Complexity and the Hilbert space dimension of 3D gravity

This paper employs quantum dynamical Krylov complexity to demonstrate that the Hilbert space dimension of a black hole in 2+1-dimensional Anti-de Sitter space equals the exponential of its Bekenstein-Hawking entropy, derived from the late-time saturation of state spread in a chaotic $SL(2,\mathbb{R})$ system.

The Gist

Imagine you are trying to figure out the size of a giant, invisible library. This library contains every possible state a black hole can be in. In physics, this "library" is called a Hilbert space, and the "books" inside are the different ways the black hole can exist.

The big question the authors of this paper are asking is: How many books are in this library?

For a long time, physicists have struggled to count these books because the rules of gravity and quantum mechanics make the library seem infinite. If the library is infinite, it's hard to understand how black holes work or how information is stored inside them.

Here is how the authors solved this puzzle, using a few creative metaphors:

1. The "Shuffling" Game (Complexity)

Instead of trying to count the books one by one, the authors decided to watch a single book "shuffle" around the library over time.

• The Setup: They start with a specific book (a quantum state) and let time pass. As time goes on, this book spreads out, touching more and more other books in the library.
• The Measure: They measure how "spread out" the book gets. This is called Spread Complexity.
• The Analogy: Imagine dropping a drop of red ink into a clear glass of water. At first, it's just a tiny dot. As time passes, the ink spreads out until it colors the whole glass. The "complexity" is a measure of how much of the glass the ink has reached.

2. The Infinite vs. The Finite Problem

When the authors first did the math using standard gravity rules, the ink kept spreading forever. It never stopped. This suggested the library was infinite, which doesn't make sense for a black hole with a finite amount of energy.

Why did this happen? The standard math they used was like looking at the library from very far away. From that distance, the shelves look like a smooth, continuous wall. But if you zoom in, you realize the shelves are actually made of individual, distinct planks (discrete energy levels). The standard math missed these individual planks.

3. The "Ghostly Bridge" (Wormholes)

To fix this, the authors looked at something called non-perturbative effects. In the language of the paper, this involves "wormholes."

• The Metaphor: Imagine two separate rooms in the library. Standard math says they are totally disconnected. But the authors realized there are "ghostly bridges" (wormholes) connecting these rooms that only show up when you look at the whole system together.
• The Effect: These bridges change the rules of the game. They force the ink to stop spreading once it has touched every single book in the library. The ink doesn't just keep spreading into an infinite void; it hits a wall because the library is actually finite.

4. The Final Count

Once they accounted for these "ghostly bridges," the math changed. The ink stopped spreading at a specific point.

• The Result: The point where the spreading stopped (the saturation point) told them exactly how many books were in the library.
• The Answer: The number of books is exponential to the black hole's entropy (a measure of its disorder or information). In simple terms: If the black hole has an entropy of $S$, the size of the library is $e^S$.

Summary

The paper claims that by watching how a quantum state "spreads" through time and accounting for subtle, hidden connections (wormholes) in the fabric of space, they can finally count the number of possible states a black hole can have.

They found that the library is finite, not infinite. The size of this library is directly tied to the black hole's entropy, confirming a long-held belief in physics that the "size" of a black hole's quantum world is determined by its surface area (entropy).

In a nutshell: They used a "spreading ink" test to measure the size of a black hole's internal universe, and by fixing a hidden "bridge" in their math, they proved the universe inside the black hole is finite and calculable.

Technical Summary

Technical Summary: Complexity and the Hilbert space dimension of 3D gravity

Problem Statement
A central challenge in formulating a theory of quantum gravity is determining the size and structure of the Hilbert space describing gravitating objects, particularly black holes. While previous approaches have utilized extremal supersymmetric black holes in string theory or Euclidean gravity path integrals to estimate entropy, a third approach involves computing the dimension of the span of states explored by a generic initial vector evolving over time. In chaotic systems, this "spread" of a state over the Krylov basis reveals the dimension of the accessible Hilbert space. The specific problem addressed here is applying this framework to 2+1-dimensional Anti-de Sitter (AdS$_3$) gravity to determine the Hilbert space dimension of near-extremal eternal black holes, overcoming the difficulty that effective field theory descriptions often yield continuous, unbounded spectra suggesting infinite-dimensional Hilbert spaces.

Methodology
The authors employ a quantum dynamical approach using Krylov complexity (specifically "spread complexity"). The methodology proceeds as follows:

1. Krylov Basis Construction: Starting with an initial Thermofield Double (TFD) state $|\text{TFD}(t)\rangle$, the authors construct a Krylov basis $\{|K_n\rangle\}$ recursively via the Hamiltonian $H$. The evolution of the state is characterized by Lanczos coefficients ($a_n, b_n$) derived from the return amplitude (overlap of the state with itself).
2. Spectral Density and Lanczos Coefficients: The authors utilize the spectral density $\rho(E)$ of near-extremal 3D gravity. Initially, they consider the Maloney–Witten–Keller (MWK) density derived from the Euclidean gravitational path integral (GPI). They find that the resulting Lanczos coefficients match those of a quantum particle moving on the $SL(2, \mathbb{R})$ group manifold.
3. Orthogonal Polynomials: The spread complexity $C_S(t)$ is expressed in terms of orthogonal polynomials (specifically generalized Laguerre polynomials $L_n^{(1/2)}$) constructed from the Lanczos coefficients and the spectral density.
4. Incorporating Non-Perturbative Effects: Recognizing that the continuous spectral density from the GPI implies unbounded complexity growth, the authors incorporate non-perturbative effects. They interpret the GPI as computing coarse-grained ensemble averages. To resolve the discreteness of the fundamental spectrum, they introduce the spectral correlator $\overline{\rho(E')\rho(E)}$. This correlator includes contributions from Euclidean wormholes connecting two boundaries, which induce Random Matrix Theory (RMT) universality (specifically the Gaussian Unitary Ensemble or GUE sine kernel).
5. Complexity Integral: The spread complexity is re-evaluated using the density-density correlator rather than a simple product of densities. This involves integrating a "complexity kernel" $A(E, E')$ (derived from the orthogonal polynomials) against the spectral correlator.

Key Contributions and Results

• Mapping to $SL(2, \mathbb{R})$: The authors demonstrate that the Lanczos coefficients for near-extremal 3D gravity match those of a particle on the $SL(2, \mathbb{R})$ group manifold, with a scaling dimension $h=3/4$. This leads to monotonically increasing spread complexity $C_S(t) \propto t^2$ at early times, consistent with the Schwarzian limit of JT gravity.
• Resolution of the Continuum Problem: The paper shows that using the continuous spectral density from the GPI alone results in complexity that grows indefinitely, implying an infinite-dimensional Hilbert space. This is identified as an artifact of the coarse-grained effective theory.
• Saturation via Wormholes: By including the connected wormhole contribution to the path integral, the spectral correlator acquires a sine-kernel form (level repulsion) characteristic of RMT. This modification regularizes the complexity integral at late times.
• Finite Hilbert Space Dimension: The calculation yields a saturation value for the spread complexity at late times. This saturation value is found to be:
  $$C_S(\infty) \propto e^{S_0}$$
  where $S_0$ is the Bekenstein-Hawking entropy of the BTZ black hole. This result explicitly demonstrates that the Hilbert space dimension of the black hole is finite and exponential in the entropy.

Significance and Claims
The paper claims to introduce a new physical method for computing the Hilbert space dimension of complex interacting systems directly from the saturating value of spread complexity. By bridging the gap between the continuous effective description (GPI) and the discrete microscopic spectrum (via RMT universality and wormholes), the authors provide a mechanism to extract finite-dimensional Hilbert space properties from gravitational path integrals.

The authors position this work as a direct computation of complexity in the bulk gravity theory, rather than relying solely on a dual field theory description. They note that while their result for spread complexity is structurally similar to non-perturbative wormhole length calculations in 2D JT gravity, the 3D case involves summing over geometries with a single torus boundary, which includes configurations not reducible to 2D geometries. Consequently, the 3D result represents a distinct formula derived from the specific spectral properties of 3D gravity. The work suggests that the saturation of complexity is a robust signature of the underlying finite-dimensional nature of quantum gravity, even when the effective theory appears continuous.