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International Conference on Quantum Information

Quantum limits in image processing

Claude Fabre, V. Delaubert, Nicolas Treps, Hans‐A. Bachor, Philippe Réfrégier

Published Jan 1, 2007• 1 citation
Publisher

Abstract

We determine the general limit to the maximum achievable sensitivity in the estimation of a parameter from the information contained in an optical image in the presence of quantum noise, either coherent or squeezed.
DOI
10.1364/icqi.2007.ithc1

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•Nov 1, 2018•2 citations
We determine the limit to the maximum achievable sensitivity in the estimation of a scalar parameter from the information contained in an optical image in the presence of quantum noise. This limit, based on the Cramer-Rao bound and valid for any image processing protocol, is calculated for a shot noise limited image, for a locally squeezed light, and for a single-mode squeezed light in a well-defined “noise mode”. In addition, we exhibit an image processing protocol that allows us to reach the limits in the different cases.
The fundamental quantum information processing task of estimating the phase of a qubit is considered. Following quantum measurement, the estimation efficiency is evaluated by the classical Fisher information which determines the best performance limiting any estimator and achievable by the maximum likelihood estimator. The estimation process is analyzed in the presence of decoherence represented by essential quantum noises that can affect the qubit and belonging to the broad class of unital quantum noises. Such a class especially contains the bit-flip, the phase-flip, the depolarizing noises, or the whole family of Pauli noises. As the level of noise is increased, we report the possibility of non-standard behaviors where the estimation efficiency does not necessarily deteriorate uniformly, but can experience non-monotonic variations. Regimes are found where higher noise levels prove more favorable to estimation. Such behaviors are related to stochastic resonance effects in signal estimation, shown here feasible for the first time with unital quantum noises. The results provide enhanced appreciation of quantum noise or decoherence, manifesting that it is not always detrimental for quantum information processing.
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.
In this thesis we make progress towards applications of quantum estimation theory to new physical systems. We first consider two commonly visited problems in quantum metrology: source optimisation and source localisation. For the first, we focus on estimating the distances, d, between neighbouring light sources along an array, which undergoes stretching deformations. We evaluate how changing the nature of the sources impacts the estimation precision of d by using the quantum Fisher information (QFI) as a figure of merit. By comparing this quantity for arrays of single photon emitters, coherent, thermal, and entangled sources, we find that the classical coherent and thermal states outperform the single photon emitters. This would be favourable since generating classical states is less resource-expensive to create. However, a quantum enhancement is observed when entanglement is employed. In agreement with separate work, the optimal state is that which entangles the eigenstates corresponding to the maximum and minimum difference eigenvalues of the generator. We demonstrate that not all entangled states can reproduce similar precision enhancements. This insight is reminiscent of previous studies, where entanglement was concluded as a necessary but insufficient resource for quantum metrology. Next, we address the source localisation problem to detect any deformations applied to a grid of sources. Improving this detection depends on our ability to engineer grids that maximise the sensitivity of the QFI matrix. Hence, we derive the generators of local translations of unitary evolutions that describe any general grid deformation, and show that our result is a multi-parameter extension of other results in the literature. We obtain a general result for the quantum Fisher information matrix (QFIM) through these generators for any grid deformation and explore specific spatial maps, including composite stretches, shears, and rotations. Since the QFI matrix depends only on the properties of the probe state and the configuration of the emitters, we explore how we can modify both to enhance our estimation sensitivity to determine the applied grid deformation. Physically motivated, we find the best arrangement of sources that enhances the sensitivity of detection for a set number of sources. Finally, we consider the optimal estimation of a complex squeezing operation in phase space. The use of squeezed light as a quantum resource is ubiquitous in quantum optics, and a complete characterisation of a complex squeezing operation is pivotal for fundamental reasons. This is a true multi-parameter quantum estimation problem of incompatible observables. Specifically, we find that the symmetric logarithmic derivates (SLDs) for amplitude and directional squeezing do not commute. This prohibits simultaneous optimal estimates of both parameters, even in the asymptotic limit. As a result, we focus on finding separable optimal estimates. The Cramer-Rao bound is determined to provide a theoretical benchmark on the bi-variate estimation precision for general single mode Gaussian probes. Using this and the SLDs, we present a practical experimental implementation that can realise the individual fundamental precision bounds.