Distinct Feedback-Strength Requirements for Quantum-State Ensemble Preparation under Channel-Equivalent Monitoring

Abstract: Different measurements of the same environment can preserve the average quantum channel yet change the feedback strength needed to prepare a state ensemble. We establish this distinction for a multiqubit register under complete Pauli monitoring, targeting the uniform ensemble of real-amplitude pure states. With unit detection efficiency, public records, and bounded Hamiltonian feedback, fixed transverse noise requires a minimum peak strength proportional to $ε<sup>{-1}$ at distributional error $ε$. Tangent noise instead admits a construction requiring only $O[\sqrt{\log(1/ε)}]$. At fixed preparation time, we determine the optimum over all allowed feedback policies and derive a geometric lower bound beyond radial symmetry. The separation identifies the measurement realization as part of the control-resource specification for monitored ensemble preparation.