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Approximate particle number distribution from direct stochastic sampling of the Wigner function

We consider the Wigner quasi-probability distribution function of a single mode of an electromagnetic or matter-wave field to address the question of whether a direct stochastic sampling and binning of the absolute square of the complex field amplitude can yield a distribution function $\tilde{P}_n$ that closely approximates the true particle number probability distribution $P_n$ of the underlying quantum state. By providing an operational definition of the binned distribution $\tilde{P}_n$ in terms of the Wigner function, we explicitly calculate the overlap between $\tilde{P}_n$ and ${P}_n$ and hence quantify the statistical distance between the two distributions. We find that there is indeed a close quantitative correspondence between $\tilde{P}_n$ and $P_n$ for a wide range of quantum states that have smooth and broad Wigner function relative to the scale of oscillations of the Wigner function for the relevant Fock state. However, we also find counterexamples, including states with high mode occupation, for which $\tilde{P}_n$ does not closely approximate $P_n$.