  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.
