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arXiv:2412.08199 · Physical Review A

Efficient estimation of error bounds for quantum multiparametric imaging with constraints

Alexander B. Mikhalychev, Saif Almazrouei, Svetlana Mikhalycheva, Abdellatif Bouchalkha, Dmitri S. Mogilevtsev, Bobomurat J. Ahmedov

Published Mar 21, 2025• 2 citations
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Abstract

Advanced superresolution imaging techniques require specific approaches for accurate and consistent estimation of the achievable spatial resolution. Fisher information supplied to the Cram\'er-Rao bound (CRB) has proved to be a powerful and efficient tool for resolution analysis and optical setups optimization. However, the standard CRB is not applicable to constrained problems violating the unbiasedness condition, while such models are frequently encountered in quantum imaging of complex objects. Complementary to the existing approaches based on modifying CRB, we propose a practical algorithm for approximate construction of a modified Fisher information matrix, which takes the constraints into account and enables accurate estimation of errors for constrained problems by the standard CRB. We demonstrate the efficiency of the proposed technique by applying it to one-, two-, and multiparameter model problems in quantum imaging. The approach provides quantitative explanation of previous results with successful experimental reconstruction of objects with the spatial scale smaller than the theoretical limit predicted by the standard CRB.

DOI
10.1103/physreva.111.032618

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Advanced superresolution imaging techniques require specific approaches for accurate and consistent estimation of the achievable spatial resolution. Fisher information supplied to the Cram'er-Rao bound (CRB) has proved to be a powerful and efficient tool for resolution analysis and optical setups optimization. However, the standard CRB is not applicable to constrained problems violating the unbiasedness condition, while such models are frequently encountered in quantum imaging of complex objects. Complementary to the existing approaches based on modifying CRB, we propose a practical algorithm for approximate construction of a modified Fisher information matrix, which takes the constraints into account and enables accurate estimation of errors for constrained problems by the standard CRB. We demonstrate the efficiency of the proposed technique by applying it to one-, two-, and multiparameter model problems in quantum imaging. The approach provides quantitative explanation of previous results with successful experimental reconstruction of objects with the spatial scale smaller than the theoretical limit predicted by the standard CRB.
While standard Fisher Information (FI) and Cramer-Rao bound (CRB) provide essential and easily obtained information about the estimating parameter sensitivity and error bounds, they fail to address the constraints. As attempts to address the constraints by modifying the CRB proved cumbersome, a practical algorithm to modify the CRB was proposed in our previous study. Here, we further investigate the effect of a imaging system's parameters on the standard and modified CRB obtained by this algorithm in the single-parameter model not limited to biphoton systems.
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.
As a method to extract information from optical systems, imaging can be viewed as a parameter estimation problem. The fundamental precision in locating one emitter or estimating the separation between two incoherent emitters is bounded below by the multiparameter quantum Cramér-Rao bound (QCRB). Multiparameter QCRB gives an intrinsic bound in parameter estimation. We determine the ultimate potential of quantum-limited imaging for improving the resolution of a far-field, diffraction-limited optical field within the paraxial approximation. We show that the quantum Fisher information matrix (QFIm) in about one emitter’s position is independent on its true value. We calculate the QFIm of two unequal-brightness emitters’ relative positions and intensities; the results show that only when the relative intensity and centroids of two-point sources, including longitudinal and transverse directions, are known exactly, the separation in different directions can be estimated simultaneously with finite precision. Our results give the upper bounds on certain far-field imaging technology and will find wide use in applications from microscopy to astrometry.
We establish the multiparameter quantum Cram\'er-Rao bound for simultaneously estimating the centroid, the separation, and the relative intensities of two incoherent optical point sources using a linear imaging system. For equally bright sources, the Cram\'er-Rao bound is independent of their separation, which confirms that the Rayleigh resolution limit is just an artifact of the conventional direct imaging and can be overcome with an adequate strategy. For the general case of unequally bright sources, the amount of information one can gain about the separation falls to zero, but we show that there is always a quadratic improvement in an optimal detection in comparison with the intensity measurements. This advantage can be of utmost importance in realistic scenarios, such as observational astronomy.
Sensitivity limits are usually determined using the Cram\'er-Rao bound. Recently, this approach has been used to obtain the ultimate resolution limit for the estimation of the separation between two incoherent point sources. However, methods that saturate these resolution limits usually require the full measurement statistics, which can be challenging to access. In this work we introduce a simple superresolution protocol to estimate the separation between two thermal sources which relies only on the average value of a single accessible observable. We show how optimal observables for this technique may be constructed for arbitrary thermal sources and we study their sensitivities when one has access to spatially resolved intensity measurements (direct imaging) and photon counting after spatial-mode demultiplexing. For demultiplexing, our method is optimal, i.e., it saturates the quantum Cram\'er-Rao bound. We also investigate the impact of noise on the optimal observables, their measurement sensitivity, and the scaling with the number of detected photons of the smallest resolvable separation. For low signals in the image plane, we demonstrate that our method saturates the Cram\'er-Rao bound even in the presence of noise.