  We analyze the fundamental quantum limit of the resolution of an optical
imaging system from the perspective of the detection problem of deciding
whether the optical field in the image plane is generated by one incoherent
on-axis source with brightness $\epsilon$ or by two $\epsilon/2$-brightness
incoherent sources that are symmetrically disposed about the optical axis.
Using the exact thermal-state model of the field, we derive the quantum
Chernoff bound for the detection problem, which specifies the optimum rate of
decay of the error probability with increasing number of collected photons that
is allowed by quantum mechanics. We then show that recently proposed
linear-optic schemes approach the quantum Chernoff bound---the method of binary
spatial-mode demultiplexing (B-SPADE) is quantum-optimal for all values of
separation, while a method using image-inversion interferometry (SLIVER) is
near-optimal for sub-Rayleigh separations. We then simplify our model using a
low-brightness approximation that is very accurate for optical microscopy and
astronomy, derive quantum Chernoff bounds conditional on the number of photons
detected, and show the optimality of our schemes in this conditional detection
paradigm. For comparison, we analytically demonstrate the superior scaling of
the Chernoff bound for our schemes with source separation relative to that of
spatially-resolved direct imaging. Our schemes have the advantages over the
quantum-optimal (Helstrom) measurement in that they do not involve joint
measurements over multiple modes, and that they do not require the angular
separation for the two-source hypothesis to be given \emph{a priori} and can
offer that information as a bonus in the event of a successful detection.
