⚛️ high-energy theory

Comments on holographic spread complexity

This paper re-evaluates the holographic proposal linking spread complexity to bulk radial momentum by demonstrating that while the relation holds for specific coherent states in de Sitter space, it generally fails in flat and de Sitter spacetimes where spread complexity primarily measures wave packet dispersion rather than classical momentum, indicating that a semiclassical limit alone is insufficient to establish a momentum-complexity correspondence.

Original authors: Zhehan Li, Jia Tian

Published 2026-07-21

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Zhehan Li, Jia Tian

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 the universe as a giant, cosmic video game where the rules of gravity and the rules of quantum mechanics are trying to play on the same console. For decades, physicists have been trying to figure out how these two rulebooks fit together. One of the most exciting ideas in this field is "holography," which suggests that a three-dimensional universe with gravity (like the one we live in) is actually a projection of a simpler, two-dimensional world without gravity, much like a 3D movie is just a flat screen showing a special kind of light.

In this cosmic game, scientists use "entanglement" (a spooky connection between particles) and "complexity" (how hard it is to describe a quantum state) as their scoreboards. Entanglement tells us about the early stages of a system, like how quickly a cup of coffee cools down. But once the coffee is cold, entanglement stops changing, even though the universe inside the black hole keeps growing and evolving. To track this late-stage evolution, physicists turned to "complexity." Think of complexity as a measure of how tangled a knot of quantum information has become. A recent, bold idea suggested that the speed at which this complexity grows is directly linked to the momentum of a particle falling through the center of a black hole. It was a beautiful, simple connection: more speed means more complexity. But was this connection real, or just a lucky coincidence that only worked in very specific, perfect conditions?

This paper, written by Zhehan Li and Jia Tian, goes back to check that connection, acting like a detective testing a theory in different environments. The authors ask: Does the speed of a falling particle always match the growth of complexity, or does it depend on the type of universe and the specific state of the particle? They find that the answer is a resounding "it depends."

The researchers discovered that the beautiful link between momentum and complexity only holds up under very strict conditions. It works perfectly in a specific type of curved space called Anti-de Sitter (AdS) space, but only if the particle is in a special "coherent state"—a quantum state that behaves almost exactly like a classical ball rolling down a hill without wobbling or spreading out. In this scenario, the particle stays tight and focused, and its speed perfectly predicts how fast the complexity grows.

However, the paper shows that this magic trick fails in other places. When they tested the idea in flat space (like our everyday universe) or in de Sitter space (a model for our expanding universe), the connection broke down. In these environments, even if you start with a neat, focused particle, the laws of quantum mechanics cause the particle to "spread out" like a drop of ink in water. In these cases, the complexity isn't measuring how fast the particle is moving; instead, it's measuring how much the particle is fuzzing out and losing its shape. The authors prove that simply being "classical" (big and heavy) isn't enough to make the momentum-complexity link work; you need the particle to be in that special, non-spreading coherent state, and the universe itself must have the right kind of symmetry to keep it that way.

Ultimately, the paper clarifies that while the idea of linking complexity to momentum is a powerful tool, it is not a universal law of nature. It is a specific feature that emerges only when the quantum system is carefully tuned to mimic a classical particle in a very specific kind of spacetime. The authors suggest that to understand the deep secrets of black holes and the universe, we must be careful not to assume that every quantum measurement of complexity will look like a simple classical speedometer. Sometimes, it's just measuring how much the wave packet is spreading, not how fast it's going.

Problem Statement
This paper revisits the holographic proposal linking the growth rate of spread complexity to the radial momentum of a bulk probe. While previous works (e.g., [58, 62]) established a relation of the form $\dot{C} \propto p_{\text{infalling}}$ in specific holographic setups like global AdS $_3$ and the JT/SYK duality, several foundational questions remain unresolved. Specifically, the precise regime of validity for the extrapolation dictionary used in these derivations is unclear, and it is uncertain whether the correspondence relies heavily on $SL(2, \mathbb{R})$ symmetry or the AdS/CFT framework itself. Furthermore, it is not fully understood how classical probe dynamics emerge from underlying quantum dynamics, particularly regarding the conditions under which spread complexity tracks classical momentum versus other dynamical features like wave packet dispersion. The authors aim to clarify these assumptions, determine the extent to which the proposal depends on AdS/CFT, and generalize the analysis to flat and de Sitter spacetimes.

Methodology
The authors employ a direct quantization approach, bypassing reliance on the AdS/CFT correspondence for the core derivation. Their methodology involves:

Direct Bulk Quantization: They quantize free massive scalar fields and free massive particles directly in the bulk spacetime (AdS $_2$ , flat space, and dS $_2$ ) rather than starting from boundary CFT operators.
Spectral Analysis and Krylov Construction: For each setup, they determine the spectral measure of the initial state with respect to the Hamiltonian. They construct the Krylov basis using orthogonal polynomials associated with these spectral measures (e.g., Meixner polynomials for AdS, Laguerre polynomials for flat space, and continuous Hahn or shifted Meixner-Pollaczek polynomials for de Sitter).
Complexity Calculation: They compute the spread complexity $C(t)$ as the mean of the probability distribution in the Krylov basis and analyze its growth rate $\dot{C}(t)$ .
Semiclassical Limits and State Selection: They examine the semiclassical limit ( $\Delta \gg 1$ ) and compare different classes of initial states, specifically contrasting generic semiclassical wave packets in the energy representation with generalized coherent states adapted to the relevant symmetry algebras (e.g., principal-series representations in de Sitter).
Geometric Quantization: In the AdS $_2$ case, they explicitly quantize the phase space of a free massive particle to identify the spread complexity operator as a quantization of a classical complexity observable within the "complexity=anything" framework.

Key Contributions and Results

AdS Spacetime:
- The authors provide a more general derivation of the momentum-complexity relation by directly quantizing the bulk probe. They confirm that for a specific initial state (a generalized coherent state dual to a primary operator insertion), the spread complexity growth rate matches the classical radial momentum of a freely falling particle.
- They demonstrate that this relation arises because the initial state is a coherent state of the $SL(2, \mathbb{R})$ algebra, and the Hamiltonian belongs to the same algebra. Consequently, the wave packet undergoes coherent motion without dispersive spreading in the semiclassical limit.
- They identify the spread complexity operator as the quantization of a specific linear combination of angular momenta (conserved charges) on the particle's phase space, realizing it as a concrete "complexity=anything" observable.
Flat Spacetime:
- In flat space, the physical Hamiltonian does not preserve the coherent-state structure associated with the Heisenberg algebra in the same way.
- For a coherent initial state evolved by the physical Hamiltonian, the spread complexity growth rate is proportional to the dispersive broadening of the wave packet (specifically, the variance of the position), not the classical momentum. The momentum is conserved, so the "motion" detected by complexity is purely dispersive.
- This result highlights that a semiclassical limit alone is insufficient to guarantee a momentum-complexity relation; the specific choice of initial state and its compatibility with the Hamiltonian's algebra are crucial.
de Sitter Space:
- The authors analyze the static patch of de Sitter space, where the spectrum is continuous (principal-series representation of $SL(2, \mathbb{R})$ ).
- Generic Wave Packets: Semiclassical wave packets in the energy representation (modeled by Gamma distributions) exhibit quadratic growth of spread complexity, driven by dispersive broadening, which does not match the exponential growth of classical radial momentum.
- Principal-Series Coherent States: By constructing states adapted to the principal-series representation (specifically, boundary-localized states selecting a single static patch), the authors find that the Lanczos coefficients grow linearly with the Krylov index. This leads to an exponential growth of spread complexity at late times.
- Crucially, for these specific states, the growth rate $\dot{C}(t)$ exhibits the same time dependence as the classical radial momentum ( $\sinh T$ ), recovering the momentum-complexity relation in de Sitter space.

Significance and Claims
The paper claims that the holographic proposal relating spread complexity growth to radial momentum is not a universal feature of all semiclassical states but is contingent on specific conditions:

State Adaptation: The initial state must be a generalized coherent state adapted to the symmetry algebra of the system.
Hamiltonian Compatibility: The Hamiltonian must belong to the same symmetry algebra to ensure the evolution preserves the coherent state structure, preventing dispersive spreading from dominating the complexity growth.
Independence from Holography: The derivation of the momentum-complexity relation does not strictly require the AdS/CFT correspondence; it emerges from the quantum dynamics of the probe itself when the above conditions are met. The boundary theory primarily serves to suggest a convenient initial state.

The authors conclude that while spread complexity can serve as a diagnostic for classical probe dynamics, one must carefully distinguish between coherent motion and dispersive spreading. The relation $\dot{C} \propto p$ holds when the system is in a regime where the quantum state behaves as a rigid classical object moving along a geodesic, a condition that is satisfied in global AdS and specific de Sitter states but fails for generic wave packets in flat space or de Sitter energy representations. This work clarifies the necessary quantum mechanical prerequisites for the emergence of classical complexity observables.

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