I want to understand the phase space formulation of quantum mechanics better. Specifically, I am considering the following situation:
A quantum state $\rho$ on two modes can be described by its Wigner function $W_{\rho}(q_1,p_1,q_2,p_2)$. In the standard formulation of QM, one can calculate the post-measurement state on the 2nd subsystem for some projective measurement of the 1st subsystem given a certain outcome. Note that the potential entanglement between the subsystems makes this non-trivial.
I would like to do the same in the continuous-variable case: Consider a heterodyne measurement of the 1st mode where the POVM elements are the projectors $\Pi_{\alpha}=|\alpha\rangle\langle\alpha|$. Here, $|\alpha\rangle$ is a coherent state, i.e. the heterodyne measurement is a projective measurement on coherent states. I am looking for an expression for the resulting Wigner function on the 2nd mode.
Note that I know how to do this only for the so called quadrature measurements of $\hat{q}, \hat{p}$. For instance, measuring $\hat{q_1}$ on the first mode with outcome $m$, I calculate the post-measurement Wigner function on the 2nd mode as $$\int dp_1 W_{\rho}(m,p_1,q_2,p_2)$$ However, I do not have the slightest idea how to generalize this to measuring arbitrary observables.