Pedagogical proofs of Bell's theorem

[Heinera]

I don't like discussing Bell's original 1964 paper. At only six pages, it is way to succinct for many people to fully understand. The were of course enough smart people that did understand it, and so subsequently produced clearer proofs of their own.

My favorite pedagogical line of proof can be sketched like this:

1. Show that the CHSH urn experiment¹ has an upper bound of 2 for the CHSH expression when N goes to infinity. It seems that everyone one the forum agree with this, so I'll skip the proof (although it is pretty simple).

2. A CHSH urn experiment can not produce the correlations of QM, i.e. a CHSH expression 2.8284 (this is a trivial corollary of 1).

3. Any LHV model can be reduced to a CHSH urn experiment (just replace the paper slips with the sequence [A(a1, lambda), A(a2, lambda), B(b1, lambda), B(b2, lambda)] and start drawing lambdas).

4. An LHV model can not produce the correlations of QM (this is a trivial consequence of 2 and 3).

Anyone who claims Bell's theorem is wrong must disagree with one of these steps. Which is it?

¹) Explained elsewhere on this forum

[Joy Christian]

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None of the above. No proof of Bell's argument (which is not a "theorem") exists without the assumption of the additivity of expectation values. But the additivity of expectation values is an invalid assumption for any hidden variable theory, regardless of its specific characteristics such as locality or realism. Therefore Bell's argument is not valid for any hidden variable theory.

I have no time to discuss this any further. The full details of my argument can be found in my paper: https://arxiv.org/pdf/1704.02876.pdf.

A one-page summary of my argument can also be found in Section II of my paper published in IEEE Access: https://ieeexplore.ieee.org/stamp/stamp ... er=9418997.
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[FrediFizzx]

@Heine LOL! All of what you wrote was shot down in 2007 by Joy. Maybe it is time for you to get more up to date!
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[Heinera]

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None of the above.
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Does it mean you disagree with all of the above steps? In particular, that you don't think the CHSH urn experiment has an upper bound of 2?

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But the additivity of expectation values is an invalid assumption for any hidden variable theory, regardless of its specific characteristics such as locality or realism.
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The CHSH urn experiment is a counterexample that proves you wrong.

[FrediFizzx]

@Heine It is all irrelevant since Joy shot it down in 2007. There is a flaw in your argument. It is up to you to find it. Time to get more up to date.
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[Heinera]

@Heine LOL! All of what you wrote was shot down in 2007 by Joy. Maybe it is time for you to get more up to date!
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Would you kindly inform us which of the above points he shot down? All of them, even the ones involving only trivial logic? He shot down logic?

[FrediFizzx]

@Heine LOL! All of what you wrote was shot down in 2007 by Joy. Maybe it is time for you to get more up to date!

Would you kindly inform us which of the above points he shot down? All of them, even the ones involving only trivial logic? He shot down logic?

There is a flaw in your argument. Up to you to find it. We are way way past this nonsense so not interested in it very much.
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[gill1109]

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None of the above. No proof of Bell's argument (which is not a "theorem") exists without the assumption of the additivity of expectation values. But the additivity of expectation values is an invalid assumption for any hidden variable theory, regardless of its specific characteristics such as locality or realism. Therefore Bell's argument is not valid for any hidden variable theory.

I have no time to discuss this any further. The full details of my argument can be found in my paper: https://arxiv.org/pdf/1704.02876.pdf.

A one-page summary of my argument can also be found in Section II of my paper published in IEEE Access: https://ieeexplore.ieee.org/stamp/stamp ... er=9418997.
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@Joy: It is been explained to you by myself and by others numerous times that the additivity of expectation values is not an assumption. It holds true for QM. So any theory which reproduces QM statistical predictions, has to reproduce the linearity.

@Heinera: I like the proof which you just sketched. For different audiences, I use different proofs.

For schoolchildren and for non STEM-educated academics, I explain with pictures why one would expect the biggest success-rate in the Bell game to be 75%. The picture I use is of a square with corners labelled X1, Y1, X2, Y2. I ask the kids: is it possible to colour each corner red or green such that exactly three of the four edges have endpoints coloured the same, one has endpoints coloured differently? Then we look at the Bell game. Alice and Bob go to different isolated cubicles. They can take what they like with them, but no means of communication. In their cubicles, they are each told a number "1" or "2" and then they each have to name a colour "red" or "green", and their joint aim is to do this such that the colours they name are the same if the two numbers they are told are not "2, 2", different if they are told "2, 2". This is repeated many times, ie, the game has many rounds. The numbers they are told will be completely random sequences of 1's and 2's.

Apparently (according to QM), if they each took one part of the singlet state with them for each round, and used the appropriate (Tsirelson bound achieving) measurements, they could have a success rate of 85%.

You could say that this is a proof by counting. There are just 16 ways to colour the corners of the square.

For probabilists and mathematical statisticians I presently use the short proof expressed in the language of modern probability theory in https://arxiv.org/pdf/2103.00225.pdf, "Comment on “Bell’s Theorem Versus Local Realism in a Quaternionic Model of Physical Space” which I expect to appear in IEEE Access quite soon. This proof is useful since a small modification of the proof leads to the martingale results which I used to neutralise the memory loophole 20 years ago, and which were used the 2015 experimentalists in their loophole-free experiments.

This proof is a proof by algebra: arguing that X2Y2 - X2Y1 - X1Y2 - X1Y1 can only take the values +/-2, if the Xi and Yj equal +/-1. Of course, that piece of algebra can be done just by checking the 16 combinations.

So either way it is just elementary combinatorics.

[Heinera]

@Heine LOL! All of what you wrote was shot down in 2007 by Joy. Maybe it is time for you to get more up to date!

Would you kindly inform us which of the above points he shot down? All of them, even the ones involving only trivial logic? He shot down logic?

There is a flaw in your argument. Up to you to find it. We are way way past this nonsense so not interested in it very much.
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I take it that you found no flaw in the argument, and that this is your attempt to sneak out of the whole thing.

[gill1109]

@Heine It is all irrelevant since Joy shot it down in 2007. There is a flaw in your argument. It is up to you to find it. Time to get more up to date.

Joy's 2007 paper, original version, had measurement outcomes not taking the values +/-1. It was immediately shot down by lots of people. Later he attempted to fix that. In some later attempts he reproduced the Bertlmann's socks phenomenon: the hidden variable lambda only took the values +/-1, it was a fair coin toss, and A(a, lambda) = -B(a, lambda) = lambda for all a and each lambda. The model predicted perfect anticorrelation whatever the settings. Bell's good friend Reinhold Bertlmann wore one pink sock and one blue sock on each foot, putting them on in the morning at random. This was back in the 60's, it was Reinhold's youthful protest against conventional authority.

We now have a new try, with your help Fred, including a computer program which does not implement Joy's analytical formulas, while Joy's formulas certainly do not reproduce the singlet correlations. I asked a simple question about your code: do the k'th pair of outcomes only depend on the k'th pair of settings and the k'th realisation of the hidden variable? Yes or no?

Yes, it is indeed all irrelevant, but it is fun. And pedagogically important to understand why many people find it hard to accept Bell's simple logic and simple maths.

[Joy Christian]

@Heine It is all irrelevant since Joy shot it down in 2007. There is a flaw in your argument. It is up to you to find it. Time to get more up to date.

Joy's 2007 paper, original version, had measurement outcomes not taking the values +/-1. It was immediately shot down by lots of people. Later he attempted to fix that. In some later attempts he reproduced the Bertlmann's socks phenomenon: the hidden variable lambda only took the values +/-1, it was a fair coin toss, and A(a, lambda) = -B(a, lambda) = lambda for all a and each lambda. The model predicted perfect anticorrelation whatever the settings. Bell's good friend Reinhold Bertlmann wore one pink sock and one blue sock on each foot, putting them on in the morning at random. This was back in the 60's, it was Reinhold's youthful protest against conventional authority.

We now have a new try, with your help Fred, including a computer program which does not implement Joy's analytical formulas, while Joy's formulas certainly do not reproduce the singlet correlations. I asked a simple question about your code: do the k'th pair of outcomes only depend on the k'th pair of settings and the k'th realisation of the hidden variable? Yes or no?

Yes, it is indeed all irrelevant, but it is fun. And pedagogically important to understand why many people find it hard to accept Bell's simple logic and simple maths.

Pure nonsense. Revisionist history and political propaganda by a fanatic Bell-believer who does not even know the difference between a bivector and a multivector. To me, Gill is a joke.
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[Justo]

@Heinera, the urn model is interesting. As far as I know the first one to come up with the idea was no other than the novel Laureate Eugene Wigner.
It would be interesting to produce a simple model of it in an excel spreadsheet so that everyone can see it and shoot down(or not) Bell's theorem for everyone to see.
Ii is not necessary for the simulation to reproduce QM predictions. It just has to violate the bound two (modulo finite statistics).

[FrediFizzx]

@Justo Done that! It's shot down so not interesting at all.
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[Justo]

@Justo Done that! It's shot down so not interesting at all.
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Congratulations! Then I agree Richard Gill is a fool not to mention all the rest (including me). BTW, I think NATURE will be interesting in publishing it.

[Joy Christian]

@Heinera, the urn model is interesting. As far as I know the first one to come up with the idea was no other than the novel Laureate Eugene Wigner.
It would be interesting to produce a simple model of it in an excel spreadsheet so that everyone can see it and shoot down(or not) Bell's theorem for everyone to see.
Ii is not necessary for the simulation to reproduce QM predictions. It just has to violate the bound two (modulo finite statistics).

Justo, the urn model is not interesting. It does not represent how the Bell-test experiments are performed. If it did, then there would be no point in performing the expensive experiments. It is extraordinary that you think anything can violate the bound of 2 on CHSH. Nothing can. And that fact makes Bell's argument ridiculously silly. Wigner, by the way, was the Ph.D. mentor of my Ph.D. mentor Shimony. But in my opinion, both of them were wrong in believing in the validity of Bell's argument.
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[Justo]

Justo, the urn model is not interesting. It does not represent how the Bell-test experiments are performed. If it did, then there would be no point in performing the expensive experiments. It is extraordinary that you think anything can violate the bound of 2 on CHSH. Nothing can. And that fact makes Bell's argument ridiculously silly. Wigner, by the way, was the Ph.D. mentor of my Ph.D. mentor Shimony. But in my opinion, both of them were wrong in believing in the validity of Bell's argument.
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I really do not understand you when you say that nothing can violate the Bell inequality. Obviously, data from experiments violate it. Also, Esail's nonlocal model violates it. So I guess what you call the Bell inequality must be something else. Even your and Fred's model violates it since they reproduce QM predictions.
The Bell inequality contains four different terms. Those terms represent the mean value of long series of four different and independent measurements performed jointly by Alice and Bob on entangled particles. Are you calling Bell inequality to something else?
For instance minkwe reinterprets the inequality and calls Bell inequality to something else. According to him, the inequality does not represent what I described before. If that is so, then you should not call Bell inequality to that "something else" because it is very confusing. In any case, you should say that the Bell inequality does not represent what it is supposed to and what Bell derived is tautologically less than 2.
For instance, the counterfactual derivation of the BI is tautologically less than 2. But then, it does not represent the real experiments as I described before.

[Joy Christian]

Obviously, data from experiments violate it.

Demonstrate, event by event, using only the experimental data, how experiments violate the bound of 2 on the CHSH inequality. Do this, because all the rest of what you say is nonsense.
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[Justo]

Obviously, data from experiments violate it.

Demonstrate, event by event, using only the experimental data, how experiments violate the bound of 2 on the CHSH inequality. Do this, because all the rest of what you say is nonsense.
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Well, I don't have the data. I am not an experimentalist. I trust experts in the field. I can't say more than that. Besides, calling nonsense other's people arguments is not a very convincing argument itself. I suspect that you don't have a good counterargument.

[Joy Christian]

Obviously, data from experiments violate it.

Demonstrate, event by event, using only the experimental data, how experiments violate the bound of 2 on the CHSH inequality. Do this, because all the rest of what you say is nonsense.
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Well, I don't have the data. I am not an experimentalist. I trust experts in the field. I can't say more than that. Besides, calling nonsense other's people arguments is not a very convincing argument itself. I suspect that you don't have a good counterargument.

The counterargument is very simple. You are claiming that a bound on a mathematical inequality is violated --- quite magically --- by experimental data. I call that nonsense because it is nonsense. You are claiming that you trust experts. I claim that the experts have fooled you enough to believe in the nonsense that a mathematical inequality can be violated by something.
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[Heinera]

Obviously, data from experiments violate it.

Demonstrate, event by event, using only the experimental data, how experiments violate the bound of 2 on the CHSH inequality. Do this, because all the rest of what you say is nonsense.
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What? Aspect et al., for starters. And then a rush of others over a 40 year period. What is your point here?