EPR paradox: instantaneous vs very fast? An EPR quantum experiment can be explained by instantaneous collapse of the wave function regardless of the distance separating a pair of entangled particles. But do we have the certainty that the process is really instantaneous? If not, what is the current experimental limit on how fast is the process?
 A: Here's a first suggestion of the problems this model would cause: let's say you have a pair of entangled particles far apart, and two observers who each measure one. The idea of this model would be that when the first particle is measured, it sends some signal to tell the other particle what state to be in. But thanks to relativity, the order of the measurements need not be absolute: it could easily be arranged that one observer sees one person measure first while another observer sees the other one measure the pair first. Which way does the signal go, then?
Well, you could go on to say that there is some special reference frame for quantum signals to be sent in, which seems likely but at first glance is still consistent with the laws of nature as we know them. However, a fairly recent paper seems to rule this out as well.
If you accept that you can't send information faster than the speed of light, then a rather recent result shows that this implies that you can't have the collapse of entanglement propagate at any finite speed.
Here's the paper in question. Specifically, the authors show that with a entangled pair of particles you can only set lower bounds on the fastest a signal could be traveling, as you would expect, but in a suitable measurement of four entangled particles any finite speed signal leads to superluminal signalling.
Abstract: The experimental violation of Bell inequalities using space-like separated measurements precludes the explanation of quantum correlations through causal influences propagating at subluminal speed1, 2. Yet, any such experimental violation could always be explained in principle through models based on hidden influences propagating at a finite speed v>c, provided v is large enough3, 4. Here, we show that for any finite speed v with   , such models predict correlations that can be exploited for faster-than-light communication. This superluminal communication does not require access to any hidden physical quantities, but only the manipulation of measurement devices at the level of our present-day description of quantum experiments. Hence, assuming the impossibility of using non-local correlations for superluminal communication, we exclude any possible explanation of quantum correlations in terms of influences propagating at any finite speed. Our result uncovers a new aspect of the complex relationship between multipartite quantum non-locality and the impossibility of signalling.
It is still an experimental question, though, and Danu is right about the current measured limits on minimum speed. Here is the current record holder, as far as I know.
A: Experimentally, I remember seeing this paper which places a lower bound on the speed of 'spooky distance at a distance' at some four orders of magnitude higher than $c$. I think that, so far, experiments are consistent with an instantaneous effect.
A: An experimentalist's view of the conundrum:

An EPR quantum experiment can be explained by instantaneous collapse of the wave function regardless of the distance separating a pair of entangled particles. But do we have the certainty that the process is really instantaneous? If not, what is the current experimental limit on how fast is the process?

In one sentence you have two words which are being misinterpreted as far as the physics of quantum mechanical systems go , in my opinion.

*

*"Collapse" instead of "measurement". Measurement is what we absolutely have to do if we want to validate a model, whether quantum mechanical or classical mechanical. We do not say that the solution to newton's equations for the orbit of a satellite  "collapses" every time we measure its location. In an analogous way when we measure an observable that obeys the solutions of a quantum mechanical system we are measuring a one entry , the energy eigenvalue for the instance.


*"entanglement" is not a mystical occurrence. The wave function defining a quantum mechanical system has all the quantum numbers an phases in it and all the correlations between the variables imposed also by conservation laws. When one measures an instant of the problem under consideration all the correlations are attendant by the imposition of mathematical  solution for the problem. Every measurable instant the wavefunction has to obey conservation laws by construction.
The term "instantaneous"  has little meaning , since we are talking of one entity, i.e. the wavefunction. Yes it is instantaneous, as we know instantaneously that if it is a full moon rising the sun has just set.
A: The consequences of the EPR paradox are the following:


*

*The universe is non-local but obeys quantum mechanics

*The universe is local and quantum mechanics is wrong since there are hidden variables in universe. 


However in 1964 in a paper called "On the Einstein Polosky Rosen paradox", physicist John Bell presente a solution for this EPR paradox. The experiment to prove if either 1 or 2 are the correct answer, were performed in 1972 and showed that the universe is non-local and quantum mechanics is right. If you want to learn more about this, see Bell's theorem and inequalities. And by non-local we mean that if two particles are entangled, then one affects the other one instantaneously. http://plato.stanford.edu/entries/bell-theorem/
A: The EPR experiment does not imply that the laws of physics are non-local. See
https://arxiv.org/abs/quant-ph/9906007
and
https://arxiv.org/abs/1109.6223.
A: It's impossible to verify experimentally that a certain speed is instantaneous simply because we cannot measure infinity. What we can measure is that it appears to be approaching instantaneous.
So far, the collapse speed has been measured to be at least four magnitudes higher than $c$.
Thus, if it turns out that the speed is instantaneous, this will be due to some theoretical argument that simplifies theory. So far, we only know theoretically that the effect is non-local in that the effect propagates faster than $c$ and this effect has been experimentally verified as noted above.
It's worth recalling that Newton philosophically denied the possibility of instantanous action at a distance (going back to Aristotle) for gravity and which was verified in Einstein's GR. It maybe so again in this case. It's an open question for new physics.
