# Quantum-optimal detection of one-versus-two incoherent optical sources with arbitrary separation. (arXiv:1802.02300v1 [quant-ph])

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.