How to find the covariance matrix after a partial homodyne measurement?

Question

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 $1$.

Answer 1

This is done in several papers, e.g. here: http://cds.cern.ch/record/546624/files/0204052.pdf

The relevant portion is Lemma 1. Given a state $\rho$ with covariance matrix in block form

$$ \gamma_{\rho}=\begin{pmatrix}{} A & B \\ B^T & C \end{pmatrix} $$

the covariance matrix after a measurement after a projection onto the pure Gaussian state with covariance matrix $\gamma_d=\operatorname{diag}(d,1/d)$ is given by

$$ \gamma^{\prime}_{\rho}=A-B(C+\gamma_d^2)^{-1}B^T $$

This can be related to the Schur complement of $A$ in $\gamma_{\rho}$.

Homodyne detection is then the limit of projection into an infinitely squeezed state (mentioned in the article one page later). If you do the math, you will find the covariance matrix after measurement to be:

$$ \gamma^{\prime}_{\rho}=A-B(PCP)^{+}B^T $$

where $P=\operatorname{diag}(1,0)$ is a projection and $^+$ denotes the Moore-Penrose pseudoinverse.

In order to cover the "inefficient" part, I'd have to know more about what you mean by "inefficient" (how is this modelled?). However, it seems likely that you can somehow model it in the way you want to.