What Is Entanglement Beyond Bell's Photon Experiment?

[Karl G.]

Every layman discussion of entanglement usually evolves around Bell's photon experiment. But could somebody give a deeper explanation of what is meant by entanglement? (Not afraid of math!)

 

[muppet]

It means that two (or more) particles are described by a common wavefunction. If you're talking about a system of indistinguishable particles (i.e. two of the same kind), the upshot is that you can no longer talk about the probability that a particular particle will posess some particular value of a measurable quantity (such as the projection of spin onto some axis). As well as the explanation of Bell-test experiments (incidentally: Bell devised the thought experiment with electrons, and the actual experiments with photons were first performed by Alain Aspect some time later), entanglement has other consequences, such as the Pauli exclusion principle. You can't describe a pair of particles by simply taking the product of the individual wavefunctions, as you'd end up assigning each of them an identity ("This particle is found here, and this one here"). Instead, you have to form linear combinations of such products, and swap the labels of the wavefunctions and co-ordinates for each term in the sum.
I'm too tired to be bother TeXing some expressions up to show you properly, but more or less what I would have written can be found here anyway.
(N.B. Follow the link to the next page on Pauli exclusion to really see what I mean.)