Optimal local or sequential protocols for returning-interval localization

Determine optimal local or sequential measurement protocols for exact localization of fixed returning quantum intervals with length i≥2, for returning intervals whose known lengths grow with the sequence size, and for the ensemble that is uniform over all nonempty intervals.

Background

The paper establishes asymptotic benchmarks for unrestricted collective measurements, including fixed known interval lengths, growing known lengths, and unknown lengths under a uniform-over-all-intervals prior. These results do not provide efficient physically implementable local or sequential procedures.

The authors distinguish the unresolved protocol-design problem from the collective-measurement analysis and identify three concrete regimes where optimal local or sequential strategies remain unknown: fixed returning intervals of length at least two, growing known lengths, and the uniform ensemble of all nonempty intervals.