Inequivalent ways to apply semiclassical smoothing to a quantum system

Abstract

In this paper, we correct a mistake we made in [Phys. Rev. Lett. 122, 190402 (2019)] and [Phys. Rev. A 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 pseudostate (Hermitian but not necessarily positive) that gives, by the usual trace formula, the correct value for a weak measurement preformed at time t, 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 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.