The Wigner function is recovered exactly as the expectation of signed weights times delta functions along classical phase-space trajectories, with the weights carrying all non-classical corrections from the Moyal bracket.
Deffner, Quantum entropy production in phase space, Europhysics Letters103, 30001 (2013), arXiv:1307.3183 [quant-ph]
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Weighted Phase-Space Paths for Exact Wigner Dynamics
The Wigner function is recovered exactly as the expectation of signed weights times delta functions along classical phase-space trajectories, with the weights carrying all non-classical corrections from the Moyal bracket.