It is often Stated that Bell's Theorem is equivalent to the statement: No theory of Local Hidden Variables can reproduce all of the predictions of quantum mechanics. I see nowhere in Bell's Theorem the assumption of hidden variables. As far as I can tell, there is one assumption: locality, which is shown to conflict with quantum mechanics. If I am wrong on this, could someone demonstrate exactly how Bell failed to demonstrate that quantum mechanics is non-local by showing how the assumption of hidden variables is used in his theorem? To me -- and by the way Bell -- the problem is locality.
The issue is locality of what? Which quantities are assumed to be local?
If you say the reults of all experiments, hypothetical as well as actual, and say that these must be assignable definite values, then this is in conflict with quantum mechanics. But this is the assumption that is called "hidden variables", the reason is that this is the assumption Einstein made in the EPR paper. This is the assumption that if you could perform an experiment to determine a quantity, then it is legitimate to give this quantity a definite value, even if you didn't do the experiment. The hidden variables are just the names of the extra quantities that determine the outcome of those hypothetical experiments.
The results of the quantum experiments themselves are only obviously nonlocal if their results are assumed to be definite things, even in those cases where you didn't actually go on to perform the experiment. This is not accepted in the standard interpretations, although for slightly different reasons in different philosophies.
Suppose you have three polarization settings, A,B,C such that the measurement of A,B,C is always the same between the two distant spins, A and B are 99% correlated, B and C are 99% correlated, and A and C are 96% correlated. This is the starkest violation of Bell's inequality that I know. In this case, you can see the violation without a calculation--- if A and B are the same except for 1 time in a hundred, and B and C are the same except for one time in a hundred, how can A and C be different more than 2 times in a hundred? Obviously they can't, and that's Bell's inequality. But obviously they are, in quantum mechanics, because the probability difference between polarization settings goes as the square of 1 minus the (small) angle squared, so that doubling the (small) angle increases the discrepancy in the measurements by a factor of 4, not by a factor of 2.
In order to argue that this is a locality violation, when you are measuring A on particle 1 and B on particle 2, you have to assume that were you to measure A on particle 2, the result would have been the same. The only locality violations are regarding the answers to these hypothetical questions. But you aren't measuring A on particle 2, so that this is a counterfactual statement. You don't have to believe that counterfactual statements of this sort are meaningful. If you believe that the results of experiments come out of thin air, that they don't come from the laws of physics, but of an irreducible interaction between the particles and the measuring device, which is called an observation.
This is Bohr's language, so I am just stating Bohr's position that the results of experiments come out of thin air. They aren't determined by anything previous. So it makes no sense to say "were you to measure A, the result would have been such and so", because you aren't measuring A. The assumption of "counterfactual definiteness" is the central thing, that it makes sense to talk about the answers to experiments you haven't done. I don't like Bohr's way of stating it, I prefer Everett's, but the two answers are exactly the same when push comes to shove.
The assumption of hidden variables is only used to allow you to talk about what the outcome of the A measurement on particle 1 would have been for those instances where you are doing a B measurement on particle 1. You can determine what they answer would have been using the hidden variables, assuming that the hidden variables determine the outcome of the experiments. You don't have to use the hidden variables if you are willing to assume that it is ok to talk about the value you would have gotten in experiments you haven't done.
Since it is only through considering couterfactual experiments that you get a contradiction, if you don't have counterfactuals, you don't have contradictions. The contradiction is that there is no way to assign consistent definite results to all the possible counterfactuals. You could just refuse to assign answers to hypothetical measurements. Equivalently, you could assign amplitudes with those answers, not definite results. Its only when you try to assign definite results that you get a contradiction.
This argument of Bohr's is very difficult to understand, and, like everything else, it is entirely cleared up by switching to Everett's many-worlds philosophy. This explains exactly what is going on, in a detailed mechanistic way that is equivalent to Bohr except for philosophical mumbo-jumbo. Since it is equivalent to Bohr, modulo philosophy, I don't see any point in retaining Bohr's harder to understand language, except to the extent that it is interesting that you can state things this way too.
In the many-worlds point of view, the A measurement splits the observer, and it splits the observer in a different direction than the B measurement does, it entangles the observer with a different property of the particle. The relative statistics that the two observers see are only determined when the two observers come to talk, and then which copy of the observer over here meets which copy of the other observer over there is always a matter of how they are tilted relative to each other in wavefunction space. The tilt between the copies of the observers is entirely determined by which quantity they chose to measure. It is now obvious that there is no reason that Bell's inequality has to hold in this sort of thing, because the results not only of counterfactual experiments are undertermined, but even the results of actual experiments are undetermined! There is still another copy of you which got the opposite results!
Bohr's philosophy is obtained by focusing on one branch, and rejecting the other counterfactual branches as non-existing.
I suppose you could see even the Everett version as violating locality, because the world-label (and the wavefunction) are global constructs. But I don't think this is the best practice. If the world is quantum mechanical, those experiments which split observers will always be indefinite.
Semi-local classical illustration
Since many-worlds requires some slightly nonobvious philosophy, it is best to make the argument in a purely classical universe, where there are no philosophical headaches. You can violate Bell's inequality inside a semi-local classical theory, with world-splitting. The theory will be a classical analog to the Many-Worlds quantum theory, but no philosophical headaches, because the splitting will be built into the theory, and not emerge from philosophical contortions.
The theory will be local in that it will obey the principle:
It will not be local in this sense:
These variables describe the analog of the entanglement between A and B.
Assume a classical Newtonian world, particles interacting by a retarded potential (so that you have relativistic locality), but with a secret world integer W. The world variable is just a stupid parameter that tells you which world you are on.
Two particles only interact with each other when they have the exact same value of the variable "W" (so that W is not a position, exactly, but a sheet-number), and initially, there is one copy of the particles on world 0.
The particles can, in addition to the Newtonian stuff, also do a "fork", which adds a new world to the list of all existing worlds (a new integer label), and copies all the particles nonlocally at their current positions into the new world, and the particle which is forked is altered in a subtle way--- it has a coupling to a forking-detector material (this can all be done explicitly in a computer model).
The copying of particles to the new world doesn't have to happen right away--- since the new world is identical to one of the old worlds, you can use the old world variables right up to the moment of first altered interaction, which propagates outward from the fork position at the speed of light.
Further, the forks have internal variables, the phases. Any two forks have a relative phase, which is a variable on the unit circle. When you fork A and fork B at angle $\theta$, you make worlds in the proportion: $cos^2(\theta)$ AB's and A'B' , $sin^2(\theta)$ AB' and A'B's. In the preceding, A and A' are the two splits descended from A, and B and B' are the two splits descended from B, and the relative angle only tells you how correlated the two splits are with each other, by arranging more copies at the point of collision (when an A split meets a B split) in the proportion determined by their relative angle.
This world-splitting classical model reproduces the quantum situation. This is not a surprise, because it is just made up to do so.
The point of this is that the quantum mechanics is only as nonlocal as this classical model. It is not clear whether the splitting and the angles are to be considered nonlocal, since they are only nonlocal in the sense of extra data, not nonlocal in the sense of influence. All the influences travel strictly at less than the speed of light.
First of all, the assumption of hidden variable is very explicit in Bell's paper, since he even gives a name to the hidden variable: $\lambda$.
However, I agree with you with the fact that it is difficult to imagine a theory without such hidden variable, so his assumption can bee seen as reducing to locality. More explicitly, a physical theory is almost by definition something which predicts the probability of obtaining some measurement results. This definition, is nothing else than the definition of a hidden variable describing the state of the system. In Bell's paper, the state is given by the Local hidden variable $\lambda$, in quantum mechanics, the hidden variable is nothing else than the (global) ket $\left|\psi\right\rangle$ or the density matrix $\rho$ and in classical physics, the hidden variable is the position in phase-space.
No need to complicate Bell's insight more than one should. The assumptions made by Bell in order to derive a measurable quantity, through which to check the existence of hidden variables, are laid quite clearly in his book “Speakable and Unspeakable in Quantum Mechanics,” pages: 36, 55, 81-87. Also in the book by Bohm and Hilley “The Undivided Universe” pages: 140-145. The idea is quite simple but very powerful in its scope. In the context of the EPR argument involving two spatially separated particles A and B, the spin of which are to be measured, Bell made the assumption that the outcome of the spin measurements is determined by a set of hidden variables lamda_a, mu_a and. lamda_b, mu_b. The hidden variable mu_a is associated with the apparatus measuring the spin of particle A, while lamda_a is associated with the particle A itself, similarly for particle B. These variables have probability distributions r_a(lamda_a), p_a(mu_a) and similarly for the spin measurement of particle B. The rest is ‘simple’ mathematical trickery through which Bell derives his famous inequality. The general way in which Bell developed his argument, based on well known statistical correlation theory, does not require any knowledge of the actual distributions r_a(lamda_a), p_a(mu_a) etc. The clever trick on Bell’s part is that, he worked out testable consequences of general simple assumptions! The nature of the argument is such that, if the experimental correlations verify Bell’s inequality, then hidden variables exist and bear an effect on these testable correlations. If however the correlations violate Bell’s inequalities, then hidden variables do not bear any observable effects. Violation of Bell's inequality also reveals the non-local nature of quantum mechanics. As far as I am aware all Aspect type experiments, and versions of it, violate Bell’s inequality, so one may argue what is the point in insisting that they exist? Hence until a set of experiments are found to agree with Bell’s theorem, hidden variables do not seem to exist. I hope this clarifies Bell’s strategy somewhat.