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arXiv:2601.13972 · Physical Review A

Experimental Evidence-Based Sub-Rayleigh Source Discrimination

Saurabh U. Shringarpure, Yong Siah Teo, Hyunseok Jeong, Michael Evans, Luis L. Sanchez-Soto, Antonin Grateau, Alexander Boeschoten, Nicolas Treps

quant-phSubmitted Jan 20, 2026• Announced Jan 21, 2026• 0 citations
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Abstract

We propose a Bayesian evidence-based inference framework based on relative belief ratios and apply it to discriminating between one and two incoherent optical point sources using spatial-mode demultiplexing (SPADE). Unlike the Helstrom measurement, SPADE require no collective detection and its optimal for asymptotically large samples. Our method avoids ad hoc statistical constructs and relies solely on the information contained in the data, with all assumptions entering only through the likelihood model and prior beliefs. Using experimental evidence, we demonstrate the superior resolving performance of SPADE over direct imaging from a new and extensible perspective; one that naturally generalizes to multiple sources and offers a practical robust approach to analyzing quantum-enhanced superresolution.

Comments
11 pages, 7 figures

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We propose a Bayesian evidence-based inference framework based on relative belief ratios and apply it to discriminating between one and two incoherent optical point sources using spatial-mode demultiplexing (SPADE). Unlike the Helstrom measurement, SPADE require no collective detection and its optimal for asymptotically large samples. Our method avoids ad hoc statistical constructs and relies solely on the information contained in the data, with all assumptions entering only through the likelihood model and prior beliefs. Using experimental evidence, we demonstrate the superior resolving performance of SPADE over direct imaging from a new and extensible perspective; one that naturally generalizes to multiple sources and offers a practical robust approach to analyzing quantum-enhanced superresolution.
We theoretically predict and experimentally demonstrate that a reduced spatial-mode demultiplexing (SPADE) measurement using only the two lowest-order Hermite-Gaussian modes exhibits remarkable robustness against background noise in discriminating between a single source and two incoherent point sources. We establish a theoretical framework incorporating uniform background noise and derive an analytical Chernoff exponent expression, showing that SPADE consistently outperforms direct imaging (DI) across all source separations. Experimental results confirm that SPADE-based hypothesis testing approaches the quantum limit even when the background-to-signal photon ratio per pixel is 0.11. This advantage stems from SPADE's ability to concentrate source information into minimal detection modes, reducing the cumulative background noise impact. Our findings provide a practical detection scheme for applications where background noise is inevitable, such as astronomical observations and quantum sensing.
We experimentally demonstrate the simultaneous estimation of the three parameters characterizing a pair of incoherent optical sources in the sub-Rayleigh regime, enabling super-resolved scene characterization. Using spatial-mode demultiplexing (SPADE) with two demultiplexers--one deliberately shifted--we determine separations well below the diffraction limit and achieve sensitive joint estimation of separation, centroid, and relative brightness over a broad range of scene configurations in a single experimental setting. We benchmark our performance using Fisher-information-based Cramér-Rao bounds, and discuss the corresponding quantum limits. We investigate two complementary scenarios: a realistic case with slightly non-identical sources, and an idealized case of indistinguishable sources.
Abstract The Rayleigh diffraction limit imposes a fundamental restriction on the resolution of direct imaging systems, hindering the identification of incoherent optical sources, such as celestial bodies in astronomy and fluorophores in bioimaging. Recent advances in quantum sensing have shown that this limit can be circumvented through spatial demultiplexing (SPADE) and photon detection, i.e. a semi-classical detection strategy. However, the general optimality for arbitrary intensity distributions and bright sources remains unproven. In this work, we develop a general model for incoherent light with arbitrary intensity undergoing diffraction. We employ this framework to compute the quantum Chernoff exponent for generic incoherent-source discrimination problems, focusing on the sub-diffraction regime. We show that, surprisingly, SPADE measurements saturate the quantum Chernoff bound only when certain compatibility conditions are met. These findings suggest that collective measurements may actually be needed to achieve the ultimate quantum Chernoff bound for the discrimination of specific incoherent sources. For the fully general case, our analysis can still be used to find the best SPADE configurations, generally achieved through a rotation of the SPADE interferometer that depends on the discrimination task. We also simulated the efficiency of a simplified Bayesian test that we developed for this identification task and show that the saturation of the Chernoff bound is already achieved for a finite number of repetitions N ⩽ 5000 . Our results advance the theory of quantum-limited optical discrimination, with possible applications in diagnostics, automated image interpretation, and galaxy identification.
Ideal spatial demultiplexing (SPADE) is proven to be a quantum-optimal tool for exoplanet detection, i.e., asymmetric source discrimination. However, recent investigations into the related problems of separation estimation and symmetric source discrimination showed its efficiency to be limited in the presence of noise. In this work, we use analytical tools to scrutinize the practical applicability of SPADE and derive the associated optimal decision strategy for exoplanet detection in the presence of experimental imperfections. On the one hand, we find that the probability of detection of noisy SPADE has the same scaling with planet-star separation and relative brightness as conventional techniques, such as direct imaging and coronagraphs. On the other hand, we prove that, due to a superior scaling coefficient under realistic noise conditions, SPADE remains the most efficient method for practical exoplanet detection in the sub-Rayleigh regime.
Spatial-mode demultiplexing (SPADE) has recently been adopted to measure the separation in the transverse plane between two incoherent point-like sources with sub-wavelength separation. It has been argued that this approach may yield extraordinary performances in the photon-counting regime. Here, we explore SPADE as a tool for precision measurements in the regime of bright, incoherent sources. First we analyze the general problem of estimating the second moments of the source's intensity distribution, for an extended incoherent source of any shape. Our theory predicts a substantial improvement in signal-to-noise ratio (SNR) of SPADE over direct imaging in the sub-wavelength regime. Second, we present an experimental application of SPADE to the case of two point-like, bright sources. We demonstrate the use of this setup for the estimation of the transverse separation and for the estimation of the relative intensity, confirming the expected improvement in SNR.
Recent works employing tools from quantum optics and quantum metrology proposed a new passive imaging technique that allows to resolve details far below the diffraction limit. This technique is based on replacing standard spatially-resolved intensity measurements, e.g. at each pixel of a camera, with spatial-mode demultiplexing (SpaDe) measurements that allow to acquire information in a more efficient way. In this contribution, we want to provide an intuitive explanation of why such a SpaDe approach is so effective, and illustrate how we used these idea to discriminate one point source from two, and to estimate the separation between two incoherent sources.