645 research outputs found
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Non-asymptotic quantum metrology: extracting maximum information from limited data
Science relies on our practical ability to extract information from reality, since processing this information is essential for developing theories that explain our world. This thesis is precisely the study of how to extract and process information using quantum systems when a constrained amount of resources means that the available data is limited. The natural framework for this task is quantum metrology, a set of tools to model and design quantum measurement strategies. Equipped with this theory, we advocate a Bayesian approach as the appropriate formalism to study systems with a finite amount of resources, which is a non-asymptotic problem, and we propose a methodology for non-asymptotic quantum metrology. To start with, we show the consistency of taking those solutions that are optimal in the asymptotic regime of many trials as a guide to calculate a generalised measure of uncertainty in the Bayesian framework. This provides an approximate but useful way of studying the non-asymptotic regime whenever a direct Bayesian optimisation is intractable, and it avoids non-physical results that can arise when only the asymptotic theory is employed. Secondly, we construct a new non-asymptotic Bayesian bound without relying on the previous approximation by first selecting the optimal quantum strategy for a single shot, and then simulating a sequence of repetitions of this scheme, which is suitable for experiments where we do not wish or cannot correlate different trials. These methods are applied to a Mach-Zehnder interferometer, which is a single-parameter problem, and to quantum sensing networks where the nodes are either qubits or optical modes, which are multi-parameter protocols. Our results provide a detailed characterisation of how the interplay between prior information, entanglement and a limited amount of data affects the performance of quantum metrology protocols, which has important implications for the analysis of theory and experiments in this field
Tau Physics
The pure leptonic or semileptonic character of tau decays makes them a good
laboratory to test the structure of the weak currents and the universality of
their couplings to the gauge bosons. The hadronic tau decay modes constitute an
ideal tool for studying low-energy effects of the strong interactions in very
clean conditions; a well-known example is the precise determination of the QCD
coupling from tau-decay data. New physics phenomena, such as a non-zero
tau-neutrino mass or violations of (flavour / CP) conservation laws can also be
searched for with tau decays.Comment: 40 pages, latex, 10 Postscript figures. To appear in Heavy Flavours
II, eds. A.J. Buras and M. Lindner (World Scientific, 1997
Space-Time Parameter Estimation in Radar Array Processing
This thesis is about estimating parameters using an array of spatially distributed sensors. The material is presented in the context of radar array processing, but the analysis could be of interest in a wide range of applications such as communications, sonar, radio astronomy, seismology, and medical diagnosis. The main theme of the thesis is to analyze the fundamental limitations on estimation performance in sensor array signal processing. To this end, lower bounds on the estimation accuracy as well as the performance of the maximum likelihood (ML) and weighted least-squares (WLS) estimators are studied. The focus in the first part of the thesis is on asymptotic analyses. It deals with the problem of estimating the directions of arrival (DOAs) and Doppler frequencies with a sensor array. This problem can also be viewed as a two-dimensional (2-D) frequency estimation problem. The ML and WLS estimators for this problem amount to multidimensional, highly non-linear optimization problems which would be expensive to solve in real-time in a radar system. Therefore, simplifications of this problem are of great interest. It is shown in this thesis that, under some circumstances, the 2-D problem decouples into 1-D problems. This means a dramatic reduction in computational complexity with insignificant loss of accuracy. The second part contains a performance analysis of the ML DOA estimator under conditions of low signal-to-noise ratio (SNR) and a small number of data samples. It is well known that the ML estimator exhibits a threshold effect, i.e. a rapid deterioration of estimation accuracy below a certain SNR. This effect is caused by outliers and is not captured by standard analysis tools. In this thesis, approximations to the mean square estimation error and probability of outlier are derived that can be used to predict the threshold region performance of the ML estimator with high accuracy. Moreover, these approximations alleviate the need for time-consuming computer simulations when evaluating the ML performance
Space-Time Parameter Estimation in Radar Array Processing
This thesis is about estimating parameters using an array of spatially distributed sensors. The material is presented in the context of radar array processing, but the analysis could be of interest in a wide range of applications such as communications, sonar, radio astronomy, seismology, and medical diagnosis. The main theme of the thesis is to analyze the fundamental limitations on estimation performance in sensor array signal processing. To this end, lower bounds on the estimation accuracy as well as the performance of the maximum likelihood (ML) and weighted least-squares (WLS) estimators are studied. The focus in the first part of the thesis is on asymptotic analyses. It deals with the problem of estimating the directions of arrival (DOAs) and Doppler frequencies with a sensor array. This problem can also be viewed as a two-dimensional (2-D) frequency estimation problem. The ML and WLS estimators for this problem amount to multidimensional, highly non-linear optimization problems which would be expensive to solve in real-time in a radar system. Therefore, simplifications of this problem are of great interest. It is shown in this thesis that, under some circumstances, the 2-D problem decouples into 1-D problems. This means a dramatic reduction in computational complexity with insignificant loss of accuracy. The second part contains a performance analysis of the ML DOA estimator under conditions of low signal-to-noise ratio (SNR) and a small number of data samples. It is well known that the ML estimator exhibits a threshold effect, i.e. a rapid deterioration of estimation accuracy below a certain SNR. This effect is caused by outliers and is not captured by standard analysis tools. In this thesis, approximations to the mean square estimation error and probability of outlier are derived that can be used to predict the threshold region performance of the ML estimator with high accuracy. Moreover, these approximations alleviate the need for time-consuming computer simulations when evaluating the ML performance
Estimation and detection with chaotic systems
Includes bibliographical references (p. 209-214).Supported by the U.S. Air Force Office of Scientific Research under the Augmentation Awards for Science and Engineering Research Training Program Grant. F49620-92-J-0255 Supported by the U.S. Air Force Office of Scientific Research. AFOSR-91-0034-C Supported by the U.S. Navy Office of Naval Research. N00014-93-1-0686 Supported by Lockheed Sanders, Inc. under a U.S. Navy Office of Naval Research contract. N00014-91-C-0125Michael D. Richard
Quantum sensing networks for the estimation of linear functions
The theoretical framework for networked quantum sensing has been developed to a great extent in the past few years, but there are still a number of open questions. Among these, a problem of great significance, both fundamentally and for constructing efficient sensing networks, is that of the role of inter-sensor correlations in the simultaneous estimation of multiple linear functions, where the latter are taken over a collection local parameters and can thus be seen as global properties. In this work we provide a solution to this when each node is a qubit and the state of the network is sensor-symmetric. First we derive a general expression linking the amount of inter-sensor correlations and the geometry of the vectors associated with the functions, such that the asymptotic error is optimal. Using this we show that if the vectors are clustered around two special subspaces, then the optimum is achieved when the correlation strength approaches its extreme values, while there is a monotonic transition between such extremes for any other geometry. Furthermore, we demonstrate that entanglement can be detrimental for estimating non-trivial global properties, and that sometimes it is in fact irrelevant. Finally, we perform a non-asymptotic analysis of these results using a Bayesian approach, finding that the amount of correlations needed to enhance the precision crucially depends on the number of measurement data. Our results will serve as a basis to investigate how to harness correlations in networks of quantum sensors operating both in and out of the asymptotic regime
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Spectral Optimization Problems Controlling Wave Phenomena
Design problems seek a material arrangement or shape which fully harnesses the physical properties of the material(s) to create an environment in which a particular phenomena is most (or least) pronounced. Mathematically, design problems are formulated as PDE-constrained optimization problems to find the material arrangement that maximizes an objective function which expresses the desired behavior. The PDE constraint describes the relationship between the material and the phenomena of interest. The focus of this thesis is four design problems where the PDE constraint is a time-independent wave equation and the objective function governs some aspect of wave motion. We consider the shape optimization of functions of Dirichlet-Laplacian eigenvalues associated with the set of star-shaped, symmetric, bounded planar regions with smooth boundary. The boundary of such a region is represented using a Fourier-cosine series and the optimization problem is solved numerically using a quasi-Newton method. The method is applied to maximizing two particular nonsmooth functions of the eigenvalues: (a) the ratio of the n-th to first eigenvalues and (b) the ratio of the n-th eigenvalue gap to first eigenvalue. Both are generalizations of the Payne-PĂłlya-Weinberger ratio. The optimal values of these ratios and regions for which they are attained, for n †13, are presented and interpreted as a study of the range of the Dirichlet-Laplacian eigenvalues. For both spectral functions and each n, the optimal region has multiplicity two n-th eigenvalue. We consider a system governed by the wave equation with index of refraction n(x), taken to be variable within a bounded region of d-dimensional space and constant outside. The solution of the time-dependent wave equation with spatially-localized initial data spreads and decays with advancing time. The rate of spatially localized energy decay can be measured in terms of the eigenvalues of the scattering resonance problem, a non-selfadjoint eigenvalue problem consisting of the time-harmonic wave (Helmholtz) equation with outgoing radiation condition at infinity. Specifically, the rate of energy escape is governed by the complex scattering eigenfrequency which is closest to the real axis. We study the structural design problem: Find a refractive index profile n* within an admissible class which has a scattering frequency with minimal imaginary part. The admissible class is defined in terms of the compact support of n(x)-1 and pointwise upper and lower (material) bounds on n(x): 0 < n- †n(x) †n+ < â. We formulate this problem as a constrained optimization problem and prove that an optimal structure, n* exists. Furthermore, n*(x) is piecewise constant and achieves the material bounds, i.e., n*(x) is n- or n+ almost everywhere. In one dimension, we establish a connection between n*(x) and the well-known class of Bragg structures, where n(x) is constant on intervals whose length is one-quarter of the effective wavelength. Consider a system governed by the time-dependent Schroedinger equation in its ground state. When subjected to weak parametric forcing by an "ionizing field" (time-varying), the state decays with advancing time due to coupling of the bound state to radiation modes. The decay-rate of this metastable state is governed by Fermi's Golden Rule (FGR), which depends on the potential V and the details of the forcing. We pose the potential design problem: find V* which minimizes FGR (maximizes the lifetime of the state) over an admissible class of potentials with fixed spatial support. We formulate this problem as a constrained optimization problem and prove that an admissible optimal solution exists. Then, using quasi-Newton methods, we compute locally optimal potentials. These have the structure of a truncated periodic potential with a localized defect. In contrast to optimal structures for other spectral optimization problems, the optimizing potentials appear to be interior points of the constraint set and to be smooth. The multi-scale structures that emerge incorporate the physical mechanisms of energy confinement via material contrast and interference effects. An analysis of locally optimal potentials reveals local optimality is attained via two mechanisms: (i) decreasing the density of states near a resonant frequency in the continuum and (ii) tuning the oscillations of extended states to make FGR, an oscillatory integral, small. Finally, we explore the performance of optimal potentials via simulations of the time-evolution. We consider a general class of two-dimensional passive propagation media, represented as a planar graph where nodes are capacitors connected to a common ground and edges are inductors. Capacitances and inductances are fixed in time but vary in space. Kirchhoff's laws give the time dynamics of voltage and current in the system. By harmonically forcing input nodes and collecting the resulting steady-state signal at output nodes, we obtain a linear, analog device that transforms the inputs to outputs. We pose the lattice synthesis problem: given a linear transformation, find the inductances and capacitances for an inductor-capacitor circuit that can perform this transformation. Formulating this as an optimization problem, we numerically demonstrate its solvability using gradient-based methods. By solving the lattice synthesis problem for various desired transformations, we design several devices that can be used for signal processing and filtering. In addition to these spectral optimization problems, we study several problems on wave propagation, diffraction, and scattering. The focus is on the behavior of time-harmonic solutions to continuous and discrete wave equations
New Concepts in Quantum Metrology: Dynamics, Machine Learning, and Bounds on Measurement Precision
Diese kumulative Promotionsarbeit befasst sich mit theoretischer Quantenmetrologie, der Theorie von Messung und SchÀtzung unter Zuhilfenahme von Quantenressourcen.
Viele VorschlĂ€ge fĂŒr quantenverbesserte Sensoren beruhen auf der PrĂ€paration von nichtklassischen AnfangszustĂ€nden und integrabler Dynamik. Allerdings sind solche nichtklassischen ZustĂ€nde schwierig zu prĂ€parieren und gegen DekohĂ€renz zu schĂŒtzen. Alternativ schlagen wir in dieser Promotionsarbeit sogenannte quantenchaotische Sensoren vor, die auf klassischen AnfangszustĂ€nden beruhen, die einfach zu prĂ€parieren sind, wobei Quantenverbesserungen an der Dynamik vorgenommen werden. Diese Herangehensweise hat ihren Ursprung darin, dass sowohl Quantenchaos als auch Quantenmetrologie ĂŒber die Empfindlichkeit fĂŒr kleine Ănderungen in der Dynamik charakterisiert werden. Wir erforschen unterschiedliche Arten von Dynamik am Beispiel des Modells eines gestoĂenen Quantenkreisels ("kicked top"), dessen Dynamik durch nichtlineare Kontrollpulse quantenchaotisch wird. AuĂerdem zeigen wir, dass Quantenchaos in der Lage ist, schĂ€dlichen DekohĂ€renzeffekte abzuschwĂ€chen. Insbesondere prĂ€sentieren wir einen Vorschlag fĂŒr ein quantenchaotisches CĂ€siumdampf-Magnetometer.
Mit der Hilfe von BestĂ€rkendem Lernen verbessern wir Zeitpunkt und StĂ€rke der nichtlinearen Pulse im Modell des gestoĂenen Quantenkreisels mit SuperradianzdĂ€mpfung. FĂŒr diesen Fall finden wir, dass die Kontrollstrategie als eine dynamische Form der Spin-Quetschung verstanden werden kann.
Ein anderer Teil dieser Promotionsarbeit beschĂ€ftigt sich mit bayesscher QuantenschĂ€tzung und insbesondere mit dem Problem der heuristischen Gestaltung von Experimenten. Wir trainieren neuronale Netze mit einer Kombination aus ĂŒberwachtem und bestĂ€rkendem Lernen, um diese zu schnellen und starken Heuristiken fĂŒr die Gestaltung von Experimenten zu machen. Die Vielseitigkeit unserer Methode zeigen wir anhand von Beispielen zu Einzel- und MehrparameterschĂ€tzung, in denen die trainierten neuronalen Netze die Leistung der modernsten Heuristiken ĂŒbertreffen.
AuĂerdem beschĂ€ftigen wir uns mit einer lange unbewiesenen Vermutung aus dem Bereich der Quantenmetrologie: Wir liefern einen Beweis fĂŒr diese Vermutung und finden einen Ausdruck fĂŒr die maximale Quantenfischerinformation fĂŒr beliebige gemischte ZustĂ€nde und beliebige unitĂ€re Dynamik, finden Bedingungen fĂŒr optimale ZustandsprĂ€paration und optimale dynamische Kontrolle, und verwenden diese Ergebnisse, um zu beweisen, dass die Heisenberg-Schranke sogar mit thermischen ZustĂ€nden beliebiger (endlicher) Temperatur erreicht werden kann.This cumulative thesis is concerned with theoretical quantum metrology, the theory of measurement and estimation using quantum resources. Possible applications of quantum-enhanced sensors include the measurement of magnetic fields, gravitational wave detection, navigation, remote sensing, or the improvement of frequency standards.
Many proposals for quantum-enhanced sensors rely on the preparation of non-classical initial states and integrable dynamics. However, such non-classical states are generally difficult to prepare and to protect against decoherence. As an alternative, in this thesis, we propose so-called quantum-chaotic sensors which make use of classical initial states that are easy to prepare while quantum enhancements are applied to the dynamics. This approach is motivated by the insight that quantum chaos and quantum metrology are both characterized by the sensitivity to small changes of the dynamics. At the example of the quantum kicked top model, where nonlinear control pulses render the dynamics quantum-chaotic, we explore different dynamical regimes for quantum sensors. Further, we demonstrate that quantum chaos is able to alleviate the detrimental effects of decoherence. In particular, we present a proposal for a quantum-chaotic cesium-vapor magnetometer.
With the help of reinforcement learning, we further optimize timing and strength of the nonlinear control pulses for the kicked top model with superradiant damping. In this case, the optimized control policy is identified as a dynamical form of spin squeezing.
Another part of this thesis deals with Bayesian quantum estimation and, in particular, with the problem of experiment design heuristics. We train neural networks with a combination of supervised and reinforcement learning to become fast and strong experiment design heuristics. We demonstrate the versatility of this method using examples of single and multi-parameter estimation where the trained neural networks surpass the performance of well-established heuristics.
Finally, this thesis deals with a long-time outstanding conjecture in quantum
metrology: we prove this conjecture and find an expression for the maximal quantum Fisher information for any mixed initial state and any unitary dynamics, provide conditions for optimal state preparation and optimal control of the dynamics, and utilize these results to prove that Heisenberg scaling can be achieved even with thermal states of arbitrary (finite) temperature
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