As discussed in my previous article, in the QM picture the "collapse of the wave function" is needed because the entire system exists in an indeterminate limbo until a measurement is made, at which time the system suddenly takes on well-defined values.  This is, of course, crazy, so we are motivated (like EPR) to hypothesize the existence of some hidden variables which supplement the description given by QM.
Hidden Variable Theories
By Travis Norsen (March 2002, Part 3 of 5)

[OBJECTIVE SCIENCE.COM] As discussed in my previous article, in the QM picture the "collapse of the wave function" is needed because the entire system exists in an indeterminate limbo until a measurement is made, at which time the system suddenly takes on well-defined values.  This is, of course, crazy, so we are motivated (like EPR) to hypothesize the existence of some hidden variables which supplement the description given by QM. [5]  In the present case this means we are motivated to suppose that each of the particles has some well-defined (but perhaps hidden) properties.  These properties are expected to be correlated between the two particles, since the particles share an intimate moment of creation.

Let's concretize by taking a specific example of a hidden variables theory. [6]  We'll start with the simplest possible case, in which we only allow Alice and Bob to measure the spins along the z-axis.  Then, instead of describing the state of the system by the quantum mechanical wave function above, we will simply assume that there is a random mixture of two types of particle pairs:

50% of the pairs will be described by  and                   (3)

50% of the pairs will be described by  and

Hopefully the notation here is obvious.  The idea is simple.  Rather than describing the particles with an "entangled" wave function that "collapses" at the moment of measurement, we'll just assume that there is a 50-50 mixture of two different (but fully well-defined) types of pairs.  In half the pairs, Alice's particle has spin "+1" and Bob's particle has spin "-1," and vice versa for the other half.  No problem, right?  This simple theory obviously reproduces all of the quantum mechanical predictions, namely (a) that each experimenter should see a random 50-50 sequence of "+1"s and "-1"s as the spins are measured, and (b) that there should be a perfect anti-correlation between the results.  Moreover, the theory does so without any of the dubious concepts of QM. 

Remember, though, that nothing forces Alice and Bob to measure only along the z-axis.  Let's now permit them to choose (independently, and randomly) between two different possible measurement axes.  We could choose, say, the z-direction and the x-direction, but instead let's be completely general and call our two arbitrary directions  and .  (So, to be clear, Alice and Bob each choose one of  or  as the direction to measure along.  That is, what we before called  will be either  or , and  will likewise be either  or .)

Let's just do what we did before and assume that, at their common moment of birth, each particle is created with certain pre-existing spin properties -- answers, if you will, to the questions that the experimenters may choose to ask down the line.  Here, since we haven't specified the relative orientation between  and , we'd better leave as arbitrary the "populations" (call them "F" for fraction) of each of the cases: 

F1 of the pairs will be described by  and

F2 of the pairs will be described by  and                                          (4)

F3 of the pairs will be described by  and

F4 of the pairs will be described by  and

The notation here is similar to that used above, except that now for each particle (A and B) we have to specify its properties for both of the measurements that Alice and Bob might decide to make, namely the spin components along  and .  Note that we aren't saying that Alice or Bob could measure both at the same time.  (This would be a gross violation of quantum mechanics.)  All we are saying is that, in principle, the information determining the result of a measurement of either spin component exists already in the local properties of that particle alone. 

Now, we just have to ask whether it's possible to choose the populations in such a way that we reproduce the quantum mechanical predictions.  Indeed, this is pretty easy to do.  In writing only these four types of pairs, we've already guaranteed that if Alice and Bob measure along the same axis, the results will automatically be completely anti-correlated.  For the other cases, where they measure along different axes, we see that we can reproduce the correct QM result if we take: 

where  is the angle between  and .  The four cases have a probability that sums to unity, which is good, and each term is positive definite.  If it's not clear why these have been chosen this way, simply go back to the previous table.  For example, the case labeled "F2" results in Alice measuring "+1" on her particle (along ) and Bob measuring "+1" on his particle (along ).  According to QM, this result (Alice and Bob each measuring "+1" along  and  respectively) is supposed to happen with a frequency given by Eq. (2), so we simply copy that result as the population fraction F2.  The other cases follow in the same way.  Again, this allows us to trivially reproduce the quantum mechanical prediction in a completely local, deterministic, common-sense manner.  

This may seem arbitrary at this point, because we aren't actually specifying any mechanism by which these population fractions are generated.  But that is precisely the point.  The idea is that if we were to seriously advocate this kind of theory, we would posit some mechanism to explain these probabilities.  For example, they might depend on some further hidden parameters involving the original decaying particle, or perhaps some influence coming from the detectors which stimulates or otherwise affects the decay, or even some process by which the particle properties evolve based on local effects as they fly apart.   

Anyway, so far the analysis is general, and we see that it's clearly possible to invent some mechanism which would explain the above populations, and thereby reproduce the quantum mechanical predictions in a completely local, deterministic, common-sense picture.  So things look good for the advocates of Local Realist Hidden Variables Theories (LRHVTs). 

Trouble, however, arises when we allow Alice and Bob to choose from among three possible directions along which to measure the spin.  We can go through the same kind of argument as before, and define populations for each of the (now) eight necessary cases:
 

F1 of the pairs will be described by  and

F2 of the pairs will be described by  and

F3 of the pairs will be described by  and

F4 of the pairs will be described by  and                                (6)

F5 of the pairs will be described by  and

F6 of the pairs will be described by  and

F7 of the pairs will be described by  and

F8 of the pairs will be described by  and

where the three numbers in parentheses refer to the spin components along , , and  respectively.  As before, we have built in the required property that the spins are anti-correlated for any case in which Alice and Bob measure along the same axis.  And again as before we will be interested in testing whether or not it is possible to reproduce the quantum mechanical result by judiciously choosing the F's.  We make a step in this direction by noticing that we can write the probability of any specific result as a sum of two of the F's.  For example, 

      (7)                                                                               

since it is precisely in cases F3 and F5 (and no others) that particle A has spin "+1" along  and particle B has spin "+1" along .  All other possibilities (e.g.,  etc.) can be written in a similar way, by simply reading off the table which cases give the desired result. 

Now we are ready to write down a condition that must hold true in our LRHVT, but which will be shown to be in conflict with the quantum mechanical prediction.  That is, we will show that it is impossible to pick the F's in such a way that we reproduce the results predicted by QM. 

Since each of the F's is a probability (and therefore positive definite), we must have inequalities like the following: 

           (8)

But we also recognize this as a statement about the relative probabilities, namely: 

    (9)

This is our first example of a so-called Bell inequality.  It follows simply from the assumption that each particle has well-defined pre-existing properties corresponding to all of the directions along which its spin might eventually be measured -- and, as mentioned before, the further assumption that the act of measurement on one particle has no effect on the second particle.  Any theory of this type (that is, any LRHVT) will produce results satisfying Bell's inequality. 

However, it is simple to show that for at least certain choices of the angles, the predictions of QM violate the Bell inequality.  For example, choose ,  and  to be co-planar, with  bisecting the angle formed by  and .  Specifically, choose the angle between  and  to be 90 degrees (so that the angle between  and   or  and  is 45 degrees).  Then, using the quantum mechanical result Eq. (2) and plugging into Bell's inequality, we have: 

      (10)

i.e.,  

      (11)                                                                                              

which is obviously false.  This shows that if we use the quantum mechanical result, we violate the Bell inequality.  So any LRHVT (which must satisfy Bell's inequality) will contradict the predictions of QM.  And when the experiments are done, the QM result is proved right.  This shows that we can't reproduce the correct statistics with the kind of theory in which the particles have ab initio well-defined properties for all possible future experiments.  This means that at least one of the assumptions mentioned in the beginning must have been wrong:  either the particles fail to have well-defined properties as they fly apart, or the measurement of the first particle "non-locally" affects the state of its entangled partner.  The first assumption (identity) is, of course, unassailable.  So it seems that the observation of Bell-inequality-violating correlations between entangled particles must be taken as evidence for some kind of super-fast "non-local" interaction. [7] 

Actually, there is still some room for skepticism here, because our discussion has focused only on one rather vague proposed type of hidden variables theory.  We weren't very careful in defining exactly what the "populations" (the F's) were and were not allowed to depend on, and whether there was some clever way to get around the conclusion of non-locality.  But it is easy enough to give a more delicate treatment which makes clear the generality of the result.


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


References and Notes:

[5]  Actually, there is some question about the appropriateness of the term "hidden variables".  The best known hidden variables theory is Bohm's causal interpretation of QM, in which the "hidden" variables are the positions of particles.  (The basic idea is that there is both a wave and a particle, and the wave guides the particle according to dynamics hidden in Schroedinger's equation.)  But, if anything here is hidden, it is certainly not the particle position -- that, after all, is what we see directly when we measure particles!  One might more plausibly say that it is the wave function itself which is "hidden", since normally measuring the particle's position gives us (at best) partial information about the wave function.  In order not to stray too much from what's actually important, however, I will leave this debate for later and use the standard (if questionable) terminology. 

[6]  The presentation of this section is taken practically verbatim from Sakurai's book "Modern Quantum Mechanics". 

[7]  Actually, "non-locality" is a horrible term for the kind of interaction that is being seen here.  One might expect that, as the sensitivity of the experiments is increased, eventually it will be discovered that the interaction involves a mechanism of finite-speed signal propagation between the entangled particles.  This speed is known to be faster than the speed of light based on current experiments.  So a better term would be simply "Superluminal interaction", but that is merely a debate over semantics.  "Non-local interaction" is the established terminology, and the goal of the present paper is not to fight that terminology, but merely to explain why the experiments entail its existence (whatever we decide to call it). 

Note also that when/if the fast but finite speed of these interactions is discovered, this will be a great thing for realists in physics.  It will be experimental proof of the breakdown of the standard interpretation of relativity theory, and will show the need for some better theory which gives a sensible physical account of the meaning of the equations of relativity.  Lorentz ether theory, for example, is a good candidate, and seems to integrate well with Bohm's causal interpretation of QM (in which the correlations between entangled particles are maintained by a "non-local" wave called the quantum potential).  When the fast but finite speed of the entanglement interaction is discovered, pragmatic and positivistic physicists will (finally) be forced to confront these issues head on, and admit that there's more to physics (just as there is more to life) than mere formalism. 

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