Joint optimal measurement for locating two incoherent optical point sources near the Rayleigh distance

The simultaneous optimization of the centroid estimation and the separation estimation of two incoherent optical point sources is restricted by a tradeoff relation through an incompatibility coefficient. At the Rayleigh distance the incompatibility coefficient vanishes and thus the tradeoff relation no longer restricts the simultaneous optimization of measurement for a joint estimation. We construct such a joint optimal measurement by an elaborated analysis on the operator algebra of the symmetric logarithmic derivative. Our work not only confirms the existence of a joint optimal measurement for this specific imaging model, but also gives a promising method to characterize the condition on measurement compatibility for general multiparameter estimation problems.