Observing Dirac's classical phase space analog to the quantum state

TL;DR

This paper experimentally measures Dirac's quasi-probability distribution for a photon's transverse quantum state, demonstrating its classical-like properties and direct observability through weak measurements.

Contribution

It provides the first experimental observation of Dirac's distribution for a quantum state using weak measurement techniques.

Findings

Dirac's distribution can be directly observed in experiments.

The distribution transforms according to Bayes' law, showing classical-like behavior.

The method applies to the transverse degree of freedom of photons.

Abstract

In 1945, Dirac attempted to develop a "formal probability" distribution to describe quantum operators in terms of two non-commuting variables, such as position x and momentum p [Rev. Mod. Phys. 17, 195 (1945)]. The resulting quasi-probability distribution is a complete representation of the quantum state and can be observed directly in experiments. We measure Dirac's distribution for the quantum state of the transverse degree of freedom of a photon by weakly measuring transverse x so as to not randomize the subsequent p measurement. Further, we show that the distribution has the classical-like feature that it transforms (e.g., propagates) according to Bayes' law.