Spade Demonstrates Superior Sub-Rayleigh Source Discrimination with Two Incoherent Points

Scientists are tackling the challenging problem of distinguishing between closely spaced light sources, a crucial task for applications ranging from astronomical imaging to advanced microscopy. Saurabh U. Shringarpure, Yong Siah Teo, and Hyunseok Jeong, from NextQuantum Innovation Research Center and Seoul National University, alongside Michael Evans from the University of Toronto, and Luis L. Sanchez-Soto et al, present compelling experimental evidence for a novel approach to this problem , sub-Rayleigh source discrimination using a technique called spatial-mode demultiplexing (SPADE). Their research, detailed in a new paper, moves beyond traditional methods by employing a Bayesian inference framework that relies solely on observed data and prior beliefs, avoiding arbitrary statistical assumptions. This innovative methodology not only demonstrates SPADE’s superior resolving power over direct imaging but also provides a robust and extensible foundation for analysing superresolution techniques and tackling scenarios with multiple light sources.

Bayesian SPADE inference for point source discrimination

This breakthrough sidesteps the limitations of the Helstrom measurement, which requires collective detection, and offers optimal performance even with asymptotically large sample sizes. The study rigorously demonstrates the efficacy of their Bayesian framework by applying relative belief (RB) analysis, a technique originally developed for quantum state estimation and incomplete tomography with limited data. The researchers constructed a system where prior probabilities are updated to posterior probabilities via Bayes’ rule, allowing for a direct comparison of prior and posterior beliefs using the RB ratio, a value exceeding one indicating data support for a given hypothesis. This framework remains reliable even with limited data, offering a meaningful quantification of statistical evidence without relying on large-sample approximations.
Experiments confirm that the RB framework provides a clear threshold at RBk= 1, effectively distinguishing between hypotheses supported by the evidence and those that are not. This contrasts sharply with traditional methods that focus on rejecting null hypotheses rather than quantifying support for alternatives. This is particularly crucial in optical imaging, where discriminating between single and multiple point sources is vital for applications such as exoplanet detection and fluorescence microscopy, especially when source separation falls below the system’s point spread function. Furthermore, the research establishes the optimality of SPADE for source discrimination, addressing vulnerabilities to imperfections like misalignments and fabrication defects that can degrade measurement fidelity. By adopting a Bayesian approach, the scientists have created a practical and robust method for analysing quantum-enhanced superresolution, offering a significant advancement over existing techniques. The work opens new avenues for developing advanced imaging systems with enhanced resolution and sensitivity, potentially revolutionising fields ranging from astronomy to biomedical imaging, and providing a powerful tool for tackling complex source discrimination problems.

Bayesian SPADE inference for point source discrimination

The study pioneered a relative-belief (RB) analysis, originally from quantum information processing, to construct optimal credible regions for quantum state estimation and incomplete tomography even with limited data. This approach necessitates a well-defined model, observed measurement data, and prior distributions, allowing for optional checks for prior bias, and crucially avoids reliance on significance levels or p-values common in conventional hypothesis testing. Prior probabilities, pr(k), were assigned to each hypothesis reflecting initial beliefs, subsequently updated to posterior probabilities, pr(k|D), via Bayes’ rule, calculated as L(D|k)pr(k) divided by the sum of L(D|k′)pr(k′) for all K hypotheses. Scientists harnessed the likelihood, L(D|k), representing the probability of observing the data given a specific hypothesis, to quantify the evidence supporting each scenario.

The experimental setup involved generating incoherent optical point sources and measuring the resulting spatial modes using the SPADE system, a configuration optimised for asymptotically large sample sizes. This technique enabled the team to resolve sources separated by distances smaller than the system’s point spread function, a regime where direct imaging typically fails. By meticulously controlling for potential imperfections like modal crosstalk, the research ensured the robustness and fidelity of the SPADE measurements, delivering a practical approach to analysing quantum-enhanced superresolution imaging.

SPADE differentiates incoherent sources via Bayesian inference

The team measured the performance of SPADE under various conditions, establishing its robustness against imperfections like misalignments and fabrication defects that typically introduce modal crosstalk and degrade measurement fidelity. Scientists calculated RB ratios to measure whether belief in a hypothesis increased due to the data, finding values greater than one indicate data support and strengthened belief. In the case of two possible states, the posterior probability of the hypothesized state was determined, while for more than two states, the evidence strength was calculated using the Heaviside step function, naturally defined as zero for the null argument. Measurements confirm a clear threshold at RBk= 1, distinguishing hypotheses supported by the evidence from those that are not.

The experimental setup employed two fiber-coupled CW lasers at 1550nm, generating coherent states of light with independent control over polarization and optical power. Each beam was collimated into free space as Gaussian beams with a waist of w1 ≃1.135mm, and the combined beams were projected through a Glan, Thomson polarizer to ensure common linear polarization. The team utilized a multiplane light converter (MPLC) with a waist of w0 ≃320μm to decompose the incoming light into the Hermite, Gauss (HG) mode basis, coupling each mode into a single-mode fiber and measuring intensities using photodiodes, specifically HG00, HG10, HG20, and HG30 were used for testing. This breakthrough delivers a transparent and evidence-driven approach to state discrimination, emphasizing the strength of support the data provide for each hypothesis.

SPADE outperforms imaging via Bayesian inference

This method, unlike traditional Helstrom measurements, does not require collective detection and performs optimally with large sample sizes. Their protocol employs a quadrant detector and photodiode to sequentially measure source position and power, calibrating for experimental imperfections using cubic spline fitting. The team assessed robustness by varying priors associated with nuisance parameters, acknowledging that results are somewhat dependent on the chosen nuisance parameter space. Future work could explore the generalisation of this framework to scenarios involving more than two sources and investigate the impact of different prior specifications on the final inference.