What’s cool here is that you are correlating the position state of each photon with its polarisation state, rather than measuring the position in way that irreversibly couples it with the environment.
After doing that, if the polarisation is inaccessible to you, you need to trace over it to get the effective density matrix, which turns it from a pure state into a mixture of position states.
But the polarisation isn’t really inaccessible, so the diagonal polariser can interact with it and transform the state back into a pure state with no definite position state.
