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Journal of Applied Mathematics and Physics

Optimal Estimate of Quantum States

Mario Mastriani

Published Jan 1, 2018• 4 citations
Publisher

Abstract

An optimal estimator of quantum states based on a modified Kalman’s filter is proposed in this work. Such estimator acts after a state measurement, allowing us to obtain an optimal estimate of the quantum state resulting in the output of any quantum algorithm. This method is much more accurate than other types of quantum measurements, such as, weak measurement, strong measurement, and quantum state tomography, among others.

DOI
10.4236/jamp.2018.66114

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An optimal estimator of quantum states based on a modified Kalman’s filter is proposed in this work. Such estimator acts after a state measurement, allowing us to obtain an optimal estimate of the quantum state resulting in the output of any quantum algorithm. This method is much more accurate than other types of quantum measurements, such as, weak measurement, strong measurement, and quantum state tomography, among others.
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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.
The Quantum Kalman Filter (QKF) emerges as an innovative approach to address the challenges of state estimation in quantum systems, adapting classical Kalman filtering to the complexities of quantum mechanics. This paper examines the theoretical framework, practical implementation hurdles, and potential applications of QKF in quantum information processing and sensing. By incorporating quantum measurement principles and stochastic dynamics, QKF offers a solution for optimal state estimation in noisy quantum environments. The study presents simulation results and comparative analyses with classical Kalman filtering for tracking single targets, underscoring the advantages and limitations of QKF in advancing quantum technology applications.
Continuous-time measurements are instrumental for a multitude of tasks in quantum engineering and quantum control, including the estimation of dynamical parameters of open quantum systems monitored through the environment. However, such measurements do not extract the maximum amount of information available in the output state, so finding alternative optimal measurement strategies is a major open problem. In this paper we solve this problem in the setting of discrete-time input-output quantum Markov chains. We present an efficient algorithm for optimal estimation of one-dimensional dynamical parameters which consists of an iterative procedure for updating a `measurement filter' operator and determining successive measurement bases for the output units. A key ingredient of the scheme is the use of a coherent quantum absorber as a way to post-process the output after the interaction with the system. This is designed adaptively such that the joint system and absorber stationary state is pure at a reference parameter value. The scheme offers an exciting prospect for optimal continuous-time adaptive measurements, but more work is needed to find realistic practical implementations.
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.