Relation between Wigner quasi-probability distribution and statistical second-moment

Question

Is there any relation between the Wigner quasi-probability distribution function $W$ and the statistical second-moment (also known as covariance matrix) of a density matrix of a continuous variable state, such as Gaussian state?

Answer 1

You mean something like $$W_{G}(\mathbf{r}) =\frac{2^{n}}{\pi^{n} \sqrt{\operatorname{Det} \sigma}} \mathrm{e}^{-(\mathbf{r}-\overline{\mathbf{r}})^{\top} \boldsymbol{\sigma}^{-1}(\mathbf{r}-\overline{\mathbf{r}})},$$ where $W_{G}(\mathbf{r})$ is the Wigner function corresponding to a Gaussian state, $\mathbf{\sigma}$ its covariance matrix, and $\overline{r}$ the vector of first moments?

If yes, then, see, for example, Eqn. (4.50) of Quantum Continuous Variables.