Weak Values and the Reconstruction Problem

Lars M. Johansen

Charlyne de Gosson, Maurice A. de Gosson

Closely associated with the notion of weak value is the problem of reconstructing the post-selected state: this is the so-called reconstruction problem. We show that the reconstruction problem can be solved by inversion of the cross-Wigner transform, using an ancillary state. We thereafter show, using the multidimensional Hardy uncertainty principle, that maximally concentrated cross-Wigner transforms corresponds to the case where a weak measurement reduces to an ordinary von Neumann measurement.

Lev Vaidman A

Yutaka Shikano

From the viewpoint of mathematics, quantum-mechanical probability seems to be controversial. To resolve this problem, we have reconstructed the Born rule using the weak value. We also discuss its meaning.

A. Matzkin

Maurice A. de Gosson, Serge M. de Gosson

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.