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

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.