Tube estimates for nonuniform targets under residual normal noise

Prove uniform first- and second-moment tube estimates for the maintained residual-normal-noise process, including the cutoff-annulus contribution, in order to establish the proposed nonuniform-target upper bound for \(\delta>0\).

Background

For a nonuniform positive target density on the target circle, the paper derives a conditional upper-bound scheme based on capture, maintenance in a stochastic tube, and tangent transport at a deterministic synchronization time. The argument requires bounds of order κδ/UN\kappa\delta/U_N and (κδ/UN)2(\kappa\delta/U_N)^2 for the conditional first and second tube moments, uniformly over the relevant parameters and including the projection cutoff region. Those estimates are not proved, so the resulting nonuniform residual-noise cost bound remains conditional.

References

These estimates must hold for the combined field $H_N+H_\parallel$, including the chosen off-target extension of $H_\parallel$. They are suggested by Eq.~(\ref{eq:S-truncatedorder}) but a full proof, including the cutoff-annulus contribution, is left open.

— Distinct Feedback-Strength Requirements for Quantum-State Ensemble Preparation under Channel-Equivalent Monitoring  (2609.31221 - Huang et al., 25 Sep 2026) in Supplemental Material, Section \texttt{Conditional residual-noise extension for regular nonuniform laws}, Eq. (\ref{eq:S-tubeassumption})