This thread is created to post a compilation of all relevant current mainstream explanations of this phenomenon, and reasonable speculation can also be included.
The mainstream explanation involves a tensor algebra over \( \mathbb C \times \mathbb C \). And of course the concept of measurement. Quantum entanglement means having to rethink what measurement is, pretty much. Otherwise, there isn't any need to speculate.
The three "pillars" of Quantum Information Science (capitalised so it looks impressive), are superposition of states, interference, and entanglement. You can show that they are the same thing, or that we (only) measure correlations (always classical) in three apparently different ways. Unfortunately, showing this involves a lot of complex mathematics. One plus for QIS is that you don't really need to know QM in its full glory, but you do need to know how to describe the above three phenomena, and that means being able to understand bra-ket notation (because everyone uses it). You should also be able to set up actual experiments or describe them mathematically. Otherwise it would be like trying to get a qualification in Computing without using any programming language or writing any code. You can be an accomplished coder without needing to understand the underlying computer architecture, likwise the "coding" aspect of QM experiments/computations is very similar, the underlying physics isn't something you need to have a deep understanding of.
One of the weird predictions (verified by experiment) is that wavefunctions can split into spatially separate parts. I haven't mentioned this "weirdness" in the posts about density matrices and the Hong-Ou-Mandel effect, but it happens. To describe this more succinctly, consider a beam of photons that interacts with a half silvered mirror (50:50 beam-splitter) at an angle of 45° which has two fully silvered mirrors in each outgoing path also at 45° so the outgoing beams from the splitter are recombined at a second half silvered mirror (also at 45° to the beams). When single photons propagate through this system, they apparently take both paths, you can show this must be true by blocking one of the paths (say with a detector).
Yes. I have shown each photon does go by both paths, without blocking either, when I measured the length of some photons. Below is part of post http://www.sciforums.com/threads/proof-of-the-existence-of-god.144082/page-40#post-3317138 (years earlier posts exist, but search will not find them.) I have measured the length of some photons and shown one photon can in a classical sense be far (four feet) from itself! Here is how you do that: Posting now below a crude "typed" drawing (in two parts): Extended light source and lens making parallel beams (0nly one shown below) but each part of the source makes a beam at very slightly different angles: * * *...............................................................()===== This beam enters beam splitter "a" shown below (this part of drawing separated for ease of construction.) * * (If I made lens () taller then parts of the light source, represented by some * , would be too far above or below lens.) Below is one of the slightly divergent beams (only one shown), leaving the lens and going to first 45 degree beam splitter "a" and going straight thru with part (of same photon) going up to hit 45 degree mirror, b, too, which makes it again traveling parallel to the entering beam. Sorry that these beam splitters and mirrors are not shown actually at 45 degrees - but that is best a "typed drawing" can do. .........................................................d ..........................b/======/======.....This is the path of "self- rejoined" photon to the screen thru another lense one focal length from it. ............................||..........................|| ............................||..........................|| ............................||..........................|| ............................||..........................|| ()===== / ======/c Lens.....................a Optically an "extended source" with lens one focal lengh from it followed by a second lense one focal length from the screen, just images the source (up side down) on the screen. Inserting these beam splitters and mirrors does not change that. It only make it possible for slightly differing path "split photons" to arrive at the screen where they would have but now they "want" to get back in phase with them selves, and do so as best a they can. Leaving dark lines where if they can not become "particles" there as they would not exist there since their waves are 180 degrees out of phase with themselves there. Note almost all the time photons can be thought of as waves, but if detected / absorbed they "die as particles" in one spot - not spread out over miles as they can be when waves. Note that the length of paths: abd and acd, are the same. I.e. when the separate SINGLE photo get back together with itself, after being at times in its flight it 4 feet from itself, it arrives at the screen, unified, at the same time, but some of the many slightly divergent beams arriving there are "out of phase" with themselves and cancel (make dark interference lines) on the screen. The diverse in angle beams following paths abd & acd are exactly the same length ONLY for paths with pairs of equal angle degree corners. (parallelograms or rectangles.) I will not go into details, but it is well known that photons ONLY interfere with themselves (and proven by using such low intensity sources that most of the time not even one exist - long exposure film, still has an interference pattern on the developed film, etc.) This is why one only needs monochromatic light, but not coherent light, source to produce interference patterns. Now here is what you do to measure the length of a photon: You rotate beam splitter a very slightly counter clockwise, so that the path ab passes to the left of mirror at b, but pull mirror b back to still be hit by that now tilted beam. You of course must also rotate mirror b slightly clockwise, so the beam leaving it follows the old path to beam splitter d again. Now the corner turned at b is not 90 degrees. Perhaps this adds 5 cm of extra length to path abd. What one sees on the screen is that there is a little light where there was none. I. e. the interference pattern on the screen is a little "washed out." This slight twisting of a & b is increased and then the pattern is more washed out. I kept repeating this until with ~30 cm extra path length for abd, the screen was with uniform illumination. Crudely speaking this implies that none of the part of the photon going via path abd had yet arrived at the screen before the full length of the part of the same photon going by path acd had already disappeared into the screen. I.e. my spectral line source was making photons that were about 30cm long.
One other physical parameter in the interferometry setup is that the paths have to be equal in length. If they aren't, this introduces a phase-difference between two photons, hence the same phase-difference between two parts of a single photon. Although Michelson and Morely weren't doing single photon experiments, this is basically the same principle they used--if one path (for two separated beams) is longer (spatially or temporally) you get interference effects, which were not detected.
What we call entanglement is something predicted by the theory. It says two particles which have interacted can exchange something although they remain unchanged. I like Penrose's example of a pair of spin-1/2 particles from the decay of a spin-0 particle. We don't worry about what these are, just how they behave. Since momentum has to sum to 0 the particles must be moving in opposite directions, and where one is spin up the other is spin down. This is entanglement of states, but, there are no correlations without measurement; unless both particles are "brought together" so measurements are made, no correlations. So can we choose any old pair of particles and look for correlations anyway? Oh sure, you bet. We can do this with distant light to get all kinds of interferometry data. Hanbury-Brown-Twiss interferometry implies entangled path-states for photons. That is, each photon follows two paths to both the detectors, the possible paths through space to each detector are in superposition.
