# Covariance matrix of the complete system when only one mode is heterodyned

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.

So the AA and AB blocks of your CM are indeed correc (i.e., $$\sigma_A'$$ and $$0$$).