Quantum Computing Asked by Kianoosh.kargar on August 20, 2021

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?

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.

Correct answer by keisuke.akira on August 20, 2021

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