Here's my understanding of the paradox (the story I tell myself about it), and why in terms of information there's no problem.
(I'm writing this from memory, so it could be rubbish, but I think I have some of the essence in here. And it's not the original EPR - Einstein, Podolsky and Rosen - formulation, which I think was based on beta decay.) Imagine launching two entangled particles, say photons, heading off to two different locations. You take a measurement on one of them, say you test for spin up or down. The result of the measurement on one immediately tells you the result you'll get on taking the same measurement on the other. Eg, you measure photon A and find spin up, so you know measuring photon B will give you spin down. OK, no big deal so far. But, quantum mechanics tells you that photon A didn't decide it was going to be spin up until you measured it. So photon B has to be told, instantaneously, what photon A had decided to give as its result. Hence the need for instant communication (faster-than-light = violates causality, 'non-localisation')
But, as explained so far, it is pretty unconvicing to say that photon A didn't decide on what result it was going to give until it was measured. There's more to it. Instead of measuring for spin up or down, you could instead measure another parameter, say for spin rotation, clockwise or anticlockwise.* If you measure for spin rotation, then if A is clockwise B will measure anticlockwise and vice versa. But, if you measure for rotation then you lose coherence (is that how to express it?) for spin up or down. So, if you measure A for rotation, B can now be spin up or down with equal probablity. Likewise if you measure A for up or down, B can be closkwise or anticlockwise with equal probability. So now what is happening is that A 'tells' B whether you measured for rotation or up/down (as well as the result). Again, it has to happen instantaneously, hence the contradiction with relativity.
Experiments have been done, apparently, which test for this - the effect that measuring A influences the outcome of measurements on B. Now, I have not yet got an understanding of how these experiments are done. I'd love to know. But, my understanding of the conclusions of the experiments is:
- they have proved that there is an influence at a distance that happens faster than the speed of light
- but, you can not communicate information faster than the speed of light in this way
Now, here's my punchline: that final point somehow ruins it all! If you are not communicating information, then what's the big issue? Surely it is just like phase velocity in, say, waveguides? There's no problem at all with phase velocity exceeding c, because you can't carry information on the phase. It is the group velocity that matters.