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Journal of Mathematical Physics

An application of filtering theory to parameter identification in quantum mechanics

Johan G. B. Beumee, HERSCHEL A. RABITZ

Published Aug 1, 1987• 1 citation
Publisher

Abstract

A method for inverting observations on quantum mechanical systems to obtain estimates of unknown parameters residing in the Hamiltonian is presented. The quantal system is represented in matrix form with respect to a chosen basis, and it is assumed that the associated expansion coefficients are truncated to a finite dimension. The uncontrollable laboratory noise will be modeled by means of an inhomogeneous white noise process so that the experimental observations are represented as stochastic variables satisfying a stochastic differential equation. It will be assumed that measurements obtained from an experiment are now equivalent to a realization of these stochastic variables. It is known from filtering theory that the minimum variance estimate of the unknown parameters in the quantal model is now given by the expectation of the unknowns conditional on this realization. This estimator can be calculated analytically from the associated a posteriori probability density if the original quantal system does not contain any random elements. This probability density for the unknown matrix elements is calculated, and it is demonstrated that for a full Hamiltonian matrix the asymptotic variance of the parameter estimator decreases as a third power in time and a fourth power in the initial conditions. Some differences with the minimum least-square method are mentioned, and a few issues of numerical implementation are discussed.

DOI
10.1063/1.527438

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A method for inverting observations on quantum mechanical systems to obtain estimates of unknown parameters residing in the Hamiltonian is presented. The quantal system is represented in matrix form with respect to a chosen basis, and it is assumed that the associated expansion coefficients are truncated to a finite dimension. The uncontrollable laboratory noise will be modeled by means of an inhomogeneous white noise process so that the experimental observations are represented as stochastic variables satisfying a stochastic differential equation. It will be assumed that measurements obtained from an experiment are now equivalent to a realization of these stochastic variables. It is known from filtering theory that the minimum variance estimate of the unknown parameters in the quantal model is now given by the expectation of the unknowns conditional on this realization. This estimator can be calculated analytically from the associated a posteriori probability density if the original quantal system does not contain any random elements. This probability density for the unknown matrix elements is calculated, and it is demonstrated that for a full Hamiltonian matrix the asymptotic variance of the parameter estimator decreases as a third power in time and a fourth power in the initial conditions. Some differences with the minimum least-square method are mentioned, and a few issues of numerical implementation are discussed.
An estimation method for unknown parameters in the initial conditions and the potential of a quantal system using the stochastic interpretation of quantum mechanics and some results in system theory are presented. According to this interpretation the possible trajectories of a particle through coordinate space may be represented by the realization of a stochastic process that satisfies a stochastic differential equation. The drift term in this equation is derived from the wave function and consequently contains all unknown parameters in the initial conditions and the potential. The main assumption of the paper is that a continuous sequence of position measurements on the trajectory of the particle can be identified with a realization of this stochastic process over the corresponding period of time. An application of the stochastic filtering theorems subsequently provides a minimum variance estimate of the unknown parameters in the drift conditional on this continuous sequence of measurements. As simple illustrations, this method is used to obtain estimates for the initial momentum of a free particle given measurements on its trajectory and to construct an estimator for the unknown parameters in a harmonic potential. It is shown that an optimal estimator exists if the stochastic processes are associated with a wave function from a potential of the Rellich type. In addition the a posteriori probability density of the parameters in the quantal system is calculated, assuming that all parameters involved prescribe a Rellich potential.
Continuous measurement of quantum systems gives rise to stochastic dynamics of the conditional quantum state, described by diffusive stochastic master equations. In this paper, we study parameter estimation for such equations when the Hamiltonian and measurement operators depend on unknown parameters. Based on multiple independent observed trajectories with a known initial state, we construct a contrast function using the deterministic averaged state and define a maximum contrast estimator for the unknown parameter. We prove strong consistency and asymptotic normality of the parameter estimator in a fixed-time, many-trajectory asymptotic regime. A key point is that the covariance matrix appearing in the asymptotic normality is given in a form that naturally leads to a consistent covariance estimator. This covariance estimator is computable from the observed data together with the deterministic averaged dynamics, so the asymptotic normality result can be used to construct standard errors and assess uncertainty for the parameter estimator.
The problem of estimating a member of a discrete-time vector process from past and present quantum mechanical measurements is considered; specifically, the minimum-variance linear estimator based on optimal present measurements is studied. Necessary and sufficient conditions that characterize the optimal processing matrix coefficients and the optimal measurements are discussed and interpreted. The optimal linear filter is compared to the optimal quantum estimator without postprocessing of past data. When the signal sequence is pairwise Gaussian and the optimal quantum measurement without postproccssing has the properties that it is linear in a specific sense and that its outcome and the corresponding element of the signal sequence are jointly Gaussian, then the optimal linear filter separates. That is, the optimal measurement can be taken to be the same as thc optimal measurement without regard to past data, and the past and present data are processed classically. Thc results are illustrated by considering the filtering of the in-phase and quadrature amplitudes of a laser field received in a single-mode cavity along with thermal noise. In this case, when the random signal sequence satisfies a linear recursion, the estimate can be computed recursively in a very efficient manner.
Precise identification of parameters governing quantum processes is a critical task for quantum information and communication technologies. In this Letter, we consider a setting where system evolution is determined by a parametrized Hamiltonian, and the task is to estimate these parameters from temporal records of a restricted set of system observables (time traces). Based on the notion of system realization from linear systems theory, we develop a constructive algorithm that provides estimates of the unknown parameters directly from these time traces. We illustrate the algorithm and its robustness to measurement noise by applying it to a one-dimensional spin chain model with variable couplings.
Multiparametric statistical model providing stable reconstruction of parameters by observations is considered. The only general method of this kind is the root model based on the representation of the probability density as a squared absolute value of a certain function, which is referred to as a psi-function in analogy with quantum mechanics. The psi-function is represented by an expansion in terms of an orthonormal set of functions. It is shown that the introduction of the psi-function allows one to represent the Fisher information matrix as well as statistical properties of the estimator of the state vector (state estimator) in simple analytical forms. A new statistical characteristic, a confidence cone, is introduced instead of a standard confidence interval. The chi-square test is considered to test the hypotheses that the estimated vector converges to the state vector of a general population and that both samples are homogeneous. The expansion coefficients are estimated by the maximum likelihood method. The method proposed may be applied to its full extent to solve the statistical inverse problem of quantum mechanics (root estimator of quantum states). In order to provide statistical completeness of the analysis, it is necessary to perform measurements in mutually complementing experiments (according to the Bohr terminology). The maximum likelihood technique and likelihood equation are generalized in order to analyze quantum mechanical experiments. It is shown that the requirement for the expansion to be of a root kind can be considered as a quantization condition making it possible to choose systems described by quantum mechanics from all statistical models consistent, on average, with the laws of classical mechanics.
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.