The Quantum Rauch-Tung-Striebel Smoothed State

Smoothing is a technique that estimates 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 utilised. 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. Additionally, the new form of the LGQ smoothing equations bring to light a property of the smoothed quantum state that was hidden in the Mayne-Fraser-Potter form, the non-differentiablilty of the smoothed mean. By identifying the non-differentiable part of the smoothed mean, I was then able to derive a necessary and sufficient condition for the quantum smoothed mean to be differentiable in the steady state regime.