# Calculating entropy in truncated Wigner

I'm trying to get some reasonable measure of the entropy of a system modelled by the truncated Wigner method. The Wigner function contains all the information about a density matrix. So, I figure it should be possible (but not necessarily easy) to calculate the von Neumann entropy: $$S=-\text{Tr}\left\{\hat{\rho}\log(\hat{\rho})\right\}$$ using a Wigner function $$W(x,p)$$. Looking at this document (section 2), it seems like the linear entropy (a linear expansion of von Neumann entropy about a pure state) can be calculated readily from the quasidistribution itself: $$S_2=1-\text{Tr}\left\{\hat{\rho}^2\right\}=1-(2\pi\hbar)\int dx\int dp \left(W(x,p)\right)^2$$ Suppose we apply the truncated Wigner method, in which one derives an equation of motion for $$W(x,p)$$, truncates higher-order terms based on a physical argument in order to get the equation of motion into Fokker-Planck form, and then stochastically unravels this Fokker-Planck equation. The observables of the system can be readily recovered from averages over the stochastic fields. But neither of these entropies are observables, even if they can be calculated from the full density matrix. Is there some method by which one or both of these entropies could be calculated from the stochastic fields in truncated Wigner?

This might be useful. Indeed, the impurity you write, $$S_2=1-h\langle W\rangle= \int\! dx dp ~(W-hW^2)$$ is exact, by an exceptional feature of the Wigner representation, and it vanishes only for pure states, thus quantifying departures from them.
It is the leading h expansion of the full quantum entropy, the exact transcription of the von Neumann entropy in phase space, $$S=- \langle \ln ~\hat \rho \rangle= -\int \! dx dp ~ W \ln _\star (hW)= \sum_{n=1}^\infty \frac{1}{n}\langle (1-hW)^n_\star\rangle ,$$ using a common notation for the star-logarithm and star-powers. (Only the leading, n =1, power lacks a star the the concomitant extra h.)