What are the measurement operators $F_k$ corresponding to a homodyne measurement?

By definition, a measurement is characterized by a set of positive-semidefinite matrices $$\{F_k\}$$ satisfying the completeness relation $$\sum_k F_k = \textbf{I}$$. I am interested in knowing how does the Homodyne measurement fit into this definition?

Edit: My understanding of Homodyne detection is no better than given in Wikipedia: homodyne detection is a method of extracting information encoded as modulation of the phase and/or frequency of an oscillating signal, by comparing that signal with a standard oscillation that would be identical to the signal if it carried null information. "Homodyne" signifies a single frequency, in contrast to the dual frequencies employed in heterodyne detection.

What are the corresponding $$\{F_k\}$$ operators for Homodyne detection?

1 Answer

One thing that I feel is worth noting is that there are two ways to understand what "measurement" means in this context.

On the one hand, you have "measurements" in the sense of POVMs. These are collections of positive semidefinite operators satisfying the normalisation condition you point out. These represent the most general way to perform some kind of measurement to the system, where "measurement" is understood as any procedure which gives as output an element of some set of possible outcomes. So, in particular, these do not give a number as output, but a stochastic outcome, with distribution given by the probabilities $$\operatorname{Tr}(F_k \rho)$$.

In other cases, one instead talk of a "measurement" in the sense of the computation of the expectation value of some observable. Here, the result of the measurement is a number. Practically speaking, to compute this expectation value you still have to perform a measurement in the former sense, via some POVM, and then perform post-processing on the collected data to obtain the final number. If the observable you consider is some Hermitian $$\mathcal O$$, then you can compute $$\langle\mathcal O\rangle$$ by doing a projective POVM on the eigenbasis of $$\mathcal O$$, and then attach numbers to each outcome according to the eigenvalues of $$\mathcal O$$.

A homodyne measurement is used to probe some properties of a quantum state of light. As such, the right question is what is the observable corresponding to it. The corresponding POVM description then just follows from that.

• Thanks, @glS. I think the corresponding observables are field quadratures $X=a+a^\dagger$ and $Y=-i(a-a^\dagger)$, but I'm still not sure how to connect it with POVM definition. Commented Dec 16, 2021 at 20:22
• @Micheal if for example the observable was $X=a+a^\dagger$, you'd need to find its eigenvectors, and from those you could get a corresponding POVM. However in these cases the operators are not finite-dimensionals, so there might be additional associated subtleties. I'm not that used to dealing with these cases, and I don't think I've ever seen these types of measurements dealt with the formalism of POVM, which is why I didn't go in more detail here.
– glS
Commented Dec 17, 2021 at 11:25