Inequivalent ways to apply semi-classical smoothing to a quantum system

In this paper, we correct a mistake we made in [Phys. Rev. Lett. $\textbf{122}$, 190402 (2019)] and [Phys. Rev. A $\textbf{103}$, 012213 (2021)] regarding the Wigner function of the so-called smoothed Weak-Valued state (SWV state). Here smoothing refers to estimation of properties at time $t$ using information obtained in measurements both before and after $t$. The SWV state is a pseudo-state (Hermitian but not necessarily positive) that gives, by the usual trace formula, the correct value for a weak measurement preformed at time $t$, $\textit{i.e.}$, its weak value. The Wigner function is a pseudo-probability-distribution (real but not necessarily positive) over phase-space. A smoothed (in this estimation sense) Wigner distribution at time $t$ can also be defined by applying classical smoothing for probability-distributions to the Wigner functions. The smoothed Wigner distribution (SWD) gives identical means for the canonical phase-space variables as does the SWV state. However, contrary to the assumption in the above references, the Wigner function of the SWV state is not the smoothed Wigner distribution.