We develop a quantum statistical framework for passive optical surface metrology. Modelling a surface as an incoherent ensemble of point emitters imaged through a diffraction-limited system, we employ techniques from quantum parameter estimation and hypothesis testing to derive ultimate bounds for jointly estimating geometrical features and for deciding the presence or absence of surface defects, and we identify optimal measurements from the geometry of the point-spread-function manifold. As a representative application, we analyse a minimal surface crack model based on three point sources and show that spatial mode sorting can simultaneously enable near-quantum-limited estimation of crack width and depth and markedly enhanced detectability of the crack, compared with direct imaging. Our results pave the way towards enhanced optical inspection and characterisation of sub-diffraction surface features by probing a limited number of spatial modes without any illumination control.
We develop a quantum statistical framework for passive optical surface metrology. Modelling a surface as an incoherent ensemble of point emitters imaged through a diffraction-limited system, we employ techniques from quantum parameter estimation and hypothesis testing to derive ultimate bounds for jointly estimating geometrical features and for deciding the presence or absence of surface defects, and we identify optimal measurements from the geometry of the point-spread-function manifold. As a representative application, we analyse a minimal surface crack model based on three point sources and show that spatial mode sorting can simultaneously enable near-quantum-limited estimation of crack width and depth and markedly enhanced detectability of the crack, compared with direct imaging. Our results pave the way towards enhanced optical inspection and characterisation of sub-diffraction surface features by probing a limited number of spatial modes without any illumination control.
Surface roughness is an important quantity to many engineering and precision manufacturing disciplines. In this paper we investigate the problem of estimating the root-mean-square roughness of a sample by passive linear optics. By adopting quantum parameter estimation methods, we determine the ultimate precision limits for estimating spatial moments of a general three-dimensional distribution of incoherent point sources in the sub-diffraction regime. Specializing this result to the axial profile, we show that the information on the first moment (mean height) and standard deviation (roughness) is bounded by a constant. While classical imaging techniques fail to achieve this bound, a quantum inspired imaging technique based on spatial mode demultiplexing is proven to be optimal for estimating the axial standard deviation. This provides a powerful and experimentally accessible route to measuring roughness of nearly smooth surface patches beyond the diffraction limit.
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.
We consider imaging tasks involving discrimination between known objects and investigate the best possible accuracy with which the correct object can be identified. Using the quantum Chernoff bound, we analytically find the ultimate achievable asymptotic error rate for symmetric hypothesis tests between \emph{any} two incoherent 2D objects when the imaging system is dominated by optical diffraction. Furthermore, we show that linear-optical demultiplexing of the spatial modes of the collected light exactly saturates this ultimate performance limit, enabling a quadratic improvement over the asymptotic error rate achieved by direct imaging as the objects become more severely diffraction-limited. We extend our results to identify the quantum limit and optimal measurement for discrimination between an arbitrary number of candidate objects. Our work constitutes a complete theoretical treatment of the ultimate quantitative limits on passive, sub-diffraction, incoherent object discrimination and is readily applicable to a multitude of real-world applications.
The ability to perform high precision measurements underpins a plethora of applications. Several techniques for force sensing, phase estimation and discrimination, as well as surface reconstruction for complex features of three-dimensional samples, have been developed in recent years. The main aim of this thesis is to investigate metrology enhancements due to quantum resources (probes and measurements), by using quantum parameter estimation and channel discrimination techniques. The thesis focuses on two main scenarios. In the first one, we deal with three-dimensional superlocalisation. By using tools from multiparameter quantum metrology, we show that a simultaneous estimation of all three components of the separation between two incoherent point sources in the paraxial approximation is achievable by a single quantum measurement, with a precision saturating the ultimate limit stemming from the quantum Cramér-Rao bound. Such a precision is not degraded in the sub-wavelength regime, thus overcoming the traditional limitations of classical direct imaging derived from Rayleigh's criterion. Our results are qualitatively independent of the point spread function of the imaging system, and quantitatively illustrated in detail for Gaussian beams. In this case, we show that a method of measuring the position of each photon at the imaging plane based on discrimination in terms of Hermite-Gaussian spatial modes reaches the quantum precision bound in the limit of infinitesimal separation. In the second part of the thesis, we investigate the role of quantum coherence as a resource for channel discrimination tasks. We consider a probe state of arbitrary dimension entering a black box, in which a phase shift is implemented, with the unknown phase randomly sampled from a finite set of predetermined possibilities. At the output, an optimal measurement is performed in order to guess which specific phase was applied in the process. We show that the presence of quantum coherence (superposition with respect to the eigenbasis of the generator of the phase shift) in the input probe directly determines an enhancement in the probability of success for this task, compared to the use of incoherent probes. We prove that such a quantum advantage is exactly quantified by the robustness of coherence, a full monotone with respect to the recently formulated resource theories of quantum coherence, whose properties and applications are developed and explored in detail.
Quantum superresolution reveals that the vanishing of separation sensitivity in conventional imaging below the Rayleigh limit does not necessarily indicate a fundamental loss of information in the optical field. However, the quantum limit for estimating the physical distance between two incoherent point sources in three-dimensional imaging systems and its dependence on the spatial structure of the point-spread function remains largely unknown. In this work, we derive the quantum-limited precision for estimating the full distance between two incoherent point sources with arbitrary intensity imbalance in a three-dimensional spatially invariant imaging system. We show that the distance information remains finite in the sub-Rayleigh regime and is governed by the second-order displacement-response tensor of the point-spread function. The eigensystem of this tensor determines the optimal relative orientation between the two sources, and reflection symmetries of the point-spread function can further provide a simplified means of identifying the optimal orientation. This geometric structure is coordinate invariant and provides a direct strategy for improving resolution by physically rotating an anisotropic imaging system to align its optimal principal response direction with the source displacement. For a general three-dimensional Gaussian point-spread function, the response tensor is proportional to the inverse spatial covariance, establishing a direct connection between quantum-limited distance precision and the geometry of Gaussian distribution.
Resolving extended incoherent objects below the diffraction limit poses an application-rich imaging challenge whose solution may enable a new generation of observational instruments and capabilities. In this work, we invoke a practical model for general imaging by approximating an arbitrary extended incoherent object as a finite grid of thermal point emitters parameterized by their brightnesses. We derive the quantum Fisher information matrix (QFIM) for simultaneous brightness estimation and show that the symmetric logarithmic derivatives weakly commute, indicating that the Helstrom bound furnishes the ultimate quantum limit on the estimation error for incoherent imaging. Furthermore, for deeply sub-diffraction scenes, we find numerical evidence of a gap between the Nagaoka-Hayashi (NH) bound and the Helstrom bound. This gap reveals that separable measurements, though more experimentally accessible, are insufficient to reach the quantum limit, and points to the prospective advantage of joint measurements acting on multiple state copies. Additionally, we show that spatial mode-demultiplexing (SPADE) often saturates the NH bound solidifying its status as a near-optimal separable measurement strategy that significantly outperforms direct imaging. Finally, we articulate two joint detection receivers implemented with bona fide quantum resources that asymptotically achieve the Helstrom bound.