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Límits quàntics en la resolució de sistemes d'imatge i Lidar

Lamaison Vidarte, Miren

Published Jun 1, 2019• 0 citations
OpenAlex

Abstract

In this project we study the ultimate precision of a Lidar system aimed at estimating the position and velocity of a target with quantum light (entangled photons). By comparison, a standard Lidar system determines the position and velocity of an object using a beam of coherent light. Using Quantum Estimation Theory we will show that the use of entangled pairs of photons can improve the precision of the measurement of position and velocity simultaneously. Several bounds for the error (typical deviation) in the estimation of the parameters are found. We study which of them is the one that gives us more information about the limits of the uncertainty in the estimation. In particular, we have found that the Holevo Cramer-Rao bound is the best one can get for this case. Finally, an idea of how this bound could be experimentally achieved is given.

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The energy-time uncertainty relation puts a fundamental limit on the precision of lidars for the estimation of range and velocity. The precision in the estimation of the range (through the time of arrival) and the velocity (through Doppler frequency shifts) of a target are inversely related to each other and are dictated by the bandwidth of the signal. Here, we use the theoretical toolbox of multiparameter quantum metrology to determine the ultimate precision of the simultaneous estimation of range and velocity. We consider the case of a single target as well as a pair of closely separated targets. In the latter case, we focus on the relative position and velocity. We show that the tradeoff between the estimation precision of position and velocity is relaxed for entangled probe states and is completely lifted in the limit of perfect photon time-frequency correlations. In the regime where the two targets are close to each other, the relative position and velocity can be estimated nearly optimally and jointly, even without entanglement, using the measurements determined by the symmetric logarithmic derivatives.

•Jul 1, 2021•25 citations

Quantum-enhanced Doppler lidar

Maximilian Reichert, Roberto Di Candia, Moe Z. Win +1 more

Abstract We propose a quantum-enhanced lidar system to estimate a target’s radial velocity, which employs squeezed and frequency-entangled signal and idler beams. We compare its performance against a classical protocol using a coherent state with the same pulse duration and energy, showing that quantum resources provide a precision enhancement in the estimation of the velocity of the object. We identify three distinct parameter regimes characterized by the amount of squeezing and frequency entanglement. In two of them, a quantum advantage exceeding the standard quantum limit is achieved assuming no photon losses. Additionally, we show that an optimal measurement to attain these results in the lossless case is frequency-resolved photon counting. Finally, we consider the effect of photon losses for the high-squeezing regime, which leads to a constant factor quantum advantage higher than 3 dB in the variance of the estimator, given a roundtrip lidar-to-target-to-lidar transmissivity larger than 50%.

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Arturo Villegas

(English) Resolution enhancement in modern optical metrology techniques has been possible due to significant technological improvements. The accurate and precise control of wavelength, bandwidth, and power of light sources; homogeneity, high-quality composition materials and surface smoothness of optical elements; and highly stable nanopositioners and optomechanical components, allow more sensitive imaging and sensing, in certain scenarios even beyond the standard diffraction limit. This has motivated a more detailed study of the fundamental resolution limitations of an optical system. The chosen approach to address this problem is to consider optical imaging as a parameter estimation problem. With this in mind, the theory of quantum estimation and statistical inference provides the tools to determine the estimation precision limits. Starting by considering the state of light as a quantum state that carries information of interest; the Crámer-Rao lower bound provides a fundamental limit for the achievable precision. This lower bound is directly associated to optical resolution in practical terms. In this thesis, we present an overview of the useful tools of quantum estimation theory that can be applied to optical metrology. We focus on the Crámer-Rao lower bound, and provide methods to calculate it. Since the bound depends on specific characteristics of the system, we explore three specific possibilities. First we present with an example the validity regimes for different bounds; explicitly to calculate the parameters of entangled photon pairs in in the frame of a quantum Lidar System. Second, we show the dependency of the lower bound on the photonic model selection, showing a discrepancy between N copies of a single photon and a multimode coherent state with average photon number N. two particular models. Third, we study the effects of lossy environments in the informational content of the quantum state. Additionally, we present the conditions that a measuring strategy must satisfy to allow attainability of the fundamental limit. If the measurement strategy is not feasible or it is not possible to implement, one can evaluate the resolution improvement of the available technique by comparing to the standard and fundamental limits. For specific scenarios of interest, measurement methods based on the use of spatial modes of light allow to asymptotically attain the resolution limit. This has drawn attention to the information carried by specific modes, and has motivated the design of measurement strategies based on probing or sensing using spatial modes. In this thesis we include an overview of spatial modes of light, their generation and detection, and their use for optical sensing. We present a method for optical beam localization in the transverse plane using spatial mode information. Moreover, we propose a technique to retrieve the full modal decomposition of an arbitrary beam; which combined with the adequate set of modes allows to estimate certain parameters of an optical state with the maximum precision possible. Finally, motivated by the informational content of an optical beam that is not easily accessible, we explore the use of artificial intelligence to extract information about spatial features of an object from its diffraction pattern; without the need of solving the inverse problem nor using a physical model of scattering. (Español) Las mejoras de resolución en técnicas modernas de metrología óptica se deben al significativo progreso tecnológico. 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En términos prácticos este límite inferior está diréctamente asociado con la resolución óptica. En esta tesis, presentamos un resumen de las herramientas útiles de la teoría de estimación cuántica que pueden ser aplicadas a la metrología óptica. Nos enfocamos en el límite inferior de Crámer-Rao y proveemos métodos para calcularlo. Dado que el límite depende de características específicas del sistema, exploramos tres posibilidades específicas. Primero, presentamos con un ejemplo los regímenes de validez de diferentes límites; explícitamente para calcular los parámetros de pares de fotones entrelazados en un en el marco de un sistema Lidar cuántico. Segundo, mostramos la dependencia en límite inferior de la selección del modelo fotónico, mostrando una discrepancia entre N copias de un fotón individual y un estado coherente multimodo con N fotones promedio. entre dos modelos particulares. 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•Apr 26, 2023•0 citations

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Lidar is a well-known optical technology for measuring a target's range and radial velocity. We describe two lidar systems that use entanglement between transmitted signals and retained idlers to obtain significant quantum enhancements in simultaneous measurements of these parameters. The first entanglement-enhanced lidar circumvents the Arthurs-Kelly uncertainty relation for simultaneous measurements of range and radial velocity from the detection of a single photon returned from the target. This performance presumes there is no extraneous (background) light, but is robust to the round-trip loss incurred by the signal photons. The second entanglement-enhanced lidar---which requires a lossless, noiseless environment---realizes Heisenberg-limited accuracies for both its range and radial-velocity measurements, i.e., their root-mean-square estimation errors are both proportional to $1/M$ when $M$ signal photons are transmitted. These two lidars derive their entanglement-based enhancements from the use of a unitary transformation that takes a signal-idler photon pair with frequencies ${\ensuremath{\omega}}_{S}$ and ${\ensuremath{\omega}}_{I}$ and converts it to a signal-idler photon pair whose frequencies are $({\ensuremath{\omega}}_{S}+{\ensuremath{\omega}}_{I})/2$ and $({\ensuremath{\omega}}_{S}\ensuremath{-}{\ensuremath{\omega}}_{I})/2$. Insight into how this transformation provides its benefits is provided through an analogy to continuous-variable superdense coding.

arXiv:1705.06793•Physical Review A•Oct 16, 2017•74 citations

Quantum Measurement Bounds beyond the Uncertainty Relations

Vittorio Giovannetti, Seth Lloyd, Lorenzo Maccone

In quantum mechanics, the Heisenberg uncertainty relations and the Cramér-Rao inequalities typically limit the precision in the estimation of a parameter through the standard deviation of a conjugate observable. Here we extend these relations by giving a bound to the precision of a parameter in terms of the expectation value of the conjugate observable. This has both foundational and practical consequences: in quantum optics it resolves a controversy over which is the ultimate precision in interferometry.

arXiv:1109.5661•Physical Review Letters•Jun 26, 2012•70 citations
In this project we study the ultimate precision of a Lidar system aimed at estimating the position and velocity of a target with quantum light (entangled photons). By comparison, a standard Lidar system determines the position and velocity of an object using a beam of coherent light. Using Quantum Estimation Theory we will show that the use of entangled pairs of photons can improve the precision of the measurement of position and velocity simultaneously. Several bounds for the error (typical deviation) in the estimation of the parameters are found. We study which of them is the one that gives us more information about the limits of the uncertainty in the estimation. In particular, we have found that the Holevo Cramer-Rao bound is the best one can get for this case. Finally, an idea of how this bound could be experimentally achieved is given.
The energy-time uncertainty relation puts a fundamental limit on the precision of lidars for the estimation of range and velocity. The precision in the estimation of the range (through the time of arrival) and the velocity (through Doppler frequency shifts) of a target are inversely related to each other and are dictated by the bandwidth of the signal. Here, we use the theoretical toolbox of multiparameter quantum metrology to determine the ultimate precision of the simultaneous estimation of range and velocity. We consider the case of a single target as well as a pair of closely separated targets. In the latter case, we focus on the relative position and velocity. We show that the tradeoff between the estimation precision of position and velocity is relaxed for entangled probe states and is completely lifted in the limit of perfect photon time-frequency correlations. In the regime where the two targets are close to each other, the relative position and velocity can be estimated nearly optimally and jointly, even without entanglement, using the measurements determined by the symmetric logarithmic derivatives.
Abstract We propose a quantum-enhanced lidar system to estimate a target’s radial velocity, which employs squeezed and frequency-entangled signal and idler beams. We compare its performance against a classical protocol using a coherent state with the same pulse duration and energy, showing that quantum resources provide a precision enhancement in the estimation of the velocity of the object. We identify three distinct parameter regimes characterized by the amount of squeezing and frequency entanglement. In two of them, a quantum advantage exceeding the standard quantum limit is achieved assuming no photon losses. Additionally, we show that an optimal measurement to attain these results in the lossless case is frequency-resolved photon counting. Finally, we consider the effect of photon losses for the high-squeezing regime, which leads to a constant factor quantum advantage higher than 3 dB in the variance of the estimator, given a roundtrip lidar-to-target-to-lidar transmissivity larger than 50%.
(English) Resolution enhancement in modern optical metrology techniques has been possible due to significant technological improvements. The accurate and precise control of wavelength, bandwidth, and power of light sources; homogeneity, high-quality composition materials and surface smoothness of optical elements; and highly stable nanopositioners and optomechanical components, allow more sensitive imaging and sensing, in certain scenarios even beyond the standard diffraction limit. This has motivated a more detailed study of the fundamental resolution limitations of an optical system. The chosen approach to address this problem is to consider optical imaging as a parameter estimation problem. With this in mind, the theory of quantum estimation and statistical inference provides the tools to determine the estimation precision limits. Starting by considering the state of light as a quantum state that carries information of interest; the Crámer-Rao lower bound provides a fundamental limit for the achievable precision. This lower bound is directly associated to optical resolution in practical terms. In this thesis, we present an overview of the useful tools of quantum estimation theory that can be applied to optical metrology. We focus on the Crámer-Rao lower bound, and provide methods to calculate it. Since the bound depends on specific characteristics of the system, we explore three specific possibilities. First we present with an example the validity regimes for different bounds; explicitly to calculate the parameters of entangled photon pairs in in the frame of a quantum Lidar System. Second, we show the dependency of the lower bound on the photonic model selection, showing a discrepancy between N copies of a single photon and a multimode coherent state with average photon number N. two particular models. Third, we study the effects of lossy environments in the informational content of the quantum state. Additionally, we present the conditions that a measuring strategy must satisfy to allow attainability of the fundamental limit. If the measurement strategy is not feasible or it is not possible to implement, one can evaluate the resolution improvement of the available technique by comparing to the standard and fundamental limits. For specific scenarios of interest, measurement methods based on the use of spatial modes of light allow to asymptotically attain the resolution limit. This has drawn attention to the information carried by specific modes, and has motivated the design of measurement strategies based on probing or sensing using spatial modes. In this thesis we include an overview of spatial modes of light, their generation and detection, and their use for optical sensing. We present a method for optical beam localization in the transverse plane using spatial mode information. Moreover, we propose a technique to retrieve the full modal decomposition of an arbitrary beam; which combined with the adequate set of modes allows to estimate certain parameters of an optical state with the maximum precision possible. Finally, motivated by the informational content of an optical beam that is not easily accessible, we explore the use of artificial intelligence to extract information about spatial features of an object from its diffraction pattern; without the need of solving the inverse problem nor using a physical model of scattering. (Español) Las mejoras de resolución en técnicas modernas de metrología óptica se deben al significativo progreso tecnológico. El control preciso de la longitud de onda, ancho de banda y potencia de las fuentes de luz; la homogeneidad, los componentes de alta calidad y lisura superficial de los elementos ópticos; mas la alta estabilidad de nanoposicionadores y componentes optomecánicos, permite imagenología y detección más sensible, en ciertos escenarios, incluso más allá del límite de difracción estándar. Esto ha motivado un estudio más detallado de los limites fundamentales de un sistema óptico. El enfoque elegido para dirigirse a este problema consiste en considerar la imagenología óptica como un problema de estimación de parámetros. Con esto en mente, la teoría de estimación cuántica e inferencia estadística provee las herramientas para determinar los límites de precisión en la estimación. Comenzando por considerar el estado de luz como un estado cuántico que porta información de interés; el límite inferior de Crámer-Rao provee un límite fundamental para la precisión alcanzable. En términos prácticos este límite inferior está diréctamente asociado con la resolución óptica. En esta tesis, presentamos un resumen de las herramientas útiles de la teoría de estimación cuántica que pueden ser aplicadas a la metrología óptica. Nos enfocamos en el límite inferior de Crámer-Rao y proveemos métodos para calcularlo. Dado que el límite depende de características específicas del sistema, exploramos tres posibilidades específicas. Primero, presentamos con un ejemplo los regímenes de validez de diferentes límites; explícitamente para calcular los parámetros de pares de fotones entrelazados en un en el marco de un sistema Lidar cuántico. Segundo, mostramos la dependencia en límite inferior de la selección del modelo fotónico, mostrando una discrepancia entre N copias de un fotón individual y un estado coherente multimodo con N fotones promedio. entre dos modelos particulares. Tercero, estudiamos los efectos de ambientes con pérdidas en el contenido de información de un estado cuántico. Adicionalmente, presentamos las condiciones que una estrategia de medición debe satisfacer para alncanzar el límite fundamental. Si la estrategia de medición no es factible, o no se puede implementar, se puede evaluar la mejora en la resolución de la técnica disponible comparando con el límite fundamental y el estándar. Para escenarios específicos de interés, métodos de medición basados en el uso de modos espaciales de luz permiten alcanzar asintóticamente el límite de resolución. Esto ha atraído la atención hacia la información que portan modos específicos, y ha motivado el diseño de estrategias de medición basadas en sondeo y detección usando modos espaciales. En esta tesis incluímos un resumen sobre modos espaciales de luz, su generación y detección, y su uso en detección óptica. Presentamos un método para localización de haces de luz en el plano tranversal usando informacion de los modos espaciales. Más aún, proponemos una técnica para recuperar la descomposición modal completa de un haz arbitrario, lo cual combinado con el conjunto adecuado de modos permite estimar ciertos parámetros de un estado óptico con la máxima precisión posible. Finalmente, motivados por la información contenida en haces ópticos que no es fácilmente accesible, exploramos el uso de inteligencia artificial para extraer información sobre las características espaciales de objetos, de su patrón de difracción; sin la necesidad de resolver el problema inverso ni utilizar un modelo físico de dispersión.
Lidar is a well-known optical technology for measuring a target's range and radial velocity. We describe two lidar systems that use entanglement between transmitted signals and retained idlers to obtain significant quantum enhancements in simultaneous measurements of these parameters. The first entanglement-enhanced lidar circumvents the Arthurs-Kelly uncertainty relation for simultaneous measurements of range and radial velocity from the detection of a single photon returned from the target. This performance presumes there is no extraneous (background) light, but is robust to the round-trip loss incurred by the signal photons. The second entanglement-enhanced lidar---which requires a lossless, noiseless environment---realizes Heisenberg-limited accuracies for both its range and radial-velocity measurements, i.e., their root-mean-square estimation errors are both proportional to 1/M1/M1/M when MMM signal photons are transmitted. These two lidars derive their entanglement-based enhancements from the use of a unitary transformation that takes a signal-idler photon pair with frequencies ${\ensuremath{\omega}}_{S}$ and ${\ensuremath{\omega}}_{I}$ and converts it to a signal-idler photon pair whose frequencies are $({\ensuremath{\omega}}_{S}+{\ensuremath{\omega}}_{I})/2$ and $({\ensuremath{\omega}}_{S}\ensuremath{-}{\ensuremath{\omega}}_{I})/2$. Insight into how this transformation provides its benefits is provided through an analogy to continuous-variable superdense coding.
In quantum mechanics, the Heisenberg uncertainty relations and the Cramér-Rao inequalities typically limit the precision in the estimation of a parameter through the standard deviation of a conjugate observable. Here we extend these relations by giving a bound to the precision of a parameter in terms of the expectation value of the conjugate observable. This has both foundational and practical consequences: in quantum optics it resolves a controversy over which is the ultimate precision in interferometry.