Quantum mechanical weak values are an interference effect measured by the cross-Wigner transform of the post- and pre-selected states, leading to a complex quasi-distribution ρ ϕ, ψ ( x , p ) on phase space. We show that the knowledge of ρ ϕ, ψ ( z ) and of one of the two functions ϕ, ψ unambiguously determines the other, thus generalizing a recent reconstruction result of Lundeen and his collaborators.

arXiv:1112.5773Journal of Physics A Mathematical and TheoreticalMar 1, 201212 citations

arXiv:1109.3665 · Physics Letters A

Weak values of a quantum observable and the cross-Wigner distribution

Maurice A. de Gosson, Serge M. de Gosson

Published Dec 2, 2011• 21 citations

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DOI: 10.1016/j.physleta.2011.11.007

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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.