Post measurement state after heterodyne measurement

Question

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.

Answer 1

This is a companion commentary to this question. It appears your confusion is on framing, and not technicalities. You are essentially asking, for one mode, $q_1,p_1$, how do you represent projective measurements in phase space, as you leave the other mode, $q_2,p_2$, alone, reducing to a reduced state.

Let's focus on the case you know. In Hilbert space, the reduced post-measurement "density matrix" is $$ \operatorname{Tr}_1 (\Pi_m |\psi\rangle\langle \psi | ), $$ where $\Pi_m=|x=m\rangle \langle x=m| $, for a position measurement in tensor factor 1, or else what you wish, like the coherent-state projector, in general, in whatever variables you choose. You are computing the expectation value of some projector. The outcome will be a reduced state, partial density matrix $\rho_2$

The Wigner map translates this invertibly to phase space, \begin{align} |\psi\rangle \langle \psi| \qquad &\mapsto \qquad h~W_\psi (q,p) \\[1em] \Pi_m=|m\rangle \langle m| \qquad &\mapsto \qquad \delta (q-m) \\[1ex] \operatorname{Tr} ( \Pi_m |\psi\rangle\langle \psi | ) \qquad &\mapsto \qquad \int\!\! \mathrm dq \,\mathrm dp ~ \delta (q-m) W_\psi (q,p) = \int \mathrm dp ~ W_\psi(m,p) , \end{align} as you already know. (See eqn. (137) of this booklet of ours.) For a partial trace, your just integrating the subset of relevant phase-space variables, and you are left with a reduced Wigner function $W_2(q_2,p_2)$, corresponding to a reduced density matrix.

Proceed to apply this, mutatis mutandis to any and all projectors of your choice, whose Wigner transform is found in the standard fashion, as for the coherent state one, $\Pi_\alpha$, in the answer linked, in any variable you choose, integrating over the (partial trace) particular variables you choose, $\operatorname{Tr} (\Pi_\alpha |\psi\rangle\langle \psi |)=\int\!\! \mathrm dq \mathrm dp ~ W_\alpha (q,p) W_\psi (q,p)$.