The Reconstruction Problem and Weak Quantum Values

TL;DR

This paper explores the reconstruction of quantum states using weak values and phase space quasi-distributions, extending previous results to show that certain measurements uniquely determine quantum states.

Contribution

It generalizes a recent quantum state reconstruction method by demonstrating that phase space distributions and one of the pre- or post-selected states uniquely determine the other.

Findings

Reconstruction of quantum states from phase space distributions.

Extension of previous state reconstruction results.

Unique determination of states from weak value measurements.

Abstract

Quantum Mechanical weak values are an interference effect measured by the cross-Wigner transform W({\phi},{\psi}) of the post-and preselected states, leading to a complex quasi-distribution {\rho}_{{\phi},{\psi}}(x,p) on phase space. We show that the knowledge of {\rho}_{{\phi},{\psi}}(z) and of one of the two functions {\phi},{\psi} unambiguously determines the other, thus generalizing a recent reconstruction result of Lundeen and his collaborators.