The observation of Bell-inequality-violating correlations between entangled particles must be taken as evidence for some kind of super-fast "non-local" interaction.
Bell's Inequality and the Case for Super-Fast "Entanglement" Interactions 
By Travis Norsen (April 2002, Part 4 of 5)

[OBJECTIVE SCIENCE.COM] Let's define A and B to be the measurement results obtained by Alice and Bob when each measures the particle's spin-component along some axis. [8]  These results will in general depend on both the direction along which the experimenter decides to measure the spin, and also on some general set of variables associated with the state of the particle being measured.  Let's call these, collectively, .  Formally, 

              (12) 

which just says that Alice and Bob always get either the result "+1" or "-1" when they measure along any arbitrary axis.  

The whole issue here involves the correlations between the results A and B, in particular, whether the observed correlations can be explained by something other than a "spooky" non-local interaction between the two particles.  Let us therefore consider the product of A and B.  This product will be +1 whenever Alice and Bob both measure either "+1" or "-1"; the product will be -1 whenever Alice and Bob obtain different results (one gets "+1" and the other gets "-1").  In the spirit of our discussion above, we will start by assuming that each particle has pre-existing well-defined properties which depend only on the shared past history of the two particles.  These properties are encoded in the variable(s) , which we may imagine depend on some arbitrary internal parameters of the particles which were correlated at their mutual birth. 

Just to be absolutely clear, in writing the measurement results as functions of the (nearby) polarizer direction and  only, we are allowing any arbitrary local explanation of the particle's properties.  All we are forbidding at this point, is that Alice's measurement result be affected by Bob's choice of which axis to measure his particle along.  That is, formally,  

                                                                                              (13)

or in words:  we assume that the outcome of Alice's experiment does not depend on how Bob (who, remember, is miles away) chooses to orient his polarizer at the last instant before he measures his own particle.  This is the locality assumption.  We are assuming that the outcome of Alice's experiment will be the same, regardless of what Bob chooses to do. [9]

What follows will be a "reductio ad empirical falsehood" of this assumption. 

OK, back to the story.  We are interested in looking for correlations between A and B, so let's write down the correlation function of A and B.  This is simply the product of A and B (discussed above) averaged over all possible values of the hidden variables : 

                      (14)

where  represents the probability of a given configuration of the hidden variables .  This correlation function measures the total frequency with which Alice and Bob agree (i.e., both get "+1" or both get "-1") minus the total frequency with which they disagree (i.e., one gets "+1" while the other gets "-1").  In short, it measures correlations between the measurement outcomes in the two spatially separated measurements. 

Now, one basic experimental fact that we must reproduce is the fact that if Alice and Bob measure along the same direction (i.e., if ) then the results must be perfectly anti-correlated, i.e., the correlation function must be exactly equal to -1.  This implies that 

                                     (15) 

That is, if Alice and Bob measure along the same direction (here called ) their results must be opposite.  We can use this to rewrite the correlation function as follows: 

                    (16)

Now, motivated by our earlier discussion which showed that a LRHVT could reproduce the QM result only until we considered three possible measurement axes, let us compare the correlation function for two different pairs of angles (with one shared angle common to each pair).  That is, consider the quantity , which represents the difference in the amount of correlation between the case in which Bob measures along the  axis and the case in which he measures along the  axis.  Using the above formula (and using the fact that since , the square of A is always +1) we arrive at: 

            (17)

If that's not obvious, one gets from the first line to the second by inserting a cleverly chosen factor of +1 in the form .  [10] 

Now, the probability distribution for the 's is a positive definite function, as is the factor in square brackets in the last line.  The remaining factor in the integrand, namely , takes on only the values -1 and 1.  Hence, we may take absolute values on both sides of the previous equation and arrive at: 

                  (18) 

where to get the last line we have recognized that  is a properly normalized probability distribution (i.e., it integrates to one).  This is a general statement of Bell's inequality.  It says that the absolute value of the difference in correlations for certain pairs of angles, should always be less than or equal to the correlation function at some other angles (minus one).  This inequality should hold if the assumptions we have made leading up to it are correct -- in particular, if the measurement result on each side is independent of the choice of direction to measure along at the distant experiment. 

It is easy to show as we did earlier, that for certain choices of ,  and , Bell's inequality is violated by the QM prediction.  And more importantly, when one builds an appropriate apparatus and actually measures the various correlation functions experimentally, the results are in agreement with quantum mechanics.  Bell's inequality is violated experimentally. 

There are two somewhat different types of these experiments, depending on whether or not the polarizer angles are held fixed during the time of flight of the particles from source to detector.  In the non-delayed-choice version (that is, when the polarizers are simply held fixed) it is simple to imagine how the result of Alice's measurement might depend on the polarizer orientation of Bob's (distant) apparatus, in a fully local way.  For example, as in Little's Theory of Elementary Waves, one could imagine that the particles are stimulated from their common source under the influence of waves which had traversed the polarizers on both sides.  Hence, the particle traveling toward Alice "knows" about the orientation of Bob's polarizer, because its emission was stimulated under the influence of a wave coming from Bob's polarizer. 

These non-delayed-choice experiments therefore do not in any way rule out the LRHVTs that we might have wanted to find.  In terms of the notation of this section, in these experiments, the local hidden variables  include the distant polarizer settings, so the violation of Bell's inequality in these experiments does not prove that Alice's measurement result depends non-locally on Bob's choice of axis.  Alice's result may depend on Bob's choice, but here it is fully possible for it to do so via a completely local, slower-than-light mechanism, since Bob's choice was made a long time ago. 

The double delayed choice experiments, however, do not suffer this shortcoming.  These experiments involve a rapid and random switching of the polarizer settings on each side, during the time of flight of the particles. [11]  Hence, there is no correlation between whatever initial setting the polarizer might have had when the particle was stimulated, and the final setting that is actually used to measure the particle's spin.  So now the hidden variables  do not include the final polarizer setting, because that setting is only made at the last instant before the particles are detected -- far too late for the distant particle to "learn" about it via any slower-than-light mechanism. 

Given the analysis presented here, the experimentally-observed violation of Bell's inequalities proves that the correlations between the results of Alice's and Bob's measurements are too strong to be explained by anything in the shared past of the two particles -- that is, the correlations are too strong to be explained by any purely local theory.  In short, the experimental results prove that Alice's choice of which axis to measure the spin along does indeed affect the state of the distant particle near Bob.  Some kind of super-fast "entanglement" interaction keeps the two particles in intimate communication even as they fly apart in space, so that measuring one of them affects the other nearly instantaneously. [12] 


Further reading:
"Non-Locality" and Realism: A Brief Discussion
By Travis Norsen (May 2002, Part 5 of 5)
Historically, "non-locality" is one of the aspects of QM that has been put forward by opponents of realism as evidence that no "common-sense" physical interpretation of the theory is possible.


References and Notes:

[8]  The presentation in this section is the same as that given in several of Bell's papers, reprinted in his book "Speakable and Unspeakable in Quantum Mechanics."  The interested reader should definitely read Bell's own papers.  My hope is that the present essay will serve as an appetizer to Bell's great banquet. 

[9]  Those who disagree with the conclusion being advocated in this paper will have to find additional assumptions that might be questioned.  It is true that there are additional assumptions on the table here (e.g., that the particles have well-defined properties at all times, and that it is possible for the space-like separated polarizers to undergo independent random rotations), but it is hard to see how any of these could be reasonably called into question.  (See footnote [11].)  Also, be clear about the logic.  The conjunction of several premises leads to a result which is observed to be false.  Hence, one of the premises must be false.  It is my contention that the assumption of locality is the only one which can reasonably be doubted.  

[10]  Some people have suggested that there is an error here in the derivation, because it makes no sense to insert the result of a measurement along  into a term in our formula that already involves the results of measurements along different axes.  After all, measurements along these different axes don't commute, and it is impossible to measure them both simultaneously.  But one must remember that here we are assuming a LRHVT in which even these non-measured properties do in fact have well-defined values.  As mentioned previously, this doesn't mean that we could in fact simultaneously measure both values, but the whole point of this kind of theory is to assume that the properties exist prior to measurement.  Ultimately, this objection amounts to the choice to sacrifice the identity assumption rather than the locality assumption -- to say that no non-local interaction is needed because the whole program of hidden variables is misguided.  I therefore disagree fundamentally with this attempt to avoid the implication of non-locality.  The principle of identity cannot be touched. 

[11]  In order to avoid the conclusion that the entangled particles are able to interact super-luminally, some people have suggested that, in fact, the polarizer settings are not random, and that pre-set correlations between the choices of polarizer settings could be used to explain the Bell-violating correlations in a fully local manner.  As an objection, this is absolutely true, but it is extremely unlikely that the answer lies here.  In principle, one could generate random numbers at each polarizer based on properties of photons streaming to the earth from distant galaxies in different directions in the sky.  So then, for the "random" numbers to be, in fact, non-random (i.e., for them to be pre-correlated), one would have to suppose that there is some pre-existing correlation in the light coming from galaxies thousands of light years apart.  This would certainly be a bigger surprise than finding that entangled particles can communicate superluminally over kilometer-scale distances.  Moreover, one could, instead of using random number generators, let Alice and Bob make real-time free-will choices of which axis to measure.  Then, to suppose that the choices are pre-correlated would be tantamount to denying that the experimenters' free choices are in fact free.  So this kind of "solution" to the strange apparent non-local interactions is no solution at all.  It replaces a surprising new phenomenon with either an inexplicable cosmic conspiracy or an outright denial of the philosophical axiom of volition.  Of course, the experiments are not in fact done this way, and it's not impossible that the results could be different if they were, but this is an awfully fishy basket in which to put one's eggs. 

A second major "loophole" which deserves some comment is the so-called "detector efficiency" loophole.  In the derivation above, we assumed that the measurement results represented a fair sample of the distribution over the hidden variables.  But in the actual experiments, less than 100% of the particle pairs are detected, so in principle one could get around the non-locality conclusion by asserting that the measured results are a biased sample -- that they don't accurately reflect the full distribution over the hidden variables.  (An advocate of this loophole would say that the full distribution actually conforms to the Bell inequality, but the observed sample violates it -- the missing particles, i.e., the ones that weren't detected due to inefficiencies, would make up the difference.)  But this is another very fishy basket.  Again, it's not impossible that there could be some reason for the observed particles to be a biased sample, but there is certainly no evidence to suggest that (let alone why) it should be so.  At best, it is arbitrary to assert that one can get around the non-locality conclusion in this way.  Anyway, there is no barrier in principle to increasing the efficiency of these experiments, so this loophole is slowly closing as the years go by. 

Finally, there's always the "possibility" that the experimenters are all simply lying to us (and/or to themselves!), and the actual experimental results in fact *are* in accordance with Bell's inequalities.  This scenario is almost unthinkable, since any clear experimental refutation of the QM prediction would guarantee a Nobel prize to the physicist who discovered it.  In short, this is the kind of hypothesis that makes even conspiracy theorists skeptical.   

For an excellent discussion of the various "loopholes" here (as well as references to the vast literature on this subject) see the article by F. Laloe ("Do we really understand Quantum Mechanics?") in a recent issue of the American Journal of Physics:  Vol. 69, pg. 655. 

[12]  This interaction, however, cannot be used by humans to send signals faster than light.  Briefly, this is because of the randomness in the measurement results.  If one knew the exact values of the "hidden variables" (e.g., the exact particle positions in Bohm's theory) then one could build a faster-than-light telephone.  However, in practice, knowledge of the hidden parameters will always be constrained by the uncertainty principle, and such telephones cannot be constructed by this mechanism.  Many positivist-leaning people infer from this that the observed non-locality is somehow unreal, or that there is no tension between non-locality and relativity.  I regard this position as deplorable, a return to the worst Bohr-style equivocation between epistemology and metaphysics.  If Bell's inequalities are indeed violated, then non-locality is a real fact about nature – regardless of its utility to human beings! 

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