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.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.

arXiv:2505.00341Physical Review AJul 21, 20252 citations

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.