Technical Summary: The Information Content of Krylov Observables: A Machine Learning Approach

1. Problem Statement

The paper addresses a fundamental question in quantum chaos and holography: How much usable information is contained in specific Krylov-space observables, and how do they relate to one another?

Specifically, the authors investigate three observables derived from the evolution of a quantum state in a Krylov basis:

1. Spread Complexity ($C(t)$): The first moment of the Krylov wavefunction, widely used as a diagnostic for quantum chaos and black hole interiors.
2. Discrete Wigner Negativity ($N(t)$): A second-moment quantity measuring non-classicality in the discrete phase space of the Krylov chain.
3. Normalized Negativity ($\chi(t) = N(t)/|S(t)|$): A proposed "second-moment infall probe" where $N(t)$ is divided by the survival amplitude $S(t)$ to isolate genuine phase-space spreading from trivial return-probability decay.

The core operational questions are:

• Does $C(t)$ uniquely determine $N(t)$ (and vice versa), or does the second moment contain information stripped away by the first?
• Can these curves recover physical parameters (e.g., thermofield double temperature, symmetry class) or reconstruct the fine structure of the Spectral Form Factor (SFF)?
• Is the informational advantage of the normalized negativity $\chi(t)$ a signature of chaotic dynamics, as physical intuition suggests?

2. Methodology

The authors employ supervised machine learning (ML) as an operational tool to quantify mutual information. The logic is that if a regression task $X \to Y$ succeeds, the information about $Y$ is contained in $X$ in a learnable form.

Data Generation

Five large labeled datasets were generated covering $\sim 57,000$ labeled evolutions:

1. Random Matrix Ensembles: GUE (complex Hermitian), GOE (real symmetric), and Poisson (uncorrelated levels) across dimensions $D \in [101, 1009]$ and temperatures $\beta \in [0, 5]$.
2. Integrable CFT Sector: The exactly solvable $SL(2, \mathbb{R})$ discrete-series sector (2d-CFT primaries) with parameters $(h, \alpha)$.
3. Chaos Interpolation: A family $H(\varepsilon) = H_{SL(2,\mathbb{R})} + \varepsilon R_0 W_{GUE}$ interpolating between the integrable chain and random matrix chaos.

Machine Learning Architecture

• Models: Tiny residual multilayer perceptrons (MLPs) with widths of 16–32 neurons, mixed with gradient-boosted decision trees.
• Protocol: Rigorous validation to avoid artifacts.
  • Splits: Random splits for ensemble data; held-out parameter values (e.g., specific $h$ values) for integrable/interpolation data to test generalization to unseen theories.
  • Metrics: Dual-metric evaluation using variance-weighted $R^2$ ($R^2_{vw}$) and mean relative $L^2$ error to detect dynamic-range biases.
  • Robustness: Results are verified to be stable across different training seeds, network widths, and dataset sizes (training on only 50% of data yields identical results).

3. Key Results

A. Random Matrix Ensembles (GUE, GOE, Poisson)

1. Temperature Recovery: Both $C(t)$ and $N(t)$ uniquely determine the thermofield temperature $\beta$ with near-perfect accuracy ($R^2 \approx 0.999$).
2. SFF Lossiness: Neither $C(t)$ nor $N(t)$ can reconstruct the fine structure of the Spectral Form Factor (SFF). The models only recover the smooth ensemble-averaged envelope (dip and ramp) with $R^2_{vw} \approx 0.18$. The sample-specific fluctuations (spectral rigidity) are irreversibly lost. Smoothing the target does not rescue this; it merely trivializes the task by erasing the dip-ramp structure.
3. Information Budget: The late-time SFF plateau (dual to the black hole interior volume) is largely encoded in the first 20% of the complexity curve ($R^2 \approx 0.86$).
4. Universality Class Identification: A single curve ($C(t)$ or $N(t)$) suffices to classify the symmetry class (GUE vs. GOE vs. Poisson) with up to 98% precision. $C(t)$ is marginally better than $N(t)$ for distinguishing GUE from GOE.

B. Integrable CFT Sector ($SL(2, \mathbb{R})$)

1. Exact Slaving: In this integrable sector, the observables are informationally equivalent due to the exact negative binomial distribution of the Krylov occupation.
2. Degeneracy Resolution: While all observables perfectly predict the composite parameter $h\alpha^2$, the negativity $N(t)$ uniquely resolves the individual parameters $(h, \alpha)$ with higher precision ($R^2 \approx 0.999$) than complexity ($R^2 \approx 0.984$).
3. Symmetry: The reconstruction asymmetry between $C$ and $N$ vanishes (within the learnability floor), confirming the theoretical expectation of exact slaving.

C. The Chaos Interpolation ($SL(2, \mathbb{R}) \to \text{Chaos}$)

This is the headline result. The authors track the "asymmetry gap" in mutual reconstruction as chaos is introduced:
$$\text{gap}_\chi = R^2(\chi \to C) - R^2(C \to \chi)$$

• The $\chi$-Surplus: As the system crosses from integrable to chaotic (measured by the level-spacing ratio $\langle r \rangle$), the gap for the normalized negativity $\chi$ switches on and grows monotonically from $+0.33$ to $+0.77$.
• The Raw Negativity Gap: The gap for the raw negativity $N$ decays to zero in the chaotic regime.
• Interpretation: The informational advantage of $\chi$ over $C$ is a signature of chaos. The division by the survival amplitude $|S(t)|$ in $\chi$ retains fine-grained, non-self-averaging spectral information (the "speckle") that is compressed away in $C(t)$.

4. Analytical Mechanism

The paper derives an analytical bound (Theorem 1) explaining the $\chi$-surplus:

• In the chaotic phase, $C(t)$ and $N(t)$ are "self-averaging" observables (smooth envelopes plus small noise).
• The survival amplitude $S(t)$, however, behaves like a complex Gaussian variable with large, non-self-averaging fluctuations (speckle) related to spectral rigidity.
• The normalized negativity $\chi = N/|S|$ inherits this speckle. Since $C$ cannot predict the speckle (as shown by the SFF lossiness result), $C$ cannot predict $\chi$ well. Conversely, $\chi$ contains the envelope information shared with $C$, allowing $\chi \to C$ to succeed.
• This asymmetry is strictly zero in the integrable limit (where $S$ is slaved to the envelope) and maximal in the chaotic limit.

5. Significance and Claims

The paper claims to provide the first operational, quantitative map of the information content of Krylov observables:

1. SFF is Irreplaceable: The fine structure of the SFF (spectral rigidity) is not recoverable from $C(t)$ or $N(t)$, even with smoothing. These moments are insufficient statistics for spectral rigidity.
2. Chaos as a Decoupler: In integrable systems, the Krylov moment tower is "slaved" (all observables carry the same info). In chaotic systems, the tower decouples, and the normalized negativity $\chi(t)$ acquires a unique informational surplus.
3. Methodological Rigor: The study demonstrates that apparent asymmetries in ML results can be artifacts of dynamic range or selection bias. By implementing strict protocols (dual metrics, held-out parameters, seed averaging), the authors isolate the physical signal (the $\chi$-surplus) from noise.
4. Practical Utility:
   • Use $C(t)$ for macroscopic labels (temperature, universality class).
   • Use $N(t)$ for resolving microscopic degeneracies in integrable sectors.
   • Use $\chi(t)$ specifically as a probe for the onset of chaos and the retention of fine-grained spectral information.

The authors emphasize that these are operational statements at finite dimensions with minimal models, serving as lower bounds on mutual information, and that the interpolation serves as a random-matrix proxy for finite-entropy holography.