Identify simple finite‑time criteria to bound σ_{Rρ}^2(t) under autonomous dynamics

Identify a simple criterion, based on finite‑time and finite‑resolution observations of high‑dimensional projectors under autonomous unitary dynamics, that provides stronger constraints on the variance σ_{Rρ}^2(t) determining pure‑state thermalization than those obtainable from second‑order correlators alone.

Background

Pure‑state thermalization in the paper is controlled by the variance σ{Rρ}^2(t), which depends on fourth‑order (out‑of‑time‑ordered) correlators. For autonomous (Hamiltonian) dynamics, the authors can extrapolate second‑order behavior from finite‑time autocorrelators, but they lack an analogous accessible criterion that would tightly constrain σ{Rρ}^2(t) from finite‑time data.

They explicitly state that they cannot presently identify a simple criterion of this type, highlighting a practical gap between the available second‑order forecasting tools and the fourth‑order quantities needed for pure‑state results without averages.