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The European Physical Journal Plus

Optical resolution from Fisher information

L. Motka, B. Stoklasa, Milena D’Angelo, Paolo Facchi A, Augusto Garuccio, Z. Hradil, Saverio Pascazio, Francesco Vincenzo Pepe, Yong Siah Teo, Jaroslav Řeháček, Luis Lorenzo Sánchez-Soto

Published Apr 30, 2016• 27 citations
Publisher

Abstract

The information gained by performing a measurement on a physical system is most appropriately assessed by the Fisher information, which in fact establishes lower bounds on estimation errors for an arbitrary unbiased estimator. We revisit the basic properties of the Fisher information and demonstrate its potential to quantify the resolution of optical systems. We illustrate this with some conceptually important examples, such as single-slit diffraction, spectroscopy and superresolution techniques.

DOI
10.1140/epjp/i2016-16130-7

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The information gained by performing a measurement on a physical system is most appropriately assessed by the Fisher information, which in fact establishes lower bounds on estimation errors for an arbitrary unbiased estimator. We revisit the basic properties of the Fisher information and demonstrate its potential to quantify the resolution of optical systems. We illustrate this with some conceptually important examples, such as single-slit diffraction, spectroscopy and superresolution techniques.
Super-resolution techniques in microscopy are opening new frontiers in biology, and super-resolution for telescope imaging may provide similar benefits for astronomy and space domain awareness. Telescope super-resolution techniques such as multi-plane light conversion and photonic lanterns have been inspired by the ability of mode sorting measurements to extract classical Fisher information closer to the quantum bound than standard telescope imaging. Fisher information analyses for super-resolution imaging systems commonly consider the case of imaging point sources, but often only consider estimating a small subset of the parameters. Furthermore, there has been no demonstration of an estimation algorithm with performance that follows the limit set by the Fisher information when all point source parameters are considered. We provide a framework for defining and analyzing super-resolution using the classical and quantum Fisher information for the multi-parameter estimation problem and demonstrate an algorithm that can achieve the classical Fisher information bound for all the parameters simultaneously.
Optical precision measurement is evolving from conventional intensity and phase readout toward a paradigm centered on mode selection, statistical estimation, and information extraction. Conventional intensity- and phase-based schemes can achieve high sensitivity, but their performance is often constrained by finite-aperture diffraction, reference stability, optical path control, and environmental perturbations in systems. Quantum estimation theory provides a physical framework for optical resolution and measurement precision by optimizing the measurement basis. Inspired by spatial mode demultiplexing, this paper extends quantum parameter estimation from incoherent point-source resolution to small-displacement measurement. Metasurface enabled mode control converts displacement induced optical-field variations into measurable changes in orthogonal modal coefficients, providing a route for extracting sub-diffraction-scale displacement information. This paper introduces Rayleigh’s curse, Fisher information, and spatial mode projection, discusses their applications in one-dimensional displacement measurement and two-dimensional super-resolution imaging, and outlines future directions in digital optics, Engineering Optics 2.0, mode-basis optimization, two-dimensional information reconstruction, and intelligent optical measurement systems for engineering applications.
Fisher information is a key notion in the whole field of quantum metrology. It allows for a direct quantification of the maximal achievable precision of the estimation of the parameters encoded in quantum states using the most general quantum measurement. It fails, however, to quantify the robustness of quantum estimation schemes against measurement imperfections, which are always present in any practical implementations. Here, we introduce a new concept of Fisher information measurement noise susceptibility that quantifies the potential loss of Fisher information due to small measurement disturbance. We derive an explicit formula for the quantity, and demonstrate its usefulness in the analysis of paradigmatic quantum estimation schemes, including interferometry and superresolution optical imaging.
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.
Fisher information is a statistical metric whose inverse represents a lower bound on the variance of an unbiased estimator. Naturally, in order to make the best estimates of a given unknown parameter, one would desire to find ways to minimize the value of this lower bound. Thus, the Fisher information can also be used to compare the quality of different estimators for a given parameter: whichever estimator has a lower value for its Fisher information must also provide better estimates. The same holds true in the framework of quantum information theory, where an analog to the Fisher information, dubbed quantum Fisher information, performs the same role as its classical counterpart. This dissertation concerns itself with both types of Fisher information, and so I present two projects in which Fisher information was a helpful tool for determining the quality of our work. In the first project, we examined potential solutions to the problem of optical loss in long-baseline interferometers, which creates a bottleneck in the resolution these arrays of telescopes can achieve. What we suggested was a novel entanglement-assisted scheme that interferes a two-mode squeezed vacuum state with the target astronomical state, and then measures the resulting state by means of homodyne detection. Ultimately, we showed that when the Fisher information is observed, our proposed scheme did not outperform the traditional schemes used in long-baseline interferometry. In the second project, we sought to write code that could determine how to best use dynamical decoupling pulses to correct dephasing error in a nitrogen-vacancy center realization of a memory and spectator qubit system. Through the use of the Python coding language, we were able to write programs that could determine the characteristic time and model the change in fidelity of the system when XY4 pulses are applied. Ultimately, we found that the quantum fidelity of our model closely mirrored the fidelity decay function our system was expected to follow in the case where XY4 pulses are applied.