We present a parameter-decoupling-informed learning model for estimating subwavelength separations of point sources, without requiring prior knowledge of source properties, control over the sources, or continued access to their light signals. The method builds on our parameter-decoupled quantum superresolution estimation theory, which circumvents the need to estimate nuisance parameters such as partial coherence, brightness imbalance, and relative phase. Our convolutional neural network further accommodates practical imperfections, including background noise, photon loss, and centroid and orientation misalignment. We validate the method using synthetic and experimentally generated images, achieving separation-estimation fidelity above $82\%$ for separations as small as approximately $0.075σ$, where $σ$ is the Gaussian intensity-PSF standard deviation. This corresponds to an illustrative separation of $\sim13.5$ nm when $σ=180$ nm, suggesting potential relevance to length scales addressed by advanced active-source microscopy. Applied to archival Hubble images of NGC~4486B, the model yields nuclear separations of 9.3-13.2~pc, consistent with the previously reported value of 12~pc. Comparisons using the same training and testing images demonstrate substantially higher fidelity than conventional direct data-driven learning in the tested cases. The method's robustness to varying source properties and imaging imperfections supports retrospective astrophysical image analysis and motivates applications in live-cell microscopy and quantum metrology where source control or renewed optical measurements are impractical.
We present a parameter-decoupling-informed learning model for estimating subwavelength separations of point sources, without requiring prior knowledge of source properties, control over the sources, or continued access to their light signals. The method builds on our parameter-decoupled quantum superresolution estimation theory, which circumvents the need to estimate nuisance parameters such as partial coherence, brightness imbalance, and relative phase. Our convolutional neural network further accommodates practical imperfections, including background noise, photon loss, and centroid and orientation misalignment. We validate the method using synthetic and experimentally generated images, achieving separation-estimation fidelity above 82% for separations as small as approximately 0.075σ, where σ is the Gaussian intensity-PSF standard deviation. This corresponds to an illustrative separation of ∼13.5 nm when σ=180 nm, suggesting potential relevance to length scales addressed by advanced active-source microscopy. Applied to archival Hubble images of NGC4486B, the model yields nuclear separations of 9.3-13.2pc, consistent with the previously reported value of 12~pc. Comparisons using the same training and testing images demonstrate substantially higher fidelity than conventional direct data-driven learning in the tested cases. The method's robustness to varying source properties and imaging imperfections supports retrospective astrophysical image analysis and motivates applications in live-cell microscopy and quantum metrology where source control or renewed optical measurements are impractical.
Quantum superresolution promises to resolve closely spaced sources beyond the diffraction limit, but existing approaches generally rely on idealized source properties or require simultaneous estimation of multiple coupled parameters of realistic sources. Here we introduce a parameter-decoupled measurement that determines the separation of realistic passive sources without estimating their unknown brightness imbalance, mutual coherence, or relative phase. A complete four-parameter quantum Fisher information analysis identifies the separation information that remains accessible in the presence of these nuisance parameters. We then construct a directly measurable log-probability invariant from projection channels whose conditional statistics depend only on the separation. Broad classes of channel pairs satisfy the resulting decoupling condition. For a Gaussian point-spread function, the lowest-order implementation asymptotically attains the four-parameter quantum limit per incident signal in the sub-Rayleigh regime. This approach converts a realistic multiparameter imaging problem into an operationally single-parameter measurement, providing a practical route toward quantum-enhanced resolution of passive spatial, temporal, and spectral signals.
Due to the wave nature of light, optical imaging is inherently subject to diffraction, which typically dictates the resolution limit for two-point sources. A practical way to describe this phenomenon is through Rayleigh’s criterion, which states that two sources are barely resolvable when the minimum of one source's diffraction pattern coincides with the maximum of the other. However, recent advances in optical quantum metrology have shown that this classical limit can be surpassed by leveraging the spatial correlations within the light field. Specifically, decomposing the light into a Hermite-Gauss mode basis provides substantially more information than traditional direct imaging, which measures local intensity [1,2,3]. In this work, we implement a spatial demultiplexing technique to estimate the separation between light sources of unequal brightness. This unknown imbalance adds complexity to the estimation task, making it inherently a multiparameter problem and thus requiring a multimode approach [4]. By optimizing the measurement strategy and introducing a shift between two demultiplexers, we achieve sub-Rayleigh resolution in distance measurements without assuming prior knowledge of the intensity imbalance or alignment. [1] Tsang, M., Nair, R., & Lu, X. M. (2016). Quantum theory of superresolution for two incoherent optical point sources. Physical Review X, 6(3), 031033. [2] Sorelli, G., Gessner, M., Walschaers, M., & Treps, N. (2021). Optimal observables and estimators for practical superresolution imaging. Physical Review Letters, 127(12), 123604. [3] Rouvière, C., Barral, D., Grateau, A., Karuseichyk, I., Sorelli, G., Walschaers, M., & Treps, N. (2024). Ultra-sensitive separation estimation of optical sources. Optica, 11(2), 166-170. [4] Řeháček, J., Hradil, Z., Koutný, D., Grover, J., Krzic, A., & Sánchez-Soto, L. L. (2018). Optimal measurements for quantum spatial superresolution. Physical review A, 98(1), 012103.
Quantum estimation has shown that spatial-mode measurements can avoid Rayleigh's curse, but the fundamental precision limits are usually formulated for a known point-spread function (PSF). We show that, for two equally bright incoherent sources sharing an unknown static, shift-invariant complex-amplitude response, the local quantum information for separation is exactly the same as in the known-PSF case. In a quantum semiparametric formulation, the equality follows from the orthogonality of the separation score to the nuisance tangent space generated by all admissible PSF perturbations. We propose two receivers to access this information: momentum-resolved Hong--Ou--Mandel interference (MR-HOM), which acquires half the small-separation quantum information per signal photon without reconstructing the PSF, and a tomography-assisted symmetric logarithmic derivative receiver (TA-SLD), which asymptotically attains the full bound under explicit recovery conditions. These results provide a route toward blind superresolution without prior PSF calibration in microscopy, astronomy, and other optical imaging systems with uncertain responses.
Resolving light sources below the diffraction limit is a fundamental task both for astronomy and microscopy. Several recent works, analysed this problem through the lens of quantum parameter estimation theory and proved that the separation between two point sources can be estimated at the quantum limit using intensity measurements after spatial-mode demultiplexing. However, most previous works have either consider low-intensity, or thermal sources. To broaden the applicability of this approach, it is important to extend these results to more general light sources. To this goal, we will present an analytical expression for the Quantum Fisher Information, determining the ultimate resolution limit, for the separation between two sources in an arbitrary Gaussian state. Applying this result to different quantum states, we can shine some light on some relevant questions. We can for example explore the role of partial coherence considering displaced and correlated thermal states, or investigate the importance of quantum correlations by considering squeezed light. In addition to the ultimate quantum limit, we will discuss a simple estimation technique, requiring access only to the mean value of a linear combination of demultiplexed intensity measurements, which is often sufficient to saturate these limits, and can easily be adapted to incorporate the most common noise sources. Finally, we will present our experimental setup that allows for the generation of the images of two sources with different photon statistics, as well as for spatial mode demultiplexing and we will discuss the first practical implementations if the above mentioned estimation techniques.
Historically, the resolution of optical imaging systems was dictated by diffraction, and the Rayleigh criterion was long considered an unsurpassable limit. In superresolution microscopy, this limit is overcome by manipulating the emission properties of the object. However, in passive imaging, when sources are uncontrolled, reaching sub-Rayleigh resolution remains a challenge. Here, we implement a quantum-metrology-inspired approach for estimating the separation between two incoherent sources, achieving a sensitivity five orders of magnitude beyond the Rayleigh limit. Using a spatial mode demultiplexer, we examine scenes with bright and faint sources, through intensity measurements in the Hermite–Gauss basis. Analyzing sensitivity and accuracy over an extensive range of separations, we demonstrate the remarkable effectiveness of demultiplexing for sub-Rayleigh separation estimation. These results effectively render the Rayleigh limit obsolete for passive imaging.
For more than a century, the diffraction limit has defined the resolution achievable by passive optical imaging systems. Although some resolution improvement can be gained through classical data processing of the image, it is limited by the noise arising from quantum nature of light. Minimizing the effect of this noise requires quantum treatment of optical imaging. By reformulating imaging as a problem of quantum measurement and estimation, it becomes possible to identify optimal detection strategies that recover spatial information previously thought inaccessible. This review summarizes the theoretical framework that underpins this development, from the formulation of quantum Cramer-Rao bounds and Chernoff bounds to the construction of receivers that attain them, such as those based on spatial-mode demultiplexing. We show how these methods can beat conventional imaging in the classification, localization, and imaging of sub-Rayleigh incoherent sources. We then discuss extensions to multiparameter and partially coherent scenarios, and highlight the unifying connections between estimation and discrimination tasks. Finally, we survey recent experimental demonstrations that approach quantum-limited resolution and outline emerging applications in microscopy, astronomy, and optical sensing.