Adapting norm‑interpolation methods to improve the prefactor in the measured Rényi bound

Ascertain whether the norm‑interpolation (complex interpolation in Schatten spaces) technique used by Dupuis (2023) can be adapted to the measured Rényi framework in order to improve the constant prefactor 3/2 in the one‑shot bound on the expected trace‑distance error stated in Theorem 4, which is expressed in terms of the measured Rényi conditional entropies H_α^{MM,↑}(X|E).

Background

The authors derive a one-shot achievability bound on the expected trace‑distance error for privacy amplification that uses measured Rényi conditional entropies and has a constant prefactor 3/2. Dupuis (2023) obtained a tighter prefactor using complex interpolation methods but for bounds stated with sandwiched Rényi divergences.

Extending interpolation-based techniques to the measured setting could potentially tighten constants while keeping the improved entropic term. However, the authors note that the norm‑interpolation approach works in spaces that may not mesh with their framework, leaving the applicability of such methods unclear.

References

The approach of, however, requires one to work in norm interpolation spaces, which makes it unclear whether an analogous method can be applied here.

— Rethinking quantum smooth entropies: Tight one-shot analysis of quantum privacy amplification  (2603.04493 - Regula et al., 4 Mar 2026) in Section 4.3 (From smoothing to Rényi divergences and error exponents), following Theorem 4