A note on the Pauli problem in light of approximate joint measurements

TL;DR

This paper demonstrates that certain phase space observables allow the reconstruction of position and momentum distributions from a single measurement, despite the overall state remaining undetermined, challenging traditional views on quantum measurement completeness.

Contribution

It introduces a class of informationally incomplete observables that still provide complete marginal distributions for position and momentum, offering new insights into quantum measurement theory.

Findings

Single observable can yield position and momentum distributions

State remains non-uniquely determined despite marginal completeness

Challenges assumptions about measurement completeness in quantum mechanics

Abstract

We show that there exist informationally incomplete phase space observables such that the Cartesian margins are informationally equivalent with position and momentum. This shows that it is possible to reconstruct the position and momentum distributions of a quantum system from the statistics of a single observable, and thus a single measurement, even though the state of the system is not uniquely determined by the statistics.