The reconstruction problem and weak quantum values

Quantum mechanical weak values are an interference effect measured by the cross-Wigner transform {W}(\phi ,\psi ) 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.