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arXiv:2112.13330 · IEEE Conference on Decision and Control (CDC)

A smoothing theory for open quantum systems: The least mean square approach

Kentaro Ohki

Published Dec 1, 2015• 8 citations
arXivHTMLPDF

Abstract

Unlike the classical smoothing theory, it is well known that quantum smoothers are, in general, not well-defined by the quantum conditional expectation. The purpose of this paper is to propose a new quantum smoothing theory based on the least mean squared errors. The least mean square estimate of quantum physical quantity composes from symmetric part and skew part, and we developed the recursive equations, respectively.

DOI
10.1109/cdc.2015.7402898

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Unlike the classical smoothing theory, it is well known that quantum smoothers are, in general, not well-defined by the quantum conditional expectation. The purpose of this paper is to propose a new quantum smoothing theory based on the least mean squared errors. The least mean square estimate of quantum physical quantity composes from symmetric part and skew part, and we developed the recursive equations, respectively.
Smoothing is a technique that is used to estimate the state of a system using measurement information both prior and posterior to the estimation time. Two notable examples of this technique are the Rauch-Tung-Striebel and Mayne-Fraser-Potter smoothing techniques for linear Gaussian systems, both resulting in the optimal smoothed estimate of the state. However, when considering a quantum system, classical smoothing techniques can result in an estimate that is not a valid quantum state. Consequently, a different smoothing theory was developed explicitly for quantum systems. This theory has since been applied to the special case of linear Gaussian quantum (LGQ) systems, where, in deriving the LGQ state smoothing equations, the Mayne-Fraser-Potter technique was utilized. As a result, the final equations describing the smoothed state are closely related to the classical Mayne-Fraser-Potter smoothing equations. In this paper, I derive the equivalent Rauch-Tung-Striebel form of the quantum state smoothing equations, which further simplify the calculation for the smoothed quantum state in LGQ systems, and I provide insight into the dynamics of the smoothed quantum state. This form of the LGQ smoothing equations brings to light a property of the smoothed quantum state that was hidden in the Mayne-Fraser-Potter form, namely the nondifferentiability of the smoothed mean. By identifying the nondifferentiable part of the smoothed mean, I am then able to derive a necessary and sufficient condition for the quantum smoothed mean to be differentiable in the steady-state regime.
State estimation is a classical problem in quantum information. In optimization of an estimation scheme, to find a lower bound to the error of the estimator is a very important step. So far, all the proposed tractable lower bounds use a derivative of the density matrix. However, sometimes, we are interested in quantities with singularity, e.g. concurrence etc. In this paper, lower bounds to a mean square error of an estimator are derived for a quantum estimation problem without smoothness assumptions. Our main idea is to replace the derivative by difference, as is done in classical estimation theory. We applied the inequalities to several examples, and derived an optimal estimator for some of them.
The inference of a hidden variable's historical value, based on observations before and after the fact, is a controversial subject in quantum mechanics. Here I address the controversy by proposing a formalism that unifies and generalizes some of the previous proposals for the task, including the quantum minimum-mean-square-error estimators proposed by Ohki, the generalized conditional expectation proposed by Accardi and Cecchini, the quantum smoothing theory proposed by Tsang, the optimal observables for parameter estimation proposed by Personick, Belavkin, and Grishanin, and the weak values proposed by Aharonov, Albert, and Vaidman. The formalism is based on Ohki's suggestion of a distance between two observables in the Heisenberg picture, which remains well defined for incompatible observables and serves as a more general foundation for quantum inference than Belavkin's nondemolition principle.
Smoothing is an estimation technique that takes into account both past and future observations and can be more accurate than filtering alone. In this Letter, a quantum theory of smoothing is constructed using a time-symmetric formalism, thereby generalizing prior work on classical and quantum filtering, retrodiction, and smoothing. The proposed theory solves the important problem of optimally estimating classical Markov processes coupled to a quantum system under continuous measurements, and is thus expected to find major applications in future quantum sensing systems, such as gravitational wave detectors and atomic magnetometers.
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 Itô calculus and its use in the modeling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics.