The Gaussian state of two modes, with quadrature operators $X_1,P_1,X_2,P_2$, is given by a displacement vector $d$ and covariance matrix

$\sigma = \begin{bmatrix} Var(X_1,X1) & Var(X_1,P_1) & Var(X_1,X_2) & Var(X_1,P_2) \\ Var(P_1,X1) & Var(P_1,P_1) & Var(P_1,X_2) & Var(P_1,P_2) \\ Var(X_2,X1) & Var(X_2,P_1) & Var(X_2,X_2) & Var(X_2,P_2) \\ Var(P_2,X1) & Var(P_2,P_1) & Var(P_2,X_2) & Var(P_2,P_2)\end{bmatrix},$

$ Var(U,V) = \frac{1}{2}\langle UV + VU\rangle - \langle U\rangle\langle V\rangle.$

A given quadrature ($X_2$ or $P_2$) of mode $2$ is measured by a homodyne detector. How do I calculated the displacement vector and the covariance matrix of mode $1$ after the measurement? I will appreciate a worked out answer. Bonus: answer for $\cos\theta X_2 + \sin\theta P_2$?

How does the covariance matrix of mode $1$ change if mode $1$ is electro-optically modified by the measured photocurrent $i$ i.e. $X_1 \to X_1 + g i$, where $g$ is some gain?

Lastly, if the homodyne measurement is inefficient can this be modelled by placing a fictitious beamsplitter before an ideal homodyne detector and discarding the ancilla mode?

Assume that this is not a single-shot experiment, rather the preparation, partial measurement on mode $2$ and measurement on mode $1$ is done many times over and the covariance matrix is reconstructed from the results of the measurements on mode $1$.

This post imported from StackExchange Physics at 2016-06-12 10:16 (UTC), posted by SE-user Abdullah Khalid