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Obviously we don't understand each other. What I always stress is that the "paths" are indistinguishable if you have entangled photon pairs. The point is that you choose the pairs, where you can't know which photon ("labelled" by its momentum) has which polarization (H and V), as becomes very clear from this figure from Wikipedia's article on SPDC:DrChinese said:Once again, not a single word of your response has the slightest connection to what I said. Which was:
"There is NO requirement that entangled photons are indistinguishable for them to be polarization entangled. For example, they can be from different PDC sources - that would distinguish them nicely.
https://arxiv.org/abs/0809.3991
CHSH measured at S=2.37 (where S<2 is required by local realism)."
Taking the photons with momenta in the directions determined by the intersection of the two cones makes it impossible to know the "specific paths" of the H-polarized and the V-polarized photon. That's why you have a superposition of two paths and that's why you have the entanglement, discussed in the experiments, including the one about entanglement swapping in the paper you quote.
It has all to do with the quantum nature of how you treat the photons 2 and 3, enabling to "project" to either of the four possible Bell states. Again it's crucial that there is no determined paths at the moment you detect these two photons after running through the PBSs to get the entanglement of photons 1 and 4 when looking at either of the four subensembles, where photons 2 and 3 are selected to be in either of the corresponding Bell states.DrChinese said:How the entangled pair 1 & 4 got entangled is not really relevant to my point, it's just another of your rabbit trails. But it certainly has nothing to do with the quantum nature of the paths of 2 and 3 (and yes, they are quantum objects). In fact, they too trace an almost perfect classical path to the Bell State Analyzer in the experiment. Of course, as I get tired of saying, entangled photons are not classical particles. And in fact their very entanglement is proof of same, with S=2.37 and therefore ruling out local realism (where S must be less than 2).
The indistinguishability of the two "paths" is the key feature of the entanglement of the two photons we discuss here.DrChinese said:So I will repeat: Indistinguishability is not necessary for entanglement, in contradiction to your statement. And in general, not referring to photons at all, entanglement is not even required to be between like quantum objects. Different quantum objects, of course, being very distinguishable as well.