Covariance Matrices and Standard form

[Ken Gallock]

Hi.
I have a question about covariance matrices (CMs) and a standard form.
In Ref. [Inseparability Criterion for Continuous Variable Systems], it is mentioned that CMs $M$ for two-mode Gaussian states can be symplectic transformed to the standard form $M_s$:
$M= \left[ \begin{array}{cc} G_1 &C \\ C^\top &G_2 \end{array} \right]\rightarrow M_s=\left[ \begin{array}{cc} nI_2 & C_s \\ C_s^\top &mI_2 \end{array} \right],$
where $C_s=\mathrm{diag}~(c_1, c_2)$.
I want to know whether or not the CMs for four-mode Gaussian states can be transformed to the standard form just like the two-mode case (every $2\times 2$ block matrices are diagonalized). I found an article saying "there does not generally exist a standard form for Gaussian states involving more than two-modes." (Ref. [Se-Wan])

 

[azntoon]

Is this still true?Thank you. Yes, it is still true that there does not generally exist a standard form for Gaussian states involving more than two modes. The standard form for two-mode Gaussian states is a special case due to the fact that two-mode states have fewer parameters than four-mode or higher-mode states. As such, it is not possible to transform four-mode or higher-mode Gaussian states into a standard form in the same way as two-mode states.