## Relation between Wigner quasi-probabability distribution and statistical second-moment

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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?

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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.

Thank you very much. It's exactly what I wanted. – Kianoosh.kargar – 2020-07-16T14:21:50.010

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