Bell’s Theorem Refuted: Bell’s 1964:(15) is False

[harry]

This is a physics forum. Does eta have any physical meaning? Gill seems to have not defined it.

Fred, This is a good question. What does the theorem mean? Fix N and a spreadsheet. Toss 2N fair coins, calculate S, the CHSH quantity based on four correlations each based on a different random subset of rows of the table. The value of S could in principle lie almost anywhere between -4 and +4. The theorem says, however, that with large probability it won't be larger than +2 by more than a few multiples of 1 / sqrt N.

For instance, let's consider eta equals quite a lot of multiples of 1 / sqrt N. For instance eta = 160 / sqrt N. We find that the chance S exceeds +2 by more than 160 / sqrt N is smaller than 8 exp(- 100). You can calculate that number on your pocket calculator. It is astronomically small.

If N = 256 million then eta = 0.01. So that's an interesting value of eta.

It is quite a large value of N. But such a value of N is commonplace in today's experiments. My theorem says something interesting about such experiments when, in the future, they are performed in rigorous experimental conditions: event ready detectors, no detection loophole, no locality loophole, fast random switching of settings. It says that under local realism it would be almost impossible to observe a value of S larger than 2.01.

I wonder what you mean with "realism" and "such experiments". This thread is drifting off topic and the discussions are a bit "schizophrenic" in the sense that some parallel discussions are very related but ignore each other. Your last remark suggests that you apply your theorem to more than simple coin experiments. It would be helpful if you clarify your "realism" and ""such experiments" in the light of the classical predictions that Fred highlighted in a parallel thread.
Thus, do you claim that your theorem applies to the kind of experiments as described in http://arxiv.org/abs/1307.2981 ?
And if yes, does that mean that you think that the authors (and their reviewers) are mistaken?
But if no, then what is the relevance of your theorem for Bell's theorem and related experiments?

[gill1109]

Yes this thread has drifted off-topic.

Gordon started a thread http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=63 called "Gill's theorem refuted". Then he started writing stuff about my theorem on this thread.

Now it is a fact that I believe that my paper contains a better proof of a stronger result than Bell's 1964. So if you want to know what *I* am talking about you had better read my paper "Statistics, causality and Bell's theorem" http://arxiv.org/abs/1207.5103

My theorem is in Section 2 and it is a little piece of elementary probability theory. If you read on into Section 3 you will find out what it has to do with physics and quantum mechanics. And in the last section of the paper you can find out what it has to do with LHV simulation models.

If you want to know what *this* thread is about you had better also (re-)read Bell (1964) carefully.

But I would also strongly recommend you also read Bell's paper "Bertlman's socks", written more than 15 years after his famous 1964 paper, containing a better mathematical proof and more clear argument as well as filling in several gaps in the earlier paper, which had caused a lot of confusion in the early years.

[gill1109]

Your last remark suggests that you apply your theorem to more than simple coin experiments. It would be helpful if you clarify your "realism" and ""such experiments" in the light of the classical predictions that Fred highlighted in a parallel thread.
Thus, do you claim that your theorem applies to the kind of experiments as described inhttp://arxiv.org/abs/1307.2981 ?
And if yes, does that mean that you think that the authors (and their reviewers) are mistaken?
But if no, then what is the relevance of your theorem for Bell's theorem and related experiments?

My theorem (and Bell's) is relevant to Bell-CHSH type experiments without loopholes. What such experiments are, you can read in "Bertlman's socks". In my paper you can read about the loopholes issue. A good experiment should be pulsed (or have event ready detectors) and it should not be vulnerable to the locality loophole and it should not be vulnerable to the detection loophole. Such experiments are just around the corner, it appears. The experimenters are very close and it is quite a race because the first loophole-free and succesful experiment will surely win the experimenters who do it the Nobel prize.

I am not interested in the experiments described in http://arxiv.org/abs/1307.2981 . The authors are clearly mixed up. Probably the reviewers and editors are also confused. A lot of people are. The same journal has published a paper showing experimental violation of Tsirelson's inequality hence contradiction with quantum theory. But nobody is citing that paper and nobody is talking about it. A lot of physicsts simply have no idea at all what it's all about.

[harry]

Yes this thread has drifted off-topic.

[..] if you want to know what *I* am talking about you had better read my paper "Statistics, causality and Bell's theorem" http://arxiv.org/abs/1207.5103 [..]

I see that your definition of "realism" goes beyond that of Einstein and Bell (see in particular the "socks" paper) - thus, perhaps what you call "non-realist" is what I and others still would call "realist"! However, that doesn't tell us if what I and many others consider "realistic" is or isn't at stake. To avoid such issues, I referred to a more appropriate modern example - and to which you indeed replied in your next message, cited here below.

But I would also strongly recommend you also read Bell's paper "Bertlman's socks", written more than 15 years after his famous 1964 paper, containing a better mathematical proof and more clear argument as well as filling in several gaps in the earlier paper, which had caused a lot of confusion in the early years.

I gave exactly the same recommendation in an earlier post today - with link provided.

Your last remark suggests that you apply your theorem to more than simple coin experiments. It would be helpful if you clarify your "realism" and ""such experiments" in the light of the classical predictions that Fred highlighted in a parallel thread.
Thus, do you claim that your theorem applies to the kind of experiments as described inhttp://arxiv.org/abs/1307.2981 ?
And if yes, does that mean that you think that the authors (and their reviewers) are mistaken?
But if no, then what is the relevance of your theorem for Bell's theorem and related experiments?

My theorem (and Bell's) is relevant to Bell-CHSH type experiments without loopholes. [..]

I am not interested in the experiments described in http://arxiv.org/abs/1307.2981 . The authors are clearly mixed up. Probably the reviewers and editors are also confused. A lot of people are. The same journal has published a paper showing experimental violation of Tsirelson's inequality hence contradiction with quantum theory. But nobody is citing that paper and nobody is talking about it. A lot of physicsts simply have no idea at all what it's all about.

That's an astonishing reply to me, as I think that Physical Review (nowadays split up in A, B etc) is one of the most respected journals around! Anyway, thanks for your clarification that you don't accept the claims of that paper - despite their evidently correct calculations.
Thus you are claiming that your theorem also applies to such experiments, right?

Interestingly, we all seem to agree that most people - including those authors - are somewhat confused about this topic. And you now indicate that you also think that not both they and you can be right. However, I suspect that their conclusions are right - in which case your expansion of scope (similar to Bell) appears to be wrong.

[FrediFizzx]

That's an astonishing reply to me, as I think that Physical Review (nowadays split up in A, B etc) is one of the most respected journals around! Anyway, thanks for your clarification that you don't accept the claims of that paper - despite their evidently correct calculations.
Thus you are claiming that your theorem also applies to such experiments, right?

Interestingly, we all seem to agree that most people - including those authors - are somewhat confused about this topic. And you now indicate that you also think that not both they and you can be right. However, I suspect that their conclusions are right - in which case your expansion of scope (similar to Bell) appears to be wrong.

Yes, that reply is more than astonishing since the experiment has been done! LOL! So you don't need to "suspect" anymore. It looks like their conclusions are right.

http://arxiv.org/abs/1406.6239
"Violation of Bell's inequality for phase singular beams"

It's too new to have been published yet but given that their theoretical paper was published, I expect this one to be published also. They are certainly clever enough in their wordings. But we will see.

[harry]

That's an astonishing reply to me, as I think that Physical Review (nowadays split up in A, B etc) is one of the most respected journals around! Anyway, thanks for your clarification that you don't accept the claims of that paper - despite their evidently correct calculations.
Thus you are claiming that your theorem also applies to such experiments, right?
[..] .

Yes, that reply is more than astonishing since the experiment has been done! LOL! So you don't need to "suspect" anymore. It looks like their conclusions are right.

http://arxiv.org/abs/1406.6239
"Violation of Bell's inequality for phase singular beams"

[..] given that their theoretical paper was published, I expect this one to be published also.[..] .

The experimental evidence of that later paper is what I meant with "their evidently correct calculations".
And it matters to me that they are not the same people and not even the same institute.
[edit:] But it could be a double error - or a conspiracy of course.

[gill1109]

That's an astonishing reply to me, as I think that Physical Review (nowadays split up in A, B etc) is one of the most respected journals around! Anyway, thanks for your clarification that you don't accept the claims of that paper - despite their evidently correct calculations.
Thus you are claiming that your theorem also applies to such experiments, right?

Interestingly, we all seem to agree that most people - including those authors - are somewhat confused about this topic. And you now indicate that you also think that not both they and you can be right. However, I suspect that their conclusions are right - in which case your expansion of scope (similar to Bell) appears to be wrong.

Look: I haven't looked at the paper yet, just skimmed through the abstract. I get the impression they don't know what they are talking about with respect to Bell. The paper may well be valuable and interesting in other respects.

Even the most respected journals around often contain misguided and/or wrong results. Nobody is perfect, lots of people know lots about one thing and not much about other things ... referees are very busy and just quickly check if something appears interesting, they don't check the claims carefully, and anyway, they may well not be competent to do so.

Who is expanding any scope? I am not expanding the scope of Bell's theorem from the context which Bell originally spelled out for it. That context is very important. Read Bertlman's socks and look carefully at the "cartoon" of the experiment. The paper you are talking about looks at entanglement in a different context and shows that, in that context, it can be explained classically. So I say: so what? The paper has simply got nothing to do with our Bell theorem discussion.

[gill1109]

Thus, do you claim that your theorem applies to the kind of experiments as described in http://arxiv.org/abs/1307.2981 ?
And if yes, does that mean that you think that the authors (and their reviewers) are mistaken?
But if no, then what is the relevance of your theorem for Bell's theorem and related experiments?

The paper you refer to has the following abstract:

We consider optical beams with topological singularities which possess Schmidt decomposition and show that such classical beams share many features of two mode entanglement in quantum optics. We demonstrate the coherence properties of such beams through the violations of Bell inequality for continuous variables using the Wigner function. This violation is a consequence of correlations between the (x,px) and (y,py) spaces which mathematically play the same role as nonlocality in quantum mechanics. The Bell violation for the LG beams is shown to increase with higher orbital angular momenta l of the vortex beam. This increase is reminiscent of enhancement of nonlocality for many particle Greenberger-Horne-Zeilinger states or for higher spins. The states with large l can be easily produced using spatial light modulators.

The abstract talks about "violations of Bell inequality for continuous variables". Well that is a contradiction in terms. Bell's inequality is about measurements with binary outcomes. I conclude that the authors may or may not be doing interesting optics, but they don't know much about Bell's theorem.

You ask: "what is the relevance of your theorem for Bell's theorem and related experiments"? My theorem is just a version of Bell's theorem. My assumptions are weaker in some respects and stronger in others; my conclusion is stronger. I explicitly assume that settings are repeatedly chosen at random. I do not assume independent and identically distributed drawings from the distribution of the hidden variables. So in my theorem, the randomness is in the settings, not in the measurement outcomes. My conclusion is stronger since I give a probability inequality for the result of N runs, N finite, whereas Bell tells us about the deterministic limit for N to infinity. Bell's conclusion can be deduced from mine on taking the limit as N converges to infinity. Bell and I are talking about the same experiments. Priyanka Chowdhury, A. S. Majumdar, G. S. Agarwal are talking about experiments with a completely different layout, different format. Their experiments are not Bell-CHSH type experiments at all. They are playing a completely different game.

[FrediFizzx]

As expected; complete denial. LOL!

Perhaps you should have read section II. "Bell Inequalities for Continuous Variable Systems" before you made your comments. But that is OK; perhaps it is too much physics for you.

[gill1109]

As expected; complete denial. LOL!

Perhaps you should have read section II. "Bell Inequalities for Continuous Variable Systems" before you made your comments. But that is OK; perhaps it is too much physics for you.

Yes, that section should be interesting. There are generalized Bell inequalities for variables taking any number of values, even continuously many ones. In my work with Stefan Zohren on the CGLMP (Collins, Gisin, Linden, Massar, Popescu) inequality, see for instance http://arxiv.org/abs/1003.0616, Europhysics Letters 90 (2010) 10002, S. Zohren, P. Reska, R. D. Gill, W. Westra, "A tight Tsirelson inequality for infinitely many outcomes", we showed that in the limit of continuously many outcomes for this most simple generalization of CHSH to the case of more than two outcomes, we get a continuous variable Bell type inequality which actually coincides with the corresponding generalized Tsirelson inequality: i.e., quantum mechanics and local realism are subject to exactly the same bound. In other words: when we go to continuous variables, Bell and Tsirelson inequalities coincide: local realism can do everything that QM can do. It is amusing that in this limit we are back with the original EPR example of measuring position and momentum Q or P on two particles, except that we measure Q or P on one of the particles, Q + P or Q - P on the other. Moreover it turns out that in order to achieve the bound, one should not use a maximally entangled pair: one should *not* use the singlet state!

In the meantime, equally off topic, did you understand now the meaning of the variable eta, Fred? The physical significance of eta was that actually in my theorem we are comparing two functions: the cumulative probability distribution function of S - 2, ie the function
"eta maps to Prob(S - 2 <= eta)", and the function "eta maps to max(0, 1 - 8 exp( - N (eta/16)^2 ))", which is another cumulative probability distribution function. In fact it is a probability distribution of well known type related to the Rayleigh distribution. The right tail of the distribution is of Gaussian type, decreasing very fast to zero; all moments are finite; it is a very nice, very "tight" probability distribution in fact. To say it in other words, the probability distribution of sqrt(N) ( S - 2) is stochastically smaller than a fixed generalized Rayleigh distribution. S can be larger than 2 but with large N, large deviations rapidly become very, very exceptional. Large deviations scale at the standard one over square root of N rate.

Back to the work of the Indian gentlemen Priyanka Chowdhury, A. S. Majumdar, G. S. Agarwal: obviously, an elementary true mathematical theorem cannot have a counter-example. Bell's theorem is truly a tautology, and it is a simple tautology at that. So if they claim to have found a classical physical system which violates Bell's theorem then, translated into mathematics they have found a mathematical system which violates Bell's theorem hence also, logically, does not satisfy the conditions of Bell's theorem. So instead of chortling with joy that Bell has been proven wrong, one should take this as an opportunity to more fully grasp what Bell is about. The title of the paper "Nonlocal continuous variable correlations and violation of Bell's inequality for light beams with topological singularities" with first sentence of the abstract "We consider optical beams with topological singularities which possess Schmidt decomposition and show that such classical beams share many features of two mode entanglement in quantum optics ... " show that the authors don't understand Bell's theorem but have come up with a classical physical system which reproduces some features of quantum mechanics but not, however, in the context of Bell's theorem, ie not in a context in which the corresponding experiment would tell us anything about locality and realism. The corresponding experiment would be an experiment with one of the famous loopholes: the loophole allowing a local realistic explanation!

So it is a very nice pedagogical exercise for people like Fred and Harry to think carefully about a loophole-free Bell type experiment, and think carefully about the experiment belonging to this paper, and to identify the "relaxation" which allows violation of a Bell type inequality by a local realistic system. I would prefer that you, the students, do this exercise yourself in your own time, before I, the teacher, reveal the solution.

Actually I didn't read the paper yet because I am concentrating on some other work at the moment: http://bengeen.wordpress.com/, http://arxiv.org/abs/1407.2731.

[FrediFizzx]

LOL! I don't expect that you will ever understand the physics!

[harry]

Who is expanding any scope? I am not expanding the scope of Bell's theorem from the context which Bell originally spelled out for it. That context is very important. [..]

Bell expanded from reasoning about socks to light and ions, and it looks to me that you similarly expand from coins to the same.

Thus, do you claim that your theorem applies to the kind of experiments as described in http://arxiv.org/abs/1307.2981 ?
And if yes, does that mean that you think that the authors (and their reviewers) are mistaken?
But if no, then what is the relevance of your theorem for Bell's theorem and related experiments?

[..] The abstract talks about "violations of Bell inequality for continuous variables". Well that is a contradiction in terms. Bell's inequality is about measurements with binary outcomes. I conclude that the authors may or may not be doing interesting optics, but they don't know much about Bell's theorem.

[rearranging:]
Back to the work of the Indian gentlemen Priyanka Chowdhury, A. S. Majumdar, G. S. Agarwal: obviously, an elementary true mathematical theorem cannot have a counter-example. Bell's theorem is truly a tautology, and it is a simple tautology at that. So if they claim to have found a classical physical system which violates Bell's theorem then, translated into mathematics they have found a mathematical system which violates Bell's theorem hence also, logically, does not satisfy the conditions of Bell's theorem.
[..] the authors don't understand Bell's theorem but have come up with a classical physical system which reproduces some features of quantum mechanics but not, however, in the context of Bell's theorem, ie not in a context in which the corresponding experiment would tell us anything about locality and realism.

Thus you agree with me that what they are saying in their paper is incompatible with what you are saying in your paper so that not both can be correct. As that is quite different from what you originally commented in the parallel thread where it first came up, that clarification was certainly useful.

By the way, I remain riddled by your persistent claims about Bell's theorem which boil down to pretending that "locality" and "reality" belong to the realm of mathematical tautology - such that you even conclude that others who don't share your peculiar opinion must be wrong because their definitions of concepts slightly differs from yours! In my book an "elementary mathematical tautology" can only tell us about numerical relationships, and I dare say that most physicists would hardly be interested in "Bell's theorem" if that's what they meant with it. Interestingly, Wikipedia clarifies that such fruitless disagreements are due to a general lack of consensus: https://en.wikipedia.org/wiki/Definition_of_mathematics

You ask: "what is the relevance of your theorem for Bell's theorem and related experiments"? My theorem is just a version of Bell's theorem. My assumptions are weaker in some respects and stronger in others; my conclusion is stronger. I explicitly assume that settings are repeatedly chosen at random. I do not assume independent and identically distributed drawings from the distribution of the hidden variables. So in my theorem, the randomness is in the settings, not in the measurement outcomes. My conclusion is stronger since I give a probability inequality for the result of N runs, N finite, whereas Bell tells us about the deterministic limit for N to infinity. Bell's conclusion can be deduced from mine on taking the limit as N converges to infinity. [..]

Gill, thanks for your clarifications!

[harry]

Off topic, just curious:

[..] I am concentrating on some other work at the moment: http://bengeen.wordpress.com/, http://arxiv.org/abs/1407.2731.

Do you mean that they condemned him based on mere data snooping/fishing, without prior suspicion based on other information?

I remember also a case like that in England, where a judge had to be explained that many baby's unexpectedly die from natural causes; and similarly, that very often someone gains the lotto, but the fact that the chance for a particular individual of gaining the lotto is extremely small doesn't mean that the winner likely cheated!

[gill1109]

Off topic, just curious:

[..] I am concentrating on some other work at the moment: http://bengeen.wordpress.com/, http://arxiv.org/abs/1407.2731.

Do you mean that they condemned him based on mere data snooping/fishing, without prior suspicion based on other information?

I remember also a case like that in England, where a judge had to be explained that many baby's unexpectedly die from natural causes; and similarly, that very often someone gains the lotto, but the fact that the chance for a particular individual of gaining the lotto is extremely small doesn't mean that the winner likely cheated!

Yes. Exactly the same.

Well - a patient died unexpectedly. He had been wrongly diagnosed. There was a paranoid doctor. There was an atmosphere of fear and suspicion, chaos; disfunctional hospital. There was one nurse who was different from all the others, stood out in the crowd.

That's all it takes.

[gill1109]

By the way, I remain riddled by your persistent claims about Bell's theorem which boil down to pretending that "locality" and "reality" belong to the realm of mathematical tautology - such that you even conclude that others who don't share your peculiar opinion must be wrong because their definitions of concepts slightly differs from yours! In my book an "elementary mathematical tautology" can only tell us about numerical relationships, and I dare say that most physicists would hardly be interested in "Bell's theorem" if that's what they meant with it. Interestingly, Wikipedia clarifies that such fruitless disagreements are due to a general lack of consensus: https://en.wikipedia.org/wiki/Definition_of_mathematics

For me, at the moment, and following Bell hinself, "Bell's theorem" is the mathematical triviality: "LHV => CHSH inequality". It's a little theorem of elementary calculus. Or in my version, of elementary probability theory.

However, for other people, "Bell's theorem" is a metaphysical claim: "quantum theory is non-local" (or: "non local-realist"). In fact Clauser et al (ie: CHSH) were the first to give this metaphysical claim the name "Bell's theorem".

Bell derived a simple mathematical consequence of some simple mathematical assumptions. He called it his "theorem". It's indeed a tautology.

Bell also thought deeply about what his "theorem" might mean in the light (a) of quantum theory, (b) in the light of experiment. He said that there were at least four logically reasonable metaphysical stances which one might take. He later admitted the possibility of a fifth. Obviously there could be more.

[gris]

Mathematics is (IMO) the evolving ultimate unambiguous thus formal language of logic that allows taking implicitly clear shortcuts.