  Superresolution refers to the estimation of parameters of an image with an
accuracy beyond standard classical techniques such as direct detection. In
seminal work by Lu et al., a measurement to estimate the separation distance of
two point sources (with a known centroid) was shown to achieve the quantum
Cramer-Rao bound. This work made implicit use of reflection symmetry of the
sources. Here we present a framework that uses more general symmetry in a
constellation to construct a quantum measurement that achieves the quantum
Cramer-Rao bound in estimation of parameters. We show how this technique can be
used to estimate parameters simultaneously in symmetric point-source
constellations with more than two point sources. In order to use symmetry
explicitly, we make use discrete point spread functions in momentum space that
maintain this symmetry. This framework allows us to use techniques from quantum
computing such as Fourier transforms and linear optical circuits to implement
the optimal measurement. To our knowledge, this is first work that shows for
more than two point sources achievable quantum limits of estimation and modal
transformations.

Quantum Physics;Optics