Finite-time emergence of asymptotic n-freeness

Establish whether, in physical quantum systems with finite Hilbert-space dimension D, asymptotic n-freeness of observables—defined by the vanishing of mixed free cumulants κ_{2n}(A(t), B, …) in the large-D limit—is achieved at finite times rather than only in the infinite-time average. Concretely, determine conditions under which there exists a finite time after which κ_{2n}(A(t), B, …) ≈ O(D^{-1}) holds for all orders n, indicating the onset of n-freeness at finite times.

Background

Within the paper’s Free Probability formulation of the full Eigenstate Thermalization Hypothesis (ETH), mixed free cumulants of alternating observables vanish under infinite-time averaging, assuming a generic non-resonance condition. This implies asymptotic freeness at infinite times.

The authors note that, while the infinite-time average is a useful mathematical tool, physical interest lies in whether asymptotic n-freeness is actually reached at finite times. They introduce a freeness time-scale in their kicked-top paper and numerically demonstrate exponential approach to freeness in that model, but the general finite-time emergence across physical systems remains conjectural.