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.Comment: 5 pages, 2 figure