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harry wrote: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

harry wrote:Off topic, just curious:

gill1109 wrote:[..] 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 wrote:[..] I am concentrating on some other work at the moment: http://bengeen.wordpress.com/, http://arxiv.org/abs/1407.2731.

gill1109 wrote: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. [..]

gill1109 wrote:

harry wrote: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.

gill1109 wrote: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. [..]

FrediFizzx wrote: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.

harry wrote: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?

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.

harry wrote: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 wrote:

harry wrote: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.[..] .

harry wrote: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.

gill1109 wrote: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 [..]

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 wrote:

harry wrote: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.

harry wrote: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?

gill1109 wrote:

gill1109 wrote:

FrediFizzx wrote: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.

gill1109 wrote:

FrediFizzx wrote: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.

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

gill1109 wrote:This tells me that if we performed Christian's experiment, with random choice of settings for each of 10 000 runs, then the chance is not more than 1 in 100 that Christian would win his bet with me.

Joy Christian wrote:I have recently won the 10,000 Euros offered by Richard Gill for theoretically producing the 2n angular momentum vectors, and , appearing in the equation (16) of my proposed experiment. He had foolishly claimed that it was mathematically impossible to construct such 2n vectors and had challenged me to produce them as a "proof of concept" for my proposed experiment. I defeated his challenge in May and June of 2014 by explicitly producing the 2n vectors in these four simulations: (1), (2), (3), and (4) (the reason there are four simulations instead of just one is because he kept moving the goalpost each time I, or someone else, defeated his challenge). The correlations in these simulations are calculated as



together with



where is the number of experiments performed. These results strongly suggest that my proposed experiment will be a spectacular success. It will reproduce the manifestly local-realistic and yet strong correlations exactly as I have predicted in my papers, confirming my hypothesis that physical space we live in respects the geometry and topology of , not .

Gordon Watson wrote:

gill1109 wrote:

Gordon Watson wrote:General conclusion:

Gill's theorem is trivial for all η; ie, for any η ≥ 0. So Gill's claim (p.3) to have established "a new version of the famous Bell-CHSH inequality" is absurd. Further, to the extent that Gill's theorem is in any way associated with the Bell-CHSH inequality: it is refuted by commonsense local realism (CLR); see the Opening Post (OP) in this thread -- viewtopic.php?f=6&t=62#p2612

PS: Richard, I will be pleased to be corrected, but I note: Your theorem is refuted by my commonsense local realism (CLR) -- which, incidentally, appears to be a philosophy very similar to your own: given your rejection of nonlocality.

E and OE: Gordon

Gordon: try picking N = 10^6 and eta = 0.1
Perhaps you will discover that my theorem does say some interesting things.

You have shown that for some values of N and eta it merely tells us something trivially true. OK. That's not a problem. Please look at some interesting values of N and eta, now.

Richard, using your data (N = 10^6 and η = 0.1) and not being a wasteful academic; ie, I'm bound by DIL (Dill's Inefficiency Lemma):

I'd rather pick N < 10^3.

Would that be OK?

Gordon