Achieving quantum-limited sub-Rayleigh identification of incoherent sources with arbitrary intensities

TL;DR

This work develops a Gaussian-state model for arbitrary-intensity incoherent light undergoing diffraction and uses it to derive the quantum Chernoff bound for binary object discrimination. In the subdiffraction regime with Gaussian PSFs, the authors show SPADE saturates the quantum bound only under compatibility conditions (e.g., commuting covariances), and they propose a rotated Hermite-Gauss SPADE (TRISPADE) that achieves optimal performance in many scenarios by aligning with the sources’ principal variances. They also derive a practical finite-N Bayesian test, requiring only simple photon-counting statistics across rotated HG modes, and demonstrate near-optimal exponential error decay for modest $N$ (e.g., $N\lesssim 5000$) with bright sources. Overall, the results generalize subdiffraction discrimination theory to arbitrary intensities, quantify SPADE’s limitations, and provide a concrete, experimentally accessible route to quantum-limited identification with potential impact on diagnostics and astronomical identification.

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 $N\leqslant 5000$. Our results advance the theory of quantum-limited optical discrimination, with possible applications in diagnostics, automated image interpretation, and galaxy identification.

Figures (4)

• Figure 1: Schematic representation of the TRISPADE scheme for optimal identification of arbitrary incoherent sources in the subdiffraction regime. In the depicted example, the setup aims to identify the source between two hypotheses: a vertical cross, or a slightly tilted version of it. As discussed in Sec. \ref{['sec:subd']}, the optimal local measurement is the projection onto a TRISPADE basis rotated of an hypotheses-dependent angle $\theta_0$. This can be achieved by simply rotating the demultiplexer.
• Figure 2: Plots of the normalized gap between the quantum Chernoff exponent $\xi_Q$ and the Chernoff exponent $\xi(\theta_0)$ associated with the rotated TRISPADE for the discrimination of 1D sources (see Section \ref{['sec:1D']}), while varying $\Delta_1+\Delta_2$, that is linear in the SPADE rotation angle $\theta_0$: to $\Delta_1+\Delta_2=0$ corresponds $\theta_0=(\theta_1+\theta_2)/2$, while to $\Delta_1+\Delta_2=\pi$ corresponds $\theta_0=(\theta_1+\theta_2)/2-\pi/2$
• Figure 3: Analysis of the Chernoff bound for the identification of identical but rotated sources for a range of values of $\Delta\theta$ and of variances $V_x,V_y$. The variances have been chosen so that $\sqrt{V_x}-\sqrt{V_y}\simeq1$ in Eq. \ref{['eq:QCBrotated']}. In particular the pairs $(V_x,V_y)=(6,12),\,(2,0.2)$ represent the variances of the images depicted on top in rows a) and b) respectively, rotated of different angles. c) Plots of the normalized gap between the quantum Chernoff exponent $\xi_Q$ and the Chernoff exponent $\xi(\theta_0)$ associated with the rotated TRISPADE (see Section \ref{['sec:rotated']}), while varying $\Delta_1+\Delta_2$, ultimately controlled by the SPADE rotation angle $\theta_0$.
• Figure 4: Plots of the simulated logarithm of the error probability $\ln P_e$ (blue circles) of the Bayesian test discussed in Sec. \ref{['sec:Bayes']}, for different values of the variances, sub-diffraction parameter and average intensity. The pink dotted and the black dashed lines are the linear fit of the first ($10\leq N\leq 510$) and last ($4510\leq N\leq 5010$) five points respectively, whose slopes are reported as $\xi_1$ and $\xi_2$ in the upper right boxes. The orange solid line represents the optimal asymptotic scaling given by the Chernoff bound, whose offset has been arbitrarily chosen so that for $N=0$ the probability of error is $1/2$.