Uniform spectral decay and the infinite-feature QMI limit

Prove a uniform spectral-decay or sufficient uniform-integrability condition for the random-feature density operators that establishes convergence of the finite-dimensional QMI to its limit as the feature dimension tends to infinity.

Background

The paper derives the limit as the number of random Fourier features grows only conditionally on a uniform geometric spectral-decay hypothesis (H). That hypothesis is not established for the deployed Gaussian construction, and entropy convergence cannot be inferred from eigenvalue convergence alone without tail control.

References

Proving (H), or a weaker uniform-integrability condition sufficient for entropy convergence, is left open, as is a quantitative lower bound relating the finite-𝐷 functional to the HSIC norm.

— Quantum mutual information statistics for detecting dependence-structure change points in time series  (2609.02787 - Kang et al., 2 Sep 2026) in Appendix A.8(iv), Status of (H)

We have not established the rate at which 𝐷 must grow, and a coupling that no coordinate pair carries at allβ€” the case in which a block test would be the only option β€” lies outside the designs used here.

— Quantum mutual information statistics for detecting dependence-structure change points in time series  (2609.02787 - Kang et al., 2 Sep 2026) in Supplementary Section B.6