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Quantum Information and Measurement (QIM)

Quantum-Optimal Object Classification in Sub-Diffraction Incoherent Imaging

Michael R. Grace, Saikat Guha

Published Jan 1, 2021• 0 citations
Publisher

Abstract

We derive quantum limits and optimal measurements for arbitrary hypothesis tests between incoherent, diffraction-limited objects, enabling discrimination of an order of magnitude more objects than direct imaging.
DOI
10.1364/qim.2021.f1a.4

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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.
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.