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I am reading this paper (open access) where, according to Figure 2, the authors claim pass a photonic mode through a Polarizing beam splitter (PBS), then measure the output using a avalanche photo-detector (APD). My understanding is that an APD simply detects the presence or absence of a photon. If we consider only the space spanned by the vacuum $|0\rangle$ and horizontally and vertically polarised photon states - $|H\rangle$ and $|V\rangle$ respectively- then we can write the action of the APD as a POVM $\{ |0\rangle \langle 0|,|H\rangle \langle H|+|V\rangle \langle V| \}$. In other words, the measurement projects into either the 'zero-photon subspace' or the 'one-photon subspace' but the measurement is not sensitive to the polarisation of the state.

A polarising beam splitter is a beam splitter which is sensitive to the polarisation of the light, transmitting horizontally polarised light and reflecting vertically polarised light (for example). I assume that passing the light through a PBS before measuring with the APD allows for phase-sensitive measurements.

My question is: what is the POVM which characterises this process (a PBS, followed by an APD)? Is it simply a measurement which can pick out the polarisation $\{|0\rangle \langle 0|,|H\rangle \langle H|,|V\rangle \langle V|\}$? How can the form of the POVM be derived?

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You can just list the possible outcomes of the setup, in this case $\{|0\rangle, |H\rangle,|V\rangle\}$, and write the corresponding projectors as you did. I would note that for this to be a POVM, i.e. in particular for the projectors to sum to the identity, you need this list to be an orthonormal basis for the space (whether this is such a basis depends on how you are defining things).

When some outcomes are not distinguished by the apparatus, like in the case without the PBS, you sum together the corresponding projectors, which is why you had $|H\rangle\!\langle H|+|V\rangle\!\langle V|$.

In summary, yes, the set you give is the correct POVM to consider in this case.

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