The Information Content of Krylov Observables: A Machine Learning Approach

Abstract

We employ machine learning techniques to estimate the amount of information contained in three Krylov-space observables: the spread complexity $C(t)$, the discrete Wigner negativity $N(t)$, and the normalized negativity $χ(t)=N(t)/|S(t)|$, where $S(t)$ is the survival amplitude. We study thermofield-double evolutions on GUE, GOE and Poisson spectra, and on the integrable $SL(2,R)$ Krylov chain of a two-dimensional CFT primary, with Lanczos coefficients $b_n=α\sqrt{n(n+2h-1)}$. To introduce chaos we add a GUE random matrix to the integrable chain, $H(\varepsilon)=H_{SL(2,R)}+\varepsilon R_0 W_{GUE}$, where $R_0$ is its spectral half-width. Either $C(t)$ or $N(t)$ alone determines the thermofield temperature, with $R^2\simeq0.999$. However, neither can reconstruct the fine structure of the spectral form factor. In the integrable sector all three observables carry the same information. When we turn on chaos an asymmetry appears: $χ$ predicts $C$ better than $C$ predicts $χ$. This gap grows from $+0.33$ to $+0.77$, while the same gap for the raw negativity decays to zero. The extra information comes from the survival amplitude, and $χ(t)$ carries it while $C(t)$ can still be recovered from $χ(t)$. We derive a lower bound on this gap, set by the fraction of the variance of $χ$ that comes from $S$.