Parameter estimation, model reduction and quantum filtering

This thesis explores the topics of parameter estimation and model reduction in the context of quantum filtering. The last is a mathematically rigorous formulation of continuous quantum measurement, in which a stream of auxiliary quantum systems is used to infer the state of a target quantum system. Fundamental quantum uncertainties appear as noise which corrupts the probe observations and therefore must be filtered in order to extract information about the target system. This is analogous to the classical filtering problem in which techniques of inference are used to process noisy observations of a system in order to estimate its state. Given the clear similarities between the two filtering problems, I devote the beginning of this thesis to a review of classical and quantum probability theory, stochastic calculus and filtering. This allows for a mathematically rigorous and technically adroit presentation of the quantum filtering problem and solution. Given this foundation, I next consider the related problem of quantum parameter estimation, in which one seeks to infer the strength of a parameter that drives the evolution of a probe quantum system. By embedding this problem in the state estimation problem solved by the quantum filter, I present the optimal Bayesian estimator for a parameter when given continuous measurements of the probe system to which it couples. For cases when the probe takes on a finite number of values, I review a set of sufficient conditions for asymptotic convergence of the estimator. For a continuous-valued parameter, I present a computational method called quantum particle filtering for practical estimation of the parameter. Using these methods, I then study the particular problem of atomic magnetometry and review an experimental method for potentially reducing the uncertainty in the estimate of the magnetic field beyond the standard quantum limit. The technique involves double-passing a probe laser field through the atomic system, giving rise to effective non-linearities which enhance the effect of Larmor precession allowing for improved magnetic field estimation. I then turn to the topic of model reduction, which is the search for a reduced computational model of a dynamical system. This is a particularly important task for quantum mechanical systems, whose state grows exponentially in the number of subsystems. In the quantum filtering setting, I study the use of model reduction in developing a feedback controller for continuous-time quantum error correction. By studying the propagation of errors in a noisy quantum memory, I present a computation model which scales polynomially, rather than exponentially, in the number of physical qubits of the system. Although inexact, a feedback controller using this model performs almost indistinguishably from one using the full model. I finally review an exact but polynomial model of collective qubit systems undergoing arbitrary symmetric dynamics which allows for the efficient simulation of spontaneous-emission and related open quantum system phenomenon

Artifact Removal Methods in EEG Recordings: A Review

To obtain the correct analysis of electroencephalogram (EEG) signals, non-physiological and physiological artifacts should be removed from EEG signals. This study aims to give an overview on the existing methodology for removing physiological artifacts, e.g., ocular, cardiac, and muscle artifacts. The datasets, simulation platforms, and performance measures of artifact removal methods in previous related research are summarized. The advantages and disadvantages of each technique are discussed, including regression method, filtering method, blind source separation (BSS), wavelet transform (WT), empirical mode decomposition (EMD), singular spectrum analysis (SSA), and independent vector analysis (IVA). Also, the applications of hybrid approaches are presented, including discrete wavelet transform - adaptive filtering method (DWT-AFM), DWT-BSS, EMD-BSS, singular spectrum analysis - adaptive noise canceler (SSA-ANC), SSA-BSS, and EMD-IVA. Finally, a comparative analysis for these existing methods is provided based on their performance and merits. The result shows that hybrid methods can remove the artifacts more effectively than individual methods

An introduction to quantum filtering

This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. We describe the quantum It\^o calculus and its use in the modelling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics.Comment: 41 pages, 1 figur

Continuous Measurement and Stochastic Methods in Quantum Optical Systems

This dissertation studies the statistics and modeling of a quantum system probed by a coherent laser field. We focus on an ensemble of qubits dispersively coupled to a traveling wave light field. The first research topic explores the quantum measurement statistics of a quasi-monochromatic laser probe. We identify the shortest timescale that successive measurements approximately commute. Our model predicts that for a probe in the near infrared, noncommuting measurement effects are apparent for subpicosecond times. The second dissertation topic attempts to find an approximation to a conditional master equation, which maps identical product states to identical product states. Through a technique known as projection filtering, we find such a equation for an ensemble of qubits experiencing a diffusive measurement of a collective angular momentum projection, and global rotations. We then test the quality of the approximation through numerical simulations. In the presence of strong randomized rotations, the approximation reproduces the exact expectation values to within 95%. The final topic applies the projection filter to the problem of state reconstruction. We find an initial state estimate based on a single continuous measurement of an identically prepared atomic ensemble. Given the ability to make a continuous collective measurement and simultaneously applying time varying controls, it is possible to find an accurate estimate given based upon a single measurement realization. Here we explore the fundamental limits of this protocol by studying an idealized model for pure qubits, which is limited only by measurement backaction. Using the exact dynamics to produce simulated measurements, we then numerically search for a maximum likelihood estimate based on the approximate expression. Our estimation technique nearly achieves an average fidelity bound set by an optimum POVM.Comment: PhD Dissertatio

Study of Multi-Modal and Non-Gaussian Probability Density Functions in Target Tracking with Applications to Dim Target Tracking

The majority of deployed target tracking systems use some variant of the Kalman filter for their state estimation algorithm. In order for a Kalman filter to be optimal, the measurement and state equations must be linear and the process and measurement noises must be Gaussian random variables (or vectors). One problem arises when the state or measurement function becomes a multi-modal Gaussian mixture. This typically occurs with the interactive multiple model (IMM) technique and its derivatives and also with probabilistic and joint probabilistic data association (PDA/JPDA) algorithms. Another common problem in target tracking is that the target\u27s signal-to-noise ratio (SNR) at the sensor is often low. This situation is often referred to as the dim target tracking or track-before-detect (TBD) scenario. When this occurs, the probability density function (PDF) of the measurement likelihood function becomes non-Gaussian and often has a Rayleigh or Ricean distribution. In this case, a Kalman filter variant may also perform poorly. The common solution to both of these problems is the particle filter (PF). A key drawback of PF algorithms, however, is that they are computationally expensive. This dissertation, thus, concentrates on developing PF algorithms that provide comparable performance to conventional PFs but at lower particle costs and presents the following four research efforts. 1. A multirate multiple model particle filter (MRMMPF) is presented in Section-3. The MRMMPF tracks a single, high signal-to-noise-ratio, maneuvering target in clutter. It coherently accumulates measurement information over multiple scans via discrete wavelet transforms (DWT) and multirate processing. This provides the MRMMPF with a much stronger data association capability than is possible with a single scan algorithm. In addition, its particle filter nature allows it to better handle multiple modes that arise from multiple target motion models. Consequently, the MRMMPF provides substantially better root-mean-square error (RMSE) tracking performance than either a full-rate or multirate Kalman filter tracker or full-rate multiple model particle filter (MMPF) with a same particle count. 2. A full-rate multiple model particle filter for track-before-detect (MMPF-TBD) and a multirate multiple model particle filter for track-before-detect (MRMMPF-TBD) are presented in Section-4. These algorithms extend the areas mentioned above and track low SNR targets which perform small maneuvers. The MRMMPF-TBD and MMPF-TBD both use a combined probabilistic data association (PDA) and maximum likelihood (ML) approach. The MRMMPF-TBD provides equivalent RMSE performance at substantially lower particle counts than a full-rate MMPF-TBD. In addition, the MRMMPF-TBD tracks very dim constant velocity targets that the MMPF-TBD cannot. 3. An extended spatial domain multiresolutional particle filter (E-SD-MRES-PF) is developed in Section-5. The E-SD-MRES-PF modifies and extends a recently developed spatial domain multiresolutional particle filter prototype. The prototype SD-MRES-PF was only demonstrated for one update cycle. In contrast, E-SD-MRES-PF functions over multiple update cycles and provides comparable RMSE performance at a reduced particle cost under a variety of PDF scenarios. 4. Two variants of a single-target Gaussian mixture model particle filter (GMMPF) are presented in Section-6. The GMMPF models the particle cloud as a Gaussian finite mixture model (FMM). MATLAB simulations show that the GMMPF provides performance comparable to a particle filter but at a lower particle cost

Chaotic time series prediction using wavelet transform and multi-model hybrid method

In order to further improve the prediction accuracy of the chaotic time series and overcome the defects of the single model, a multi-model hybrid model of chaotic time series is proposed. First, the Discrete Wavelet Transform (DWT) is used to decompose the data and obtain the approximate coefficients (low-frequency sequence) and detailed coefficients (high-frequency sequence) of the sequence. Secondly, phase space reconstruction is performed on the decomposed data. Thirdly, the chaotic characteristics of each sequence are judged by correlation integral and Kolmogorov entropy. Fourthly, in order to explore the deeper features of the time series and improve the prediction accuracy, a sequence of Volterra adaptive prediction models is established for the components with chaotic characteristics according to the different characteristics of each component. For the components without chaotic characteristics, a JGPC prediction model without chaotic feature sequences is established. Finally, the multi-model fusion prediction of the above multiple sequences is carried out by the LSTM algorithm, and the final prediction result is obtained through calculation, which further improves the prediction accuracy. Experiments show that the multi-model hybrid method of Volterra-JGPC-LSTM is more accurate than other comparable models in predicting chaotic time series