Recently we propose “measurement theory”, which includes measurements in classical and quantum systems and is constructed in terms of a C∗-algebra. The purpose of this paper is to study (1): “regression analysis” (2): “Kalman filter” (3): “measurement error model” in measurement theory. And, we show that this approach is applicable to very general situations.
Recently we propose “measurement theory”, which includes measurements in classical and quantum systems and is constructed in terms of a C∗-algebra. The purpose of this paper is to study (1): “regression analysis” (2): “Kalman filter” (3): “measurement error model” in measurement theory. And, we show that this approach is applicable to very general situations.
Recently, we proposed measurement theory ( or. quantum language) as a linguistic turn of quantum mechanics (with the Copenhagen interpretation). This theory has a great power of scientific descriptions. In fact, we have continued asserting that even statistics can be described in terms of measurement theory. Thus, we believe that quantum language is future statistics (i.e., statistics will develop into quantum language). However, now we think that our arguments were too abstract and philosophical, that is, we should have presented concrete examples much more. Thus, in this paper, we show that the calculation of Kalman filter is more understandable in terms of quantum language than in terms of usual statistics. For this, we devote ourselves to statistical measurement theory, in which the Bertrand paradox is discussed.
Although regression analysis has a great history, we consider that it has always continued being confused. For example, the fundamental terms in regression analysis (e.g., "regression", "least-squares method", "explanatory variable", "response variable", etc.) seem to be historically conventional, that is, these words do not express the essence of regression analysis. Recently, we proposed quantum language (or, classical and quantum measurement theory), which is characterized as the linguistic turn of the Copenhagen interpretation of quantum mechanics. We believe that this language has a great power of description, and therefore, even statistics can be described by quantum language. Therefore, in this paper, we discuss the regression analysis and the generalized linear model (i.e., multiple regression analysis) in quantum language, and clarify that the terms "explanatory variable" and "response variable" is respectively characterized as a kind of causality and the measured value.
State variable and parameter estimations are important for signal sensing and feedback control in both traditional engineering systems and quantum systems. The Kalman filter, which is one of the most popular signal recovery techniques in classical systems for decades, has now been connected to the stochastic master equations of linear quantum mechanical systems. Various studies have invested effort on mapping the state evolution of a quantum system into a set of classical filtering equations. However, establishing proper evolution models with symmetry to classical filter equation for quantum systems is not easy. Here, we review works that have successfully built a Kalman filter model for quantum systems and provide an improved method for optimal estimations. We also discuss a practical scenario involving magnetic field estimations in quantum systems, where non-linear Kalman filters could be considered an estimation solution.
Abstract Single‐beam atomic magnetometers herald a new era of high‐precision magnetic field sensing, with applications spanning fundamental physics to biomagnetism. Nevertheless, their utility is often curtailed by quantum sensor noise, encompassing both technical and quantum‐mechanical noise. This research delves into the potential of the Kalman filter as a tool to subdue quantum sensor noise, thereby augmenting the sensitivity of single‐beam atomic magnetometers. Quantum‐mechanical noise is integrated into the system model as the process noise and measurement noise, and a discrete Kalman filter equipped with a time delay variable is employed. The findings reveal that the time delay variable significantly influences temporal signal tracing, while the discrete Kalman filter enhances sensitivity performance in frequency domain analysis, bypassing the typical sensitivity and time resolution trade‐off encountered in coherent sensing strategies. Partial‐knowledge signal scenarios are also taken into account, wherein a polynomial model is proposed to expansively render the discrete Kalman filter more relevant and adaptable to real‐world situations. Collectively, through experimentation involving sine‐like, non‐Gaussian, and medical magnetocardiography (MCG) signals, our results underscore the promising potential of the Kalman filter in enhancing the sensitivity of atomic magnetometers for practical sensing applications.
The term measurement theory refers to that part of a physical theory in which the empirical and operational content of the concepts of the theory is determined. Measurements are analyzed both as operational procedures defining the → observables of the theory and as physical processes which are themselves subject to the laws of physics. In classical physics, measurements are performed in order to deter-mine the values of one or several observables of the physical system un-der consideration. Classical physics allowed the idealized notion that every physical quantity has a definite value at any time, and that this value can be determined with certainty by measurement without influ-encing the object system in a significant way. By contrast, in quantum mechanics both features fail to hold without strong qualifications. Ac-cordingly, in their seminal paper of 1935 [1], Einstein, Podolsky and Rosen used elements of this description as a sufficient criterion of phys-ical reality, applicable both in classical and quantum mechanics:
The purpose of this tutorial is to give a brief introduction to linear quantum control systems. The mathematical model of linear quantum control systems is presented first, then some fundamental control-theoretic notions such as stability, controllability and observability are given, which are closely related to several important concepts in quantum information science such as decoherence-free subsystems, quantum non-demolition variables, and back-action evasion measurements. After that, quantum Gaussian states are introduced, in particular, an information-theoretic uncertainty relation is presented which often gives a better bound for mixed Gaussian states than the well-known Heisenberg uncertainty relation. The quantum Kalman filter is presented for quantum linear systems, which is the quantum analogy of the Kalman filter for classical (namely, non-quantum-mechanical) linear systems. The quantum Kalman canonical decomposition for quantum linear systems is recorded, and its application is illustrated by means of a recent experiment. As single- and multi-photon states are useful resources in quantum information technology, the response of quantum linear systems to these types of input is presented. Finally, coherent feedback control of quantum linear systems is briefly introduced, and a recent experiment is used to demonstrate the effectiveness of quantum linear systems and networks theory.