Discrimination and estimation of incoherent sources under misalignment

Spatially resolving two incoherent point sources whose separation is well below the diffraction limit dictated by classical optics, has recently been shown possible using techniques that demultiplex the incoming radiation into various spatial orthogonal modes. However, spatial mode demultiplexing is position sensitive requiring prior information as to the sources center. We show that the lack of such prior information---which leads to misalignment of the demultiplexing measurements with respect to the objects centroid---effectively restores the diffraction limit in two complementary tasks; the estimation of the separation between the two sources and the task of discriminating between one versus two incoherent point sources. Specifically, we determine how the performance of spatially demultiplexing measurements decreases as a function of their misalignment for both these tasks and use techniques form quantum estimation theory to propose alternative measurement strategies. Surprisingly, we discover that the same measurement strategy is optimal for both tasks if the misalignment is known, and show how to implement it using passive linear optics. The implementation of this measurement also allows us to propose a strategy for correcting the effects of misalignment in a realistic scenario where the latter is not known.