kurt101 said:
What I mean by beam splitting polarizers is a calcite crystal that separates light into two plane polarized beams.
Ok, that helps a lot. "Beam splitter" means the kind of thing that's in the MZI. "Calcite crystal" is much better. However, as you will see below, in many respects what it does is very similar to what the beam splitter in the MZI does.
The two beams coming out of a calcite crystal still go in different directions, so you would still need mirrors or something similar to bring them back together in a second crystal. This "recombining" operation would end up outputting a single beam from the second crystal that was identical to the input beam coming into the first crystal, if no other operations were done in between to either beam, just as with the MZI, the single output beam from the second beam splitter (or "recombiner") is identical to the original input beam to the first splitter, if no other operations were done in between.
kurt101 said:
it is my understanding that the beam splitter in the MZI splits the incoming light randomly (if you were to try and detect which way the photon went)
If there is just a single input photon, then yes, the beam splitter normally used in the MZI would have a 50-50 chance of having a photon detected in either output beam, if you were running that experiment. However, it's extremely difficult to produce single photons. Usually the MZI is run with a beam of light in the usual sense, i.e., a continuous beam of light of a certain intensity; in that case, each output beam has half the intensity of the input beam.
kurt101 said:
where as the crystal splitter splits the light based on its polarization.
Yes, and if an unpolarized beam is input into the crystal, each output beam has half the intensity of the input beam, just as with the beam splitter in the MZI. Also, if you were to use a (very difficult) single photon source to fire a single photon into the crystal, and the single photon were unpolarized, there would be a 50-50 chance of detecting the photon in either output beam, just as with the beam splitter in the MZI.
The main
difference with a calcite crystal as compared to the beam splitters used in the MZI is that it is
possible to prepare a light source to produce a beam that will
not be split by the crystal--just prepare the source to produce light in one of the two polarization eigenstates that the crystal separates (for example, +45 or -45). AFAIK there is no way to prepare such a source for the MZI beam splitter--any single input beam will be split into two.