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Bradley A. Chase
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This dissertation explores the topics of parameter estimation and model reduction in the context of quantum filtering. Chapters 2 and 3 provide a review of classical and quantum probability theory, stochastic calculus and filtering. Chapter 4 studies the problem of quantum parameter estimation and introduces the quantum particle filter as a practical computational method for parameter estimation via continuous measurement. Chapter 5 applies these techniques in magnetometry and studies the estimators uncertainty scalings in a double-pass atomic magnetometer. Chapter 6 presents an efficient feedback controller for continuous-time quantum error correction. Chapter 7 presents an exact model of symmetric processes of collective qubit systems.
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We investigate cryptographic quantum parameter estimation with a high-dimensional system that allows only Bob (Receiver) to access the result and achieve optimal parameter precision from Alice (Sender). Eavesdropper (Eve) only can disturb the parameter estimation of Bob, but she can not obtain the information of parameter. We analyze the security and show that the high-dimensional system can help to utilize the resource to obtain better precision than the two-dimensional system. Finally, we generalize it to the case of multi-parameter.
We present a general framework for sensitivity optimization in quantum parameter estimation schemes based on continuous (indirect) observation of a dynamical system. As an illustrative example, we analyze the canonical scenario of monitoring the position of a free mass or harmonic oscillator to detect weak classical forces. We show that our framework allows the consideration of sensitivity scheduling as well as estimation strategies for non-stationary signals, leading us to propose corresponding generalizations of the Standard Quantum Limit for force detection.
We provide a general framework for handling the effects of a unitary disturbance on the estimation of the amplitude $lambda$ associated to a unitary dynamics. By computing an analytical and general expression for the quantum Fisher information, we prove that the optimal estimation precision for $lambda$ cannot be outperformed through the addition of such a unitary disturbance. However, if the dynamics of the system is already affected by an external field, increasing its strength does not necessary imply a loss in the optimal estimation precision.
In this article we derive a measure of quantumness in quantum multi-parameter estimation problems. We can show that the ratio between the mean Uhlmann Curvature and the Fisher Information provides a figure of merit which estimates the amount of incompatibility arising from the quantum nature of the underlying physical system. This ratio accounts for the discrepancy between the attainable precision in the simultaneous estimation of multiple parameters and the precision predicted by the Cramer-Rao bound. As a testbed for this concept, we consider a quantum many-body system in thermal equilibrium, and explore the quantum compatibility of the model across its phase diagram.
Quantum metrology holds the promise of an early practical application of quantum technologies, in which measurements of physical quantities can be made with much greater precision than what is achievable with classical technologies. In this review, we collect some of the key theoretical results in quantum parameter estimation by presenting the theory for the quantum estimation of a single parameter, multiple parameters, and optical estimation using Gaussian states. We give an overview of results in areas of current research interest, such as Bayesian quantum estimation, noisy quantum metrology, and distributed quantum sensing. We address the question how minimum measurement errors can be achieved using entanglement as well as more general quantum states. This review is presented from a geometric perspective. This has the advantage that it unifies a wide variety of estimation procedures and strategies, thus providing a more intuitive big picture of quantum parameter estimation.
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