Optical quantum super-resolution imaging and hypothesis testing

The study addresses two fundamental imaging problems under the diffraction limit: (i) deciding whether a scene contains one source or two (with potentially large intensity imbalance), and (ii) estimating the angular separation between two incoherent thermal sources. Classical direct imaging (DI) is limited by the Rayleigh criterion, with minimum resolvable angular separation roughly λ/D. When sources are closely spaced and/or one is much dimmer (e.g., exoplanet near a bright star), DI suffers severe performance degradation due to the point-spread function overlap. Quantum techniques can outperform classical limits by exploiting full amplitude and phase information and by selecting near-optimal measurements. The work situates these challenges within quantum hypothesis testing (to minimize false negatives under a bounded false-positive rate) and quantum parameter estimation (to reach the quantum Cramér-Rao bound for separation estimation). The purpose is to provide an experimentally simple, two-mode interferometric method that both enhances detection of a dim secondary source and achieves super-resolved separation estimates beyond the Rayleigh limit, with performance matching quantum limits in relevant regimes.

Prior work shows quantum measurements can enhance imaging via reduced noise, ghost imaging, quantum-enhanced nonlinear microscopy, quantum lithography, and sensing. For resolving two incoherent sources, theory established that the quantum Fisher information (QFI) for separation estimation can remain finite and independent of separation, eliminating the classical Rayleigh penalty. Spatial-mode demultiplexing (SPADE) and related mode-sorting approaches can approach these limits but are experimentally demanding, sensitive to centroid misalignment and cross-talk, and less suitable for large baselines. Classical DI yields a classical relative entropy (CRE) for one-vs-two-source discrimination that scales quadratically with the relative intensity of the weak source ε, making exoplanet detection challenging at small ε and small separations. Quantum hypothesis testing via the quantum relative entropy (QRE) offers substantial gains, with scaling that improves by a factor ~1/ε over DI, and mode-sorting can achieve linear scaling in ε. Recent research on coherent detection of incoherent light and on interferometric approaches indicates that linear interferometry can, in principle, attain quantum limits for incoherent imaging, motivating simplified, scalable measurement architectures.

Theory and hypothesis testing: The task is to discriminate between H0 (single source of intensity N at position x0) and H1 (two sources with total intensity N, relative intensities 1−ε and ε, and angular separation θ=s/z0). The corresponding single-photon density operators are ρ0=|ψ_star⟩⟨ψ_star| and ρ1=(1−ε)|ψ_star⟩⟨ψ_star|+ε|ψ_planet⟩⟨ψ_planet|. In the highly attenuated regime with n detected photons, the quantum Stein lemma bounds the type-II error βn under a fixed type-I error αn<δ via βn≈exp[−n D(ρ0||ρ1)], where D(·||·) is the QRE, already optimized over all measurements. For DI with Gaussian-approximated PSF of width σ, CRE ≈ (exp(θ²/σ²)−1) ε² /2 for θ≲σ and ε≪1, revealing weak scaling in ε. Measurement design: Replace a lens with two spatial collectors at positions d1 and d2 (baseline d=d1−d2) at distance z0 from the sources. Light is interfered through an adjustable phase shift α and a 50:50 beam splitter, followed by photon counting at outputs a and b. This two-mode interferometer yields output detection probabilities that depend on α, a visibility parameter v, and the phase ϕ related to θ and geometry. For discrimination, the CRE for this measurement is maximized at a specific α and matches the QRE, making the setup optimal for the state discrimination task in the considered regime. Parameter estimation: For two equally bright sources (ε=0.5), the QFI for angular separation obeys Ig=k² d²/4, independent of θ, so the quantum Cramér-Rao bound is constant with separation, enabling super-resolution below the Rayleigh limit. The same two-mode interferometric measurement saturates both the QRE for discrimination and the QFI for estimation around α≈0 or π. The beam splitter and phase shift transform the field operators to produce detection probabilities Pa,b=(1/2)[1±v cos(α) cos(ϕ)]. Estimation of ϕ (hence θ) is performed from the detection statistics. Estimator and data processing: A Bayesian/maximum-likelihood procedure updates the probability density P(ϕ) after each detection using P(μ|ϕ,α,v) (μ∈{a,b}). P(ϕ) is represented via a Fourier series whose coefficients are updated from the observed clicks based on the model. The maximum of P(ϕ) yields ϕ̂, and the separation estimate is θ_est=2|ϕ̂|/(k d). Precision is quantified by the mean squared error (MSE), which equals the variance for unbiased estimators. Experimental setup: A fiber-coupled VCSEL at 848.2 nm (0.11 nm FWHM), pulsed at 1 MHz, feeds electro-optic phase and amplitude modulators driven by random waveforms to produce pseudo-thermal light (validated via HBT). Coherent reference pulses alternate with thermal pulses to provide interferometric stabilization via feedback from the detectors. Pseudo-thermal light is sent through 8 m multimode fiber to wash out spatial correlations, then collimated and passed through a custom mask with two circular pinholes (10–50 µm diameter, separations 15 µm to ~1 cm) to realize two pseudo point sources. Two single-mode PM fibers (collectors) separated by 5.3 mm at 1 m distance couple into a balanced interferometer with an adjustable air-gap for loss balancing and phase tuning. Outputs are detected by Si-SPADs; a TCSPC unit records timestamps (1 ps resolution). Active feedback maintains interferometric visibility >99% during acquisition. For each configuration, 25 measurements with 5 s integration time are taken to reduce Poisson noise. An ND filter on one pinhole introduces controlled brightness imbalance for hypothesis testing experiments.

• The two-mode interferometric measurement optimally discriminates between one vs two incoherent sources: by tuning the phase α, the classical relative entropy (CRE) of the measurement matches the quantum relative entropy (QRE), achieving the quantum Stein lemma error exponent for small separations.
• Scaling improvement for detection of a dim secondary source: Whereas DI yields CRE ∝ ε² (for small θ), the quantum strategy provides an effective 1/ε improvement, achieving linear-in-ε scaling with an almost-optimal measurement. Experimental CRE follows the two-mode QRE for ε ≳ 10⁻²; around ε ~ 10⁻³, experimental imperfections reduce the relative entropy but still surpass DI by approximately two orders of magnitude.
• Super-resolved separation estimation with constant precision: For equal-brightness sources, the QFI Ig=k² d²/4 is independent of θ, enabling constant precision below the Rayleigh limit. The same two-mode measurement around α≈0 or π saturates the QCRB.
• Quantitative performance: Using a baseline of 5.3 mm at 1 m standoff, the system resolves sources separated by 15 µm with 1.7% relative RMSE. Across tested separations, the experimentally achieved MSE is within a factor of 2 of the QCRB. For θ=1.48×10⁻⁴ rad, the RMSE is within 1.7% of the true value, outperforming shot-noise-limited DI by two to three orders of magnitude for the same 5.3 mm “aperture.”
• Data volume and visibility: For each θ, 25 estimates were obtained with about n≈60,000 detected photons per estimate. Experimental visibility-phase factor cos(α) was between 0.96 and 0.985, consistent with the region where the Fisher information approaches the QFI.
• Practicality: The setup is minimal—a two-collector interferometer with photon counting—and is compatible with large-baseline instruments, avoiding complexities of SPADE and large mode-sorting optics.

The findings demonstrate a simple two-mode interferometric architecture that addresses both the detection of a weak secondary source and the super-resolved estimation of source separation, achieving quantum-limited performance. By matching the CRE to the QRE, the method substantially reduces type-II error rates for dim companion detection at sub-Rayleigh separations compared with DI. For equal-brightness sources, the approach achieves constant precision independent of θ, saturating the QCRB and overcoming the Rayleigh limit. The setup’s simplicity and robustness make it attractive for astronomical applications and other scenarios where source engineering (e.g., fluorophore control) is not possible. The approach is compatible with existing stellar interferometers and could enable improved discrimination and localization of binaries or exoplanet-like companions at separations far below the diffraction limit of comparable-aperture DI. The requirement for phase stabilization is practical with reference channels or guide stars, and the method benefits from high visibility and sufficient photon counts over stabilization windows. Compared with intensity interferometry, which avoids phase links but has low success probabilities, the proposed method uses every detected photon and achieves higher precision given a phase-stabilized link.

This work provides a unified, experimentally simple two-mode interferometric strategy that (i) optimally discriminates between single and binary incoherent sources with substantial gains over direct imaging, and (ii) super-resolves angular separations for equal-brightness sources at quantum-limited precision. The demonstration uses a 5.3 mm baseline to resolve 15 µm separations at 1 m with 1.7% accuracy, achieving two to three orders of magnitude improvement over DI of comparable size and MSE within a factor of two of the QCRB. The method’s compatibility with large-baseline instruments and minimal optical complexity suggest practical pathways toward integration in astronomical interferometry and other super-resolution tasks. Future directions include hypothesis testing among multiple sources of different brightness, composite hypothesis testing, and extending to multi-mode interferometers optimized for more complex source configurations.

• Requires active phase stabilization; performance degrades with sub-optimal α and reduced visibility. Experimental data were collected with cos(α)≈0.96–0.985; higher visibility (~99%) would further improve precision for very small separations.
• At very small ε (~10⁻³), experimental imperfections reduce the achievable relative entropy below the ideal quantum limit, though still outperforming DI.
• Precision depends on sufficient photon counts within stability windows; environmental factors (vibrations, temperature) can limit phase stability.
• Broad, continuous spectra of astronomical sources limit interferometric visibility; narrow bandpass filtering mitigates this at the cost of photon flux.
• Background noise and losses reduce detected counts; while losses mainly affect data rate (in the single-photon post-selected regime), practical implementations must manage detector dark counts, timing jitter, and optical loss.
• Implementation in large-baseline systems will require robust, scalable phase-referencing or guide-star techniques.