ddd123 said:
On my part, I don't know what to think. On one hand, the long-range correlations are there because of measurement, and avoiding collapse doesn't practically account for compound measurements (you have to believe it would work if you could do the practically impossible calculation of treating the whole measurement device quantum mechanically). On the other hand, collapse is frame-dependent, although the consequences are the same whatever frame you choose in the end, so it seems to beg for a deeper explanation.
This discussion is a mess, and I'm sorry that I got involved into it again. The physics is very clear, and there is no problem.
An experiment happens in the lab and not in Hilbert space. Let's take the example of the Aspect-like experiment. There, via parametric downconversion a polarization-entangled photon pair is produced by shining a laser into a certain type of birefringent crystal. The interaction of the em. field produced by the laser is local (according to QED). It is localized in the sense that it takes place in the crystal and thus the extension of the interaction region is at most the size of the crystal (a few ##\mathrm{cm}^3## I'd say). Via some optical devices you have a two-photon state with the polarization part given as
$$|\psi \rangle=\frac{1}{\sqrt{2}} (|HV \rangle - |VH \rangle).$$
According to the usual rules of probabilities to get the polarization state of the single photons you have to trace over the other photon, i.e., you have
$$\hat{\rho}_A=\mathrm{Tr}_B |\psi \rangle \langle \psi |=\frac{1}{2} \hat{1},$$
and the same for ##\hat{\rho}_B##. As you see the single-photon polarizations are completely undetermined, i.e., you have unpolarized photons.
Now Alice (A) and Bob (B) perform a polarization measurement with the polarizer in H direction at very far distant places, such that according to the finite signal propagation (maximal speed is the speed of light) the measurement of A's photon's polarization cannot affect B's photon's polarization at the moment he is measuring it. Within QED this is ensured by the locality of the interaction of the photons with the measurement device and the microcausality of QED (it's built in into the theory by construction!).
Now, although both photons are precisely unpolarized due to the entanglement of the prepared photon pair the polarization measurements are strictly correlated. According to the rules of QT the probabilities for the four possible outcomes (VV, HH, VH, HV) are
$$P_{HH}=|\langle HH|\psi \rangle|^2=P_{VV}=|\langle VV|\psi \rangle|^2=0, \quad $$P_{VH}=P_{HV}=\frac{1}{2}.$$
So although the photons are completely unpolarized there's a correlation for the pairs. You never find both H polarized or both V polarized but always with perpendicular polarizations. If A measures H B measures V and vice versa.
According to this description this correlation is due to the preparation of the two-photon state in the very beginning and not due to the polarization measurement of A or B. Note that also A and B can find this correlation only by exchanging information according to their precise measurement protocols, i.e., both must keep track of the times they register the photons to know which two photons come from one pair and then afterwards they can check the correlation. In no way can you propagate instantaneously information by such a setup.
Also note that there's no collapse of the state as a whole via the measurement of either A or B. It's only such that if A finds H, she knows that B's photon will be found to have polarization B, but for Bob that doesn't change anything, i.e., the only thing he knows is that he will find with probability 50% either H or V. Also A finds with 50% probability H. So everything is consistent, and there is no spooky action at a distance, which is implied by the assumption of a collapse, but as you see, we don't need the collapse to understand the correlations. Further according to QT you cannot say more about the outcome of these measurements than the said probabilities, and the understanding is that the polarization of the single photons is really maximally indetermined.
In the hope that there may be a way to mimic these QT probabilities with a deterministic theory one came up with the idea of hidden variables which take determined values in any case and they are just unknown to A and B. Now Bell has shown that this assumption together with locality of interactions (between the photons and the polarization measurement devices) leads to an inequality for certain correlation functions, which is violated by QT (here QED which uses local interactions only by construction!). Indeed in corresponding experiments with such entangled photons (here you have to set the relative angle of the polarizers to another value than 0 or ##\pi/2##) that the Bell inequality is violated with an astonishing significance (google for Zeilinger to find the details) and with the same significance the QT prediction is confirmed. The conclusion is that there is at least no deterministic hidden-variable theory with local interactions that is in accordance with QT and the observations. Only QT (in this case QED) admits the locality of interactions at the same time with the strong correlations described by entanglement which in the sense of violating Bell's inequality are stronger than the correlations possible for local hidden-variable theories.
