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