3
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ Thank you very much. It's exactly what I wanted. $\endgroup$ Jul 16 '20 at 14:21
  • 1
    $\begingroup$ Great, can you please accept this answer then? $\endgroup$ Jul 16 '20 at 15:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.