Optimal condition for measurement observable via error-propagation

Abstract

Propagation of error is a widely used estimation tool in experiments, where the estimation precision of the parameter depends on the fluctuation of the physical observable. Thus which observable is chosen will greatly affect the estimation sensitivity. Here we study the optimal observable for the ultimate sensitivity bounded by the quantum Cram\'er-Rao theorem in parameter estimation. By invoking the Schr\"odinger-Robertson uncertainty relation, we derive the necessary and sufficient condition for the optimal observables saturating the ultimate sensitivity for single parameter estimate. By applying this condition to Greenberg-Horne-Zeilinger states, we obtain the general expression of the optimal observable for separable measurements to achieve the Heisenberg-limit precision and show that it is closely related to the parity measurement. However, Jose {\em et al} [Phys. Rev. A {\bf 87}, 022330 (2013)] have claimed that the Heisenberg limit may not be obtained via separable measurements. We show this claim is incorrect.