  We consider the problem, where a camera is tasked with determining one of two
hypotheses: first with an incoherently-radiating quasi-monochromatic point
source and the second with two identical closely spaced point sources. We are
given that the total number of photons collected over an integration time is
assumed to be the same under either hypothesis. For the one-source hypothesis,
the source is taken to be on-axis along the line of sight and for the
two-source hypothesis, we give ourselves the prior knowledge of the angular
separation of the sources, and they are assumed to be identical and located
symmetrically off-axis. This problem was studied by Helstrom in 1973, who
evaluated the probability of error achievable using a sub-optimal optical
measurement, with an unspecified structured realization. In this paper, we
evaluate the quantum Chernoff bound, a lower bound on the minimum probability
of error achievable by any physically-realizable receiver, which is
exponentially tight in the regime that the integration time is high. We give an
explicit structured receiver that separates three orthogonal spatial modes of
the aperture field followed by quantum-noise-limited time-resolved photon
measurement and show that this achieves the quantum Chernoff bound. In other
words, the classical Chernoff bound of our mode-resolved detector exactly
matches the quantum Chernoff bound for this problem. Finally, we evaluate the
classical Chernoff bound on the error probability achievable using an ideal
focal plane array---a signal shot-noise limited continuum photon-detection
receiver with infinitely many infinitesimally-tiny pixels---and quantify its
performance gap with the quantum limit.
