Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information

TL;DR

This paper derives a universal lower bound on the average resolution of shift parameter estimates, incorporating prior information, and extends Heisenberg limits to arbitrary generators and measurement scenarios.

Contribution

It introduces a rigorous, universally applicable bound on shift parameter estimation that accounts for prior information and generalizes Heisenberg limits for various measurement contexts.

Findings

Establishes a lower bound of k_I/<2|G|> for average resolution.

Provides a specific bound for phase sensing with small phase intervals.

Discusses extensions to noisy scenarios and resource measures.

Abstract

A rigorous lower bound is obtained for the average resolution of any estimate of a shift parameter, such as an optical phase shift or a spatial translation. The bound has the asymptotic form k_I/<2|G|> where G is the generator of the shift (with an arbitrary discrete or continuous spectrum), and hence establishes a universally applicable bound of the same form as the usual Heisenberg limit. The scaling constant k_I depends on prior information about the shift parameter. For example, in phase sensing regimes, where the phase shift is confined to some small interval of length L, the relative resolution \delta\hat{\Phi}/L has the strict lower bound (2\pi e^3)^{-1/2}/<2m| G_1 |+1>, where m is the number of probes, each with generator G_1, and entangling joint measurements are permitted. Generalisations using other resource measures and including noise are briefly discussed. The results rely on the derivation of general entropic uncertainty relations for continuous observables, which are of interest in their own right.