Optimal condition for measurement observable via error-propagation

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 the observable that is chosen will greatly affect the estimation sensitivity. Here we study the optimal observable for the ultimate sensitivity bounded by the quantum Cramer-Rao theorem in parameter estimation. By invoking the Schrodinger-Robertson uncertainty relation, we derive the necessary and sufficient condition for the optimal observables saturating the ultimate sensitivity for the 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 et al (2013 Phys. Rev. A 87 022330) have claimed that the Heisenberg limit may not be obtained via separable measurements. We show this claim is incorrect.