The Information Content of Krylov Observables: A Machine Learning Approach

This paper investigates the information content of Krylov-space observables—specifically spread complexity C(t), discrete Wigner negativity N(t), and normalized negativity χ(t)—to determine how much they reveal about the underlying quantum system. The author employs a machine learning approach, training small residual networks and boosted trees on approximately 57,000 labeled evolutions. These datasets span various spectral ensembles (GUE, GOE, Poisson), an integrable SL(2,R)/CFT sector, and a chaos interpolation model that transitions from integrability to random-matrix chaos.

The study reveals that while both complexity and negativity can accurately determine the thermofield temperature (R² ≈ 0.999), neither can reconstruct the fine structure of the spectral form factor (SFF). However, the SFF strictly dominates both moments, and the coarse e^S plateau is recoverable from the early stages of C(t). In the integrable sector, the observables are informationally equivalent, but the negativity is superior at resolving specific parameter degeneracies. The most significant result is that the informational advantage of χ(t) over C(t) is a direct signature of chaos; this asymmetry gap grows monotonically as the system crosses over to GUE statistics, providing an operational link between second-moment Krylov diagnostics and chaotic dynamics.

This work provides a rigorous, operational framework for evaluating the utility of Krylov observables in quantum dynamics. By quantifying the "information budget" of these curves, the paper clarifies which physical features (like temperature or symmetry class) are encoded in simple dynamical diagnostics and which require more complex probes like the SFF. The identification of the χ-surplus as a marker of chaos offers a new, computationally accessible tool for diagnosing the onset of chaotic behavior in quantum systems, particularly in the context of holography and black hole physics.