Quantum Rauch-Tung-Striebel smoothed state
Kiarn T. Laverick
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.
arXiv:2010.11027•Physical Review Research•Aug 27, 2021•4 citations