mn4j said:
You completely missed the point of the critiques of Bell's theorem. You have a "calculator" called Bell's theorem which critics claim calculates wrongly, yet you ask that I prove that the "calculator" is wrong by providing numbers that will always give the "right" answer using the same "wrong calculator"?!
You were introducing a sophisticated model that was going to violate the proof of Bell's theorem in a more general context. In this simple context, the "model" reduces to 8 numbers, and they are the simplest form of Bell's theorem - of course in much less generality. If you claim to have a violation of ALL of Bell's theorem, then you must also find a way to violate this claim.
Now, you shift the argument, not to a technicality in the general proof of Bell's theorem, but to the critique that one uses values for simultaneously unmeasurable quantities. Now, that critique is of course rather ill-posed, because that's the very content of Bell's theorem: that one cannot give a priori values to simultaneously unmeasurable quantities! It's its very message.
If one limits oneself to simultaneously measurable quantities, then OF COURSE we get out a standard probability distribution. That's no surprise. No theorem is going to go against this. But you seem to have missed entirely the scope (and limitations) of what Bell's theorem tells us.
It tells us that it is not possible to have any pre-determined values for the outcomes of all possible (a priori yet not decided) measurements that will be done on the system that will generate the outcomes of quantum predictions. We have of course to pick one to actually measure, and then of course the other outcomes will not be compatible. But as we could have taken any of the 3 possible measurements, the outcomes have to be pre-specified if they are going to be responsible for the correlations. That's the idea. So telling me that I use probabilities for incompatible measurements is no surprise, it is the essence of the setup. The idea is that hidden variables pre-determine ALL potential results, but of course I can only pick one of them. Now, it doesn't matter what mechanism is used internally to keep this information, in no matter what algebraic structure, and no matter what mechanism is responsible for generating the outcome when the measurement device has been set up in a certain direction. The only thing that matters is that this outcome is fixed and well-determined for all potential outcomes, and it has to, because we are free to make the choice.
This comes from the basic assumption that correlations can only occur if there is a common cause. This is the basic tenet which is taken in the Bell argument: correlations between measurements MUST HAVE a common cause. If there is no common cause, then measurements are statistically independent. This doesn't need to be so, of course. Correlations "could happen". But we're not used to that.
If you flip a switch, and each time you flip it, the light in the room goes on or goes out, you might be tempted to think that there is some causal mechanism between the switch and the light. You would find it strange to find a switch you can flip, with the light that goes on and off with it, and nevertheless no causal link somehow between the two events.
So we came to think of any statistical correlation as being the result of a causal link (directly, in that one event influences the other, or indirectly, in that there is a common cause). For instance, when looking at the color of my left sock, it is usually strongly correlated with the color of my right sock. That doesn't mean that my left sock's color is causally determining the color of my right sock, it simply means that there was a common cause: this morning I took a pair of socks with identical color.
Take two dice. You can throw them "independently". Somehow we assume that the outcomes will be statistically independent. But it is of course entirely possible to find a Kolmogorov distribution of dice throws that makes that they are perfectly correlated: each time dice 1 gives result A, dice 2 gives result 7 - A. If I send dice 1 to Japan, and dice 2 to South Africa, and people throw dices, and later they come back and compare the lists of their throws, then they would maybe be highly surprised to find that they are perfectly anti-correlated. Nevertheless, there's no problem in setting up a statistical description of both dice that does this. So one wonders: is there some magical causal link between them, so that when you throw dice 1, you find an outcome, and that, through hyperwobble waves, influences the mechanics of dice 2 when you throw it ?
Or, is there an a priori programmed list of outcomes in both dices, which makes them just give out these numbers ? This last possibility is what Bell inquires.
So Bell's theorem tries to inquire up to what point there can be a common cause to the correlations of the quantum measurements on a Bell pair, starting from the assumption that every correlation must have a causal origin. As direct causal link is excluded (lightlike distance between the measurement events), only indirect causal link (common cause) can be the case. So Bell inquires up to what point, IF ALL OUTCOMES ARE PRE-DETERMINED (even though they cannot be simultaneously observed), the quantum correlations can follow from a common cause, assuming that the choice of the settings of the analysers is "free" (is not part of the same causal system).
That's all. As such, the critique that Bell's theorem assigns probabilities to simultaneously unobservable outcomes is ill-posed, because it is exactly the point it is going to analyse: is it POSSIBLE to pre-assign values to each of the (incompatible) measurement outcomes and reproduce the quantum correlations ? Answer: no, this is not possible. That's what Bell's theorem says. No more, no less.
But it is impressive enough. It means that under the assumptions that correlations must always occur by causal (direct or indirect) link and the assumption of "free" choice of the settings of the analyser, the quantum correlations cannot be produced by pre-assigning values to all possible outcomes. It would have been the most straightforward, classically-looking hidden variable implementation that one could obtain to mimick quantum theory, and it is not going to work.
But of course, you can reject one of the several hypotheses in Bell. You can reject the fact that without a direct or indirect causal link, correlations cannot happen. Indeed, "events can happen". You can think of the universe as a big bag of events, which can be correlated in just any way, without necessarily any causal link.
Or you can take the hypothesis that the settings of the analysers are no free choice actually, but are just as well determined by the source of the particles as the outcomes. That's called "superdeterminism" and points out that in a fully deterministic universe, it is not possible to make "statistically independent free choices" in the settings of instruments, as these will be determined by earlier conditions which can very well be correlated with the source.
Nevertheless, in both cases, we're left with a strange universe, because it wouldn't allow us in principle to make any inference from any correlation - which is nevertheless the basis of all scientific work. Think of double-blind medical tests. If statistical correlations are found between the patients who took the new drug, and their health improvement, then we take it that there must be a causal link somehow. It is not because we can find an entirely satisfying Kolmogorov distribution, and that it "just happened" that the people that took the new drug were correlated with a less severe illness, that we say that there is no causal effect. If we cannot conclude that a correlation is directly or indirectly causally linked, we are in a bad shape to do science. But that's nevertheless what we have to assume to reject Bell's conclusions. If we assume that correlation means: causal influence, then Bell's assumptions are satisfied. And then he shows us that we cannot find any such causal mechanism that can explain the quantum correlations.
Another way to get around Bell is to assume that there CAN be a direct causal link of the measurement at observer 1 to the measurement of observer 2. That's what Bohmian mechanics assumes.
Finally, a way to get around Bell is to assume that BOTH outcomes actually happened, and that the correlations only "happen" when we bring together the two results. That's the MWI view on things.
But Bell limits oneself to showing that there cannot be a list of pre-established outcomes for all potential measurements which generates the quantum correlations. As such, it is perfectly normal, in its proof, that one makes a list of pre-established outcomes, and assigns a probability to them.
Let us know when you've got a better theory, we'll be all ears. Remember that the reason we no longer believe in epicycles is the existence of hard experimental data which agreed better with Kepler's description than it did with any epicycle theory of comparable complexity (plus Newton and Einstein correctly predicted new phenomena that would not otherwise have been expected). So far all you have is a small controversy (over which particular classes of theory have been proven unworkable) not a working alternative to the complete mainstream theory.
