You can get two photons entangled, and send them off in different directions; this is what happens in EPR experiments. Is the entanglement then somehow affected if one puts a thick slab of EM shielding material between the entangled photons? Have such experiments been made?
According to EPR experiments measurements of the entangled states are at odds with SR, so based on that I'd assume the answer is "no"/"don't know", but any citations would be appreciated!
The original goal of the EPR paper was to show that quantum mechanics is incomplete. Hence, that extra variables have to be added to complete it, contrary to what Cedric claims. The goal of EPR is to show that either nature is non-local (and thus in conflict with SR) either quantum mechanics is incomplete. Since Einstein was not ready to abandon locality and SR, he concluded that quantum mechanics is incomplete.
Later however, John Bell would show that quantum mechanics is in fact non-local. To do that, he first devised an inequality that would have to be satisfied by any local physical theory. Then, he showed that this inequality is violated for certain entangled states, thereby proving that quantum mechanics is non-local. In the 70's Alain Aspect then made an experiment to check if Bell's inequalities were violated in nature or not. There have been many similar experiments since then and they all point to nature being non-local and quantum mechanics being a good description of this non-locality.
Now, there are possible loopholes in the experiments, which I won't discuss here.
One can also object that quantum mechanics described by Schrödinger's equation is not Lorentz-invariant, so we should not expect quantum mechanics to agree with SR.
What about equations which are Lorentz-invariant? Dirac equations, Klein-Gordon, etc... That's where it gets difficult. We know that a correct description of these equations requires quantum field theories. But we only manage to treat the field theories perturbatively. Other approaches are numerical or very limited. So I don't know of any detailed study within the context of quantum field theory of entanglement and Bell inequalities. But I hope someone can come in with more information about these. My knowledge is limited in these areas.
The standard test for whether two things are really entangled with one another in the spooky-action-at-a-distance sense of the EPR picture is to see whether measurements of the states of the two particles violate one of the Bell inequalities, meaning that the correlation between the states is stronger than can be explained by any local hidden variable theory. This has been done with lots of systems having significant separation between the particles-- as long ago as 1982, Alain Aspect's group did a test with time-varying detectors that were separated by 40 feet or so, and the results were something like nine standard deviations away from the LHV limit.
More recently, Chris Monroe's group at the University of Maryland has done experiments where they entangle the states of two ions in two different ion traps, and showed Bell violation by something like 3.5 standard deviations. I wrote this up on the blog a while back, and the post includes links to the relevant papers. I'm not sure whether there's a complete lack of a straight-line path between the ions, but they're in completely separate vacuum chambers (mostly stainless steel), so I think it's fairly likely to meet the requirements of the question.
The answer to the question depends a bit on what is meant by "mediated". A composite quantum system composed of two or more quantum subsystems can be in a quantum state in which the subsystems are entangled from the beginning, i.e., from the initial state. If the composite system evolves without any interaction among the subsystems, then the form and degree of entanglement between them will not change. If there is interaction, the entanglement will, usually, change in form and/or degree. In particular, if the initial state is not entangled, then subsequent interaction between the subsystems will entangle them. But this fact, per se, doesn't depend on the kind of interaction. Any interaction, whatsoever, between the subsystems will, at least for awhile, entangle the initially non-entangled subsystems. One might well regard this as "mediation" of entanglement by interaction. But physicists, generally, do not think of entanglement, per se, as a dynamical feature of quantum mechanics. Rather, it is regarded as a kinematical/structural possibility of quantum states for composite systems which can be modified by the presence of interactions but is not ultimately, due to interactions.
Let me add that entangled states are the most common states, by far. The unentangled states of composite systems (which are just the so-called product states) are much rarer by comparison.
Pretty sure that EPR does not state entanglement is at odds with SR or if it does it is incorrect. The point of the EPR paper was that the consequences of entanglement were so strange they could not be real.
Experimental evidence however supports entanglement and has never shown any violation of SR.
I think you answered your question.
"According to EPR experiments measurements of the entangled states are at odds with SR": if you mean that we cannot consider that the result of a measurement made on one entangled particle "propagates" to another one because this propagation would violate SR principles, you have to rules out a interaction in the sense of "strong, weak, ... interaction", ie an interaction not in violation with SR.
In addition, we do not need to have such an interaction as it is directly explained by the principles of quantum mechanics.
It is like imagining that an "interaction" teaches QM to the particles.
The main outcome of the treatments of the EPR paradox is to make hidden variables theories irrelevant, so basically "quantum mechanics" wins and we don't need other explanations.
Conclusion
1. Definitely not a causal interaction - see Alain Aspect delayed choice experiments
2. Rather the amplitudes of causally separated particles are merely remain correlated due to past common origin events
3. See Smerlak and Rovelli at http://arxiv.org/abs/quant-ph/0604064 .
See also Rovelli at http://arxiv.org/abs/quant-ph/9609002 for a coherent point of view, but it's somewhat subtle.
There is no effect of the one measurement event upon the other. It is not until the results from both measurements are brought together for comparison, and accumulated statistically, that it gets interesting.
The only interactions relevant to the entanglement are at the source, when the singlet spin system falls apart into two spin one particles (or whatever exactly it is you're doing) and again when the measurements are correlated in one place. The latter is not often mentioned in QM, but only in discussions of the philosophy of QM.
Too many writers assume something nonlocal is going on, or some superluminal effect occurs. The classic papers on the Bell experiments typically state that one (at least) of these has to go: locality, causality, realism. It's an open discussion even today in 2010, which one.
Personally, I like to toss out causality, and understand things according to Cramer's Transactional Interpretation -- http://mist.npl.washington.edu/npl/int_rep/tiqm/TI_toc.html -- but at the end of the day I really don't know any better than anyone esle.
In a sense, entanglement acts as if spacetime doesn't exist - that's why we call it non-local. QM doesn't know about c. We could model this by calling for an extra dimension where the correlation resides, such that the two sides of the correlation are coincident in this extra dimension. But that's a purely speculative model I dreamed up.
I'm also not being strictly accurate when I say "spacetime", because time does enter in here in the sense that the first leg to be measured is the one which collapses the wavefunction (whatever that means) and destroys the entanglement. A recent paper by Wilzcek & Shapere ( http://arxiv.org/abs/1208.3841 : Constraints on Chronologies, and an informal review here http://www.technologyreview.com/view/428962/special-relativity-and-the-curious-physics-of/) demonstrates the essential role of time ordering and simultaneity; it's a deep observation which (he says) will result in a further paper, because it's about how QM and SR/GR interact. David Albert's talk at the Aharanov 80th Honorary talks about the same sort of thing ( http://ibc.chapman.edu/Mediasite/Viewer/?peid=f9b9519414844b79b36ffda1240c65061d : David Albert: “Physics and narrative”).
No.
The more or less formal definition of interaction between two systems is that you have a system 1 with Hilbert space $\cal H_1$ and a system 2 with Hilbert space $\cal H_2$. If system 1 were in isolation from other systems, it would have the Hamiltonian $H_1$ to govern its time evolution. Likewise for system 2 and $H_2$. When the systems are considered as sub-parts of a combined system, the Hilbert space for the combined system is $\cal H_1 \otimes \cal H_2$. Suppose that this combined system is, for whatever reason, governed by the Hamiltonian $H_3$. The interaction term between the two systems is defined as being the difference between this actual Hamiltonian and the Hamiltonian which would have obtained if the sub-systems had no interaction, which would've been $H_1 \otimes I_2 + I_1\otimes H_2$ where $I_i$ is the identity operator on the Hilbert space $\cal H_i$. That is, the interaction term $H_{\mathrm{int}}$ is, by definition, equal to
$$ H_3 -( H_1 \otimes I_2 + I_1\otimes H_2) .$$
The two subsystems can be entangled even if this interaction term is identically zero for all time from $- \infty$ to $ \infty$. They could also be unentangled (i.e., in a separable state) even if this interaction term is quite violent or weird or whatever. Entanglement has nothing to do with interaction.
As a practical matter, if you want to prepare a state, you have to have some interaction between something and your system, and if you want to prepare an interesting entangled state, you have to use an interesting interaction. But this referes to the interaction between your preparation apparatus and the two sub-systems, not to any interaction between the sub-systems themselves.
Raskolnikov is the only one who seems to grasp the essence of Bell's theorem. Einstein's whole point in the EPR paper (which was actually not directly written by Einstein ... see his autobiographical notes where he summarizes the EPR argument in a terse way) was to add hidden variables to quantum mechanics to eradicate spooky action in the theory. His idea of a Bertleman's socks explanation fails, as proven by Bell. Then, how can we explain how two entangled particles always coordinate their behavior? They must somehow communicate with each other.
