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Physical Review A

Bayesian analysis of the measurements of the evolution systems for which the quantities measured are subject to a master equation

Masahiro Agu

Published Jul 1, 1980• 4 citations
Publisher

Abstract

A Bayesian analysis of measurement process is formulated with respect to the physical quantity being subject to the master equation. The dynamical equation of the a posteriori probability distribution of the physical quantity is derived with the use of the system size expansion. The coupled nonlinear equations of the most probable path with the results of measurements taken into account and of the variance of the fluctuations around the path are derived, which are the extensions of those of the Kalman's filter in the linear control theory to the nonlinear physical system on the basis of a physically reasonable assumption, the "local nature of transition." The formula of the entropy reduction and the $H$ theorem which are associated with the measurement process are also derived.

DOI
10.1103/physreva.22.269

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A Bayesian analysis of measurement process is formulated with respect to the physical quantity being subject to the master equation. The dynamical equation of the a posteriori probability distribution of the physical quantity is derived with the use of the system size expansion. The coupled nonlinear equations of the most probable path with the results of measurements taken into account and of the variance of the fluctuations around the path are derived, which are the extensions of those of the Kalman's filter in the linear control theory to the nonlinear physical system on the basis of a physically reasonable assumption, the "local nature of transition." The formula of the entropy reduction and the HHH theorem which are associated with the measurement process are also derived.
Bayesian analysis of the physical-measurement process is formulated with respect to the physical quantity being subject to the master equation. Here nonlinear dynamics of the measurement system and its own fluctuations are taken into account with the use of a nonlinear Langevin equation. The coupled nonlinear equations of the most probable path and of the variance of the fluctuations around the path are derived with the results of the measurement taken into account. The theory is applied to the measurement in which the field is detected with the use of a cavity resonator or laser amplifier. The minimum uncertainty in the estimate of the most probable path of the field is determined as a function of the variance of the fluctuations of the measurement system itself.
Measurement in classical physics is examined here as a process involving the joint evolution of object-system and measuring apparatus. For this, a model of measurement is proposed which lends itself to theoretical analysis using Hamiltonian mechanics and Bayesian probability. At odds with a widely-held intuition, it is found that the ideal measurement capable of extracting finite information without disturbing the system is ruled out (by the third law of thermodynamics). And in its place a Heisenberg-like precision-disturbance relation is found, with the role of ℏ/2\hbar/2ℏ/2 played by kBT/Ωk_BT/\OmegakB​T/Ω; where TTT and Ω\OmegaΩ are a certain temperature and frequency characterizing the ready-state of the apparatus. The proposed model is argued to be maximally efficient, in that it saturates this Heisenberg-like inequality, while various modifications of the model fail to saturate it. The process of continuous measurement is then examined; yielding a novel pair of Liouville-like master equations -- according to whether the measurement record is read or discarded -- describing the dynamics of (a rational agent's knowledge of) a system under continuous measurement. The master equation corresponding to discarded record doubles as a description of an open thermodynamic system. The fine-grained Shannon entropy is found to be a Lyapunov function (i.e. S˙≥0\dot S\geq0S˙≥0) of the dynamics when the record is discarded, providing a novel H-theorem suitable for studying the second law and non-equilibrium statistical physics. These findings may also be of interest to those working on the foundations of quantum mechanics, in particular along the lines of attempting to identify and unmix a possible epistemic component of quantum theory from its ontic content. More practically, these results may find applications in the fields of precision measurement, nanoengineering and molecular machines.
Quantum and classical stochastic evolution equations are derived for the statistical state of a system subject to continuous observation of a position variable. In the classical case this results in a stochastic Fokker - Planck equation for the conditional state of the system conditioned on observation records. In the quantum case a stochastic master equation results, or a stochastic Schrödinger equation if the system starts in a pure state. In both cases the resulting equations are nonlinear. A simple modification of the model results in a linear stochastic Schrödinger equation which can also be interpreted as a conditional evolution equation for observations which give no information about the measured system but indicate the size of the random perturbations induced by the environment coupling.
Abstract In quantum physics, a stochastic master equation (SME) estimates the state (density operator) of a quantum system in the Schrödinger picture based on a record of measurements made on the system. In the Heisenberg picture, the SME is a quantum filter. For a linear quantum system subject to linear measurements and Gaussian noise, the dynamics may be described by quantum stochastic differential equations (QSDEs), also known as quantum Langevin equations, and the quantum filter reduces to a so-called quantum Kalman filter. In this article, we introduce a quantum extended Kalman filter (quantum EKF), which applies a commutative approximation and a time-varying linearization to systems of nonlinear QSDEs. We will show that there are conditions under which a filter similar to a classical EKF can be implemented for quantum systems. The boundedness of estimation errors and the filtering problem with ‘state-dependent’ covariances for process and measurement noises are also discussed. We demonstrate the effectiveness of the quantum EKF by applying it to systems that involve multiple modes, nonlinear Hamiltonians, and simultaneous jump-diffusive measurements.
One calls attention to the fact that the stochastic physical systems are not random completely. They have both random and regular components of their evolution. Dynamic system is considered to be a special case of physical system with vanishing stochastic component of evolution. Mathematical technique for description of regular evolution component of physical systems (stochastic and deterministic) is constructed. The regular component of the system S evolution is described explicitly by the statistical average system , which is a continuous dynamic system. The action for is reduced to the form of the action for a set of interacting identical deterministic systems S_d. The form of interaction of S_d describes implicitly the character of the stochastic component of S evolution. Interplay between the physical system S and the statistical average system in the measurement process is discussed. There are at least two different kind of measurement: single measurements (S-measurements) connected with measurements in S and mass measurements (M-measurements) connected with measurements in . Conventional identification of S-measurement and M-measurement is a source of many misunderstandings and paradoxes.