We report the direct continuous-in-phase sampling of a regularized
P function, the so-called nonclassicality quasiprobability, for squeezed light. Through their negativities, the resulting phase-space representation uncovers the quantum character of the state. In contrast to discrete phase-locked measurements, our approach allows an unconditional verification of nonclassicality by getting rid of interpolation errors due to fixed phases. To realize the equal phase distribution of measured quadratures, a data selection is implemented with quantum random numbers created by measuring the vacuum noise. The continuously measured squeezed field was generated in an optical parametric amplifier. Suitable pattern functions for obtaining the regularized
P function are investigated. The significance of detecting negativities in our application is determined. The sampling of nonclassicality quasiprobabilities is shown to be a powerful and universal method to visualize quantum effects within arbitrary quantum states.
We consider quantum phase-space dynamics using Wigner's representation of quantum mechanics. We stress the usefulness of the integral form for the description of Wigner's phase-space current $\mathbit{J}$ as an alternative to the popular Moyal bracket. The integral form brings out the symmetries between momentum and position representations of quantum mechanics, is numerically stable, and allows us to perform some calculations using elementary integrals instead of Groenewold star products. Our central result is an explicit, elementary proof which shows that only systems up to quadratic in their potential fulfill Liouville's theorem of volume preservation in quantum mechanics. Contrary to a recent suggestion, our proof shows that the non-Liouvillian character of quantum phase-space dynamics cannot be transformed away.
It is shown that phase-space distribution functions that characterize a single-particle quantum system obey a certain "generalized non-negativity condition," which reflects the fact that the density operator is a positive operator. A corresponding criterion is obtained for the associated characteristic functions and is found to resemble, to some extent, Bochner's theorem of classical probability theory. Necessary and sufficient conditions on a phase-space representation of quantum mechanics are also derived, which ensure that all the possible distribution functions in that representation are non-negative; but it is also shown that such distribution functions are not joint probabilities for position and momentum. In fact, our results readily provide a new proof of theorems of Wigner, and of Cohen and Margenau, which imply that quantum mechanics cannot be formulated as a stochastic theory in phase space.