Bayesian Quantum Estimation of the Super-resolution of Two Incoherent Point Sources

We address the estimation problem of the separation of two arbitrarily closed incoherent point sources from the Bayesian quantum point of view, i.e., when a prior probability distribution function (PDF) on the separation is available. In the Fisherian approach, Tsang et al showed theoretically that the Rayleigh limit can be exceeded, by using a spatial-mode demultiplexing (SPADE) measurement. We compare the SPADE, and direct imaging (DI) with the minimum mean square error (MMSE) in the Bayesian approach. For a given PDF, by varying the parameters of mean and variance, we find the SPADE will not always attain the MMSE, and may not always show an advantage over DI.