of the Hartle–Hawking State in JT Gravity

Problem and Motivation
This paper investigates the growth of quantum complexity for the Hartle–Hawking (HH) state in Jackiw–Teitelboim (JT) gravity, specifically focusing on the Wigner negativity as an operational measure of the difficulty of simulating a quantum state on a classical computer. While the AdS/CFT correspondence suggests that strongly coupled boundary quantum systems admit a dual semi-classical gravitational description, a general strategy to identify when such a description is "useful" remains an open question. The author proposes that Wigner negativity serves as a probe for this utility: states with non-negative Wigner functions are "classical" (Gaussian), while negativity quantifies the "non-stabilizer" content required for quantum computation. The study aims to determine if the length basis (dual to the bulk wormhole length) provides an efficient semi-classical description of chaotic dynamics at large $N$ (or large $e^{S_0}$ ).

Methodology
The author works within the framework of JT gravity dual to a double-scaled random matrix ensemble. They analyze the time evolution of the HH state, defined as a thermal state $|\psi_\beta\rangle \propto e^{-\beta H/2}|0\rangle$ , in the length basis (the continuous variable $x$ corresponding to the renormalized wormhole length).

Semiclassical Analysis: They treat the inverse temperature $\beta$ as a small semiclassical parameter ( $\beta \ll 1$ ). The HH state is expanded around the classical turning point of the Liouville wall potential $V(x) = 4e^{-x}$ .
Wigner Function Calculation: The Wigner function $W(x, p; t)$ is computed using the exact wavefunctions (modified Bessel functions) and their asymptotic forms. The negativity is defined as $N(t) = \int dx dp |W(x, p; t)|$ .
Topological Expansion: The author distinguishes between the "disk" contribution (leading order in $e^{S_0}$ ) and "double trumpet" contributions (connected, suppressed by $e^{-2S_0}$ ). They compare the behavior of the Wigner negativity against the Spectral Form Factor (SFF) and Spread Complexity.
Krylov Basis Comparison: The results are contrasted with the behavior of Spread Complexity (Krylov complexity) and the growth of Wigner negativity in generic bases, utilizing the connection between the Krylov basis and the geometric length basis.

Key Contributions and Results

Disk-Level Behavior (Sub-exponential Times):
- At $t=0$ , the HH state in the length basis is a minimum-uncertainty Gaussian wavepacket centered at the classical turning point $x = 2\log(2\beta/\pi)$ .
- Consequently, the Wigner function is strictly positive at $t=0$ , yielding a negativity of $N(0) = 1 + o(1)$ .
- Under time evolution, the packet reflects off the Liouville wall and then evolves freely. Since free evolution corresponds to a Clifford shear (a symplectic transformation generated by a quadratic Hamiltonian), the Wigner negativity remains exactly invariant after the reflection.
- No Linear Rise: Unlike discrete models where negativity often rises linearly, the continuum disk-level negativity exhibits no linear rise at early times. This is enforced by time-reversal symmetry ( $N(t)$ is even in $t$ ) and the strict positivity of the initial Gaussian.
- No Spectral Ramp: The disk-level negativity remains frozen at $1 + o(1)$ for all sub-exponential times. Unlike the SFF, which displays a "ramp" due to the connected double-trumpet geometry, the negativity's disconnected disk term does not decay; therefore, the connected correction (double trumpet) is never exposed to dominate the signal.
Late-Time Behavior and Quantum Plateau:
- The growth of negativity to an exponentially large value is a non-perturbative effect in $e^{-S_0}$ , driven entirely by the discreteness of the spectrum.
- At late times ( $t \gtrsim t_{sat}$ ), the state spreads over the entire accessible phase space. The negativity saturates to a plateau value:
  $N_{\text{plateau}}(\beta) = \sqrt{\frac{2}{\pi}} \sqrt{d_{\text{eff}}(\beta)}$
  where $d_{\text{eff}}(\beta) = Z(\beta)^2/Z(2\beta)$ is the inverse participation ratio of the Boltzmann distribution. This value scales as $e^{S_0/2}$ .
- The time scale for this saturation is not fixed by the disk analysis but is associated with the discreteness of the spectrum.
Spread Complexity vs. Negativity:
- The paper highlights a sharp contrast between Spread Complexity ( $C(t)$ ) and Wigner Negativity.
- Spread complexity grows quadratically ( $C(t) \sim \sigma_E^2 t^2$ ) at early times and continues to grow ballistically at the disk level, reflecting the unbounded nature of the Schwarzian spectrum.
- In contrast, Wigner negativity remains small ( $O(1)$ ) for sub-exponential times. This suggests that the wavefunction can spread significantly (increasing complexity in the Krylov basis) while the state remains effectively "classical" (low Wigner negativity) in the length basis.
Normalized Negativity:
- The author derives a seed-normalized negativity $\tilde{N} = N/P$ (where $P$ is the survival probability). At the disk level, this is exactly the inverse of the survival probability.
- While the raw negativity $N$ shows no ramp, the normalized negativity $\tilde{N}$ inherits the linear ramp of the survival probability (SFF) through its denominator once the dip time $t_{\text{dip}} \sim \beta^{1/4} e^{S_0/2}$ is reached.

Significance and Claims
The paper claims that the length basis is ideally suited for a dual, semi-classical effective description of chaotic quantum dynamics at large $e^{S_0}$ and sub-exponential times.

Efficiency of Gravity: The results support the view that gravity acts as an "efficient, classical simulation" of boundary quantum dynamics. The fact that the Wigner negativity remains $O(1)$ (classical) for a long time, despite the system being chaotic and the spread complexity growing, implies that the bulk geometric variables (length) capture the dynamics in a way that minimizes the "non-stabilizer" resource required for simulation.
Distinction from SFF: The absence of a ramp in the Wigner negativity (unlike the SFF) is a crucial finding. It demonstrates that the "ramp" in complexity measures is not universal; it depends on the specific observable and the decay of the disconnected term.
Role of Discreteness: The transition to the exponentially large plateau is identified as a non-perturbative effect arising from the discrete spectrum, marking the breakdown of the semi-classical disk approximation.

In summary, the paper establishes that for the Hartle–Hawking state in JT gravity, the Wigner negativity in the length basis remains minimal and frozen for sub-exponential times, providing strong evidence that the geometric length basis offers a low-complexity, semi-classical window into the chaotic dynamics of the dual quantum system.

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