Quantitative relation between finite-dimensional QMI and HSIC

Derive a quantitative lower bound relating the finite-dimensional quantum mutual-information functional to the Hilbert–Schmidt independence criterion norm.

Background

The paper establishes qualitative identifiability results for finite-dimensional QMI under particular random-feature constructions, while also discussing the infinite-feature limit. It does not establish a quantitative bound connecting finite-dimensional QMI to HSIC, leaving the strength of this relationship unresolved.

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)