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.


1 Answer 1


The one thing you can say for sure is what state system A is in. (And I understand you know the corresponding expression, see e.g. also Post-measurement state after homodyne measurements of part of the system).

The other thing you can say for sure is that the correlations between system A and B or gone.

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

Now what state system B is in depends on how your measurement is implemented. While the "by the book" definition of measurement in QM is that the system remains in the state you measure it in (in which case, the B system should have the CM you measured), this is really not true in many scenarios. E.g., if you detect a photon in a photon detector, it is simply gone. There is no post-measurement state (or even system), as the measurement is destructive.

So in order to say sth. meaningful about the post-measurement state on B, you have to know about the mechanism through which the measurement is implemented. If you don't the only safe statements you can make are about system A.


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