Okay, I have the following scenerio (attached picture).
The inputs for path b and c are two photons. both photons aren't entangled with each other - they're entangled to other photons. Before the beam splitter there is a wave plate that converts the photon into the 45/135 basis.
So what we end up with is taking each basis state and add it to each of the other basis states of the other pair of entangled photon when figuring out what happens at the beam splitter.
I'm a bit confused on how to work it out. Because the photons remain entangled with their original partners, how do we write the overall state after the BS?
Take |H>|45> for path b and combine it with -|V>|45> on path c. Combine only the |45>'s at the beam splitter. So we end up with -|45>i|45>(b) + -i|45>|45>(c)?