Let's say we have a two mode Gaussian state $A + B$ characterized by the covariance matrix $\sigma$ (in block matrix form:):
$$\sigma = \begin{pmatrix} \sigma_A & \sigma_{AB} \\ \sigma_{AB}^T & \sigma_B \end{pmatrix}$$
And the first moments $\textbf{r}$:
$$\textbf{r} = \begin{pmatrix} \textbf{r}_A \\ \textbf{r}_B \end{pmatrix}$$
Let's say we heterodyne the mode $B$. Upon heterodyning, the subsystem $A$ will collapse into some state with the moments $\sigma_A^{'}$ and $\textbf{r}_A^{'}$, with the outcome of the heterodyne measurement being $\alpha$.
My question is: What is the covariance matrix of the overall two-mode system? Is it:
$$\sigma = \begin{pmatrix} \sigma_A^{'} & 0 \\ 0 & \frac{1}{2} \textit{I}_2 \end{pmatrix}$$
$$\textbf{r} = \begin{pmatrix} \textbf{r}_A^{'} \\ \textbf{r}_{\alpha} \end{pmatrix}$$
My reasoning for chosing the above form is as follows: The subsystem $B$ will collapse to some coherent state $|\alpha \rangle$ upon the measurement outcome $\alpha$. Hence the block for $B$ is just $\frac{\textit{I}_2}{2}$ (which corresponds to the coherent state). $\textbf{r}_{\alpha}$ is the first moment vector for $|\alpha \rangle$. The cross terms are zero since, after measurement, there will be no entanglement in the system.
