Quantum-optimal detection of one-versus-two incoherent optical sources with arbitrary separation

TL;DR

This paper investigates the fundamental quantum limits of resolving one versus two incoherent optical sources and demonstrates that certain linear-optic schemes can approach these limits, outperforming traditional imaging methods especially at small separations.

Contribution

The paper derives the quantum Chernoff bound for the detection problem and shows that linear-optic schemes like B-SPADE are quantum-optimal across all separations, with SLIVER being near-optimal at sub-Rayleigh distances.

Findings

B-SPADE achieves quantum optimality for all source separations.

SLIVER is near-optimal for sub-Rayleigh separations.

Proposed schemes outperform direct imaging in scaling with source separation.

Abstract

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.