Beam SplitterTeleprompter

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I have been trying to understand this heralded CNOT gate on photons but I have some questions regarding polarizing beam splitters:

Beam splitterworking principle

However, this does not destroy any superposition. Let's say you had an incoming photon in the state $$ \alpha|H\rangle+\beta|V\rangle. $$ The PBS changes this state into $$ \alpha|H\rangle|u\rangle+\beta|V\rangle|l\rangle, $$ where $u$ and $l$ denote two different paths that the photon is on ("upper" and "lower" being the origin of the letters). It is still very much in superposition, but it means that you can measure the path information (i.e. put detectors on the two paths) to find out the information about the state of the photon.

Polarizingbeam splitterprinciple

Your understanding is basically right. A polarising beam splitter sends the photon in one of two different directions based on what polarisation it has. And, yes, once it is travelling along a specific direction, it has that specific polarisation.

The way I understand it, a PBS lets the photon through if the photon is polarized a certain direction, but as I understand it this would squeeze the photon back into a distinct polarization (H/V in the case of the top PBS and D/A in the case of the bottom PBS). This of course is wrong, but I don't understand how the photon passes through without destroying the superposition. This part is the most confusing part of the circuit but even if the PBS doesn't destroy the superposition how does the PBS2 create the superposition from the target. What does the operation matrix for a PBS filter look like? What is a good way to understand them?