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

Abstract

Simultaneously optimizing the estimation of the centroid and separation of two incoherent optical point sources is constrained by a tradeoff relation determined by 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.