$$V^{\rho}=S^{\rho}\left(  D^{\rho}\oplus D^{\rho}\right)  \left(  S^{\rho
}\right)  ^{T},$$ where $S^{\rho}$ is a symplectic matrix and $D^{\rho}\equiv\operatorname{diag}(\nu_{1},\ldots,\nu_{m})$ is a diagonal matrix of symplectic eigenvalues such that $\nu_{i}\geq1$ for all $i\in\left\{  1,\ldots,m\right\}  $. We say that a quantum Gaussian state is faithful if all of its symplectic eigenvalues are strictly greater than one (this also means that the state is positive definite). Faithfulness of Gaussian states is required to ensure that $G^{\rho}$ is non-singular. We can write the density operator $\rho$ of a faithful state in the following exponential form [@PhysRevA.71.062320; @K06; @H10] (see also [@H13book; @S17]): $$\begin{aligned}
&  \rho=(Z^{\rho})^{-1/2}\exp\left[  -\frac{1}{2}(\hat{x}-\mu^{\rho})^{T}G^{\rho}(\hat{x}-\mu^{\rho})\right]  ,\label{eq:exp-form}\\
&  \mathrm{with}\quad Z^{\rho}\equiv\det(\left[  V^{\rho}+i\Omega\right]
/2)\\
&  \mathrm{and}\quad G^{\rho}\equiv-2\Omega S^{\rho}\left[
\operatorname{arcoth}(D^{\rho})\right]  ^{\oplus2}\left(  S^{\rho}\right)
^{T}\Omega, \label{eq:G_rho}$$ where $\operatorname{arcoth}(x)\equiv\frac{1}{2}\ln\!\left(  \frac{x+1}{x-1}\right)  $ with domain $\left(  -\infty,-1\right)  \cup\left(
1,+\infty\right)  $. Note that we can also write$$G^{\rho}=2i\Omega\operatorname{arcoth}(iV^{\rho}\Omega),
\label{eq:more-compact-G-rho}$$ so that $G^{\rho}$ is represented directly in terms of the covariance matrix $V^{\rho}$. By inspection, the $G$ and $V$ matrices are symmetric. In what follows, we adopt the same notation for quantities associated with a density operator $\sigma$, such as $\mu^{\sigma}$, $V^{\sigma}$, $S^{\sigma}$, $D^{\sigma}$, $Z^{\sigma}$, and $G^{\sigma}$.

A two-mode Gaussian state $\rho$ with covariance matrix in standard form has a covariance matrix as follows [@DGCZ00; @S00]:$$V^{\rho}=\begin{bmatrix}
a & c\\
