Quantum Rauch-Tung-Striebel smoothed state

Kiarn T Laverick

doi:10.1103/physrevresearch.3.033196

Abstract

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.

Highlights

Estimating an unknown state, i.e., a probability density function (PDF), of physical systems using indirect measurement results has been studied in great depth [1,2,3,4,5,6,7,8,9,10,11,12,13]
One advantage of the RTS form of the linear Gaussian quantum (LGQ) state smoothing equations is that there is no need to compute the retrofiltered effect, and we only have to focus on computing the necessary quantities to obtain the smoothed state
While this does not reduce the overall number of differential equations that need to be solved to compute the smoothed quantum state, it does remove the extra step of computing the smoothed mean and covariance using the MFP equations

Summary

INTRODUCTION

Estimating an unknown state, i.e., a probability density function (PDF), of physical systems using indirect measurement results has been studied in great depth [1,2,3,4,5,6,7,8,9,10,11,12,13]. When restricting to the case of (classical) linear Gaussian (LG) systems, Kalman and Bucy [1] developed an optimal estimation technique, known as filtering, which conditions the estimate of the state on past measurement information, i.e., measurement information up until the time of estimation τ. One such technique was developed soon after the KalmanBucy filtering theory by Rauch, Tung, and Striebel [2,3] This technique, referred to as smoothing, utilizes the past measurement information as the Kalman-Bucy filter does, but it uses information gathered after the estimation time τ , i.e., the “future” measurement record, to provide a more accurate estimate of the state. I identify a necessary and sufficient condition for the mean of the smoothed quantum state to be differentiable in the steady-state regime

Classical

Quantum

DERIVING THE QUANTUM RAUCH-TUNG-STRIEBEL SMOOTHED STATE

DIFFERENTIABILITY OF THE QUANTUM SMOOTHED MEAN

CONCLUSION