The real puzzle

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

Re: The real puzzle

Postby minkwe » Sat May 03, 2014 1:55 pm

gill1109 wrote:
minkwe wrote:You still haven't told us what the upper bound is for the expression (from your LG paper), when the coincidence probability is γ=0.5:
δ > 0
δ ≥ 4 − 3/γ
| E(AC′|ΛAC′) + E(AD′|ΛAD′)| + |E(BC′|ΛBC′) − E(BD′|ΛBD′)|≤ 6/γ − 4

it is a simple calculation, just replace gamma in those expressions by 0.5 and see the ceiling collapse, and it immediately reveals how fatally flawed the paper is


With gamma equal 0.5, Your expressions become
δ > 0
δ ≥ -2
| E(AC′|ΛAC′) + E(AD′|ΛAD′)| + |E(BC′|ΛBC′) − E(BD′|ΛBD′)|≤ 8

You start by assuming in your paper that δ > 0, but then here we have an experimental situation that will give you δ ≥ -2, clearly contradicting your earlier assumption. Keeping your assumption alive, then it implies from the second expression that your theorem can only be valid for γ ≥ 0.75. Clearly then, contrary to your suggestion in the paper that that you save Bell's analysis from the coincidence loophole, you actually do not. Your theorem is not valid for precisely the experiments where coincidence matching is more important, those with γ < 0.75. Interestingly, replacing γ = 0.75 into the last expression gives you an upper bound of 4. The highest possible value, corresponding to the lowest possible γ value according to your theorem.

Then the last expression claims an upper bound of 8, even though we know for a fact that 4 is the absolute upper bound of expressions of that form.
The last expression is trivial nonsense. In fact, your theorem, had it been correct, should never give a RHS above 4. But it implies that it is possible to reach 7.9, for γ=0.5. Your theorem is just as relevant as the claim that the upper bound for the CHSH is when we know for a fact that it cannot be above 4.
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Re: The real puzzle

Postby gill1109 » Sat May 03, 2014 2:02 pm

minkwe wrote:With gamma equal 0.5, Your expressions become
δ > 0
δ ≥ -2
| E(AC′|ΛAC′) + E(AD′|ΛAD′)| + |E(BC′|ΛBC′) − E(BD′|ΛBD′)|≤ 8


I see no contradictions either between these three expressions or with anything else.

If I say "pi = 3.14..." and then later prove a theorem saying "pi >= 3" there is no contradiction.

There are values of gamma for which our theorem is interesting / useful. And values of gamma for which it is not interesting /useful. A theorem is a theorem. A tautology. The statement of a theorem is a necessary logical conclusion of the assumptions of the theorem. That's the definition of a theorem. A tautology is something which is true by definition.

So there is also a trivial bound
| E(AC′|ΛAC′) + E(AD′|ΛAD′)| + |E(BC′|ΛBC′) − E(BD′|ΛBD′)|≤ 4
so we could have restated our theorem, trivially, as
min(| E(AC′|ΛAC′) + E(AD′|ΛAD′)| + |E(BC′|ΛBC′) − E(BD′|ΛBD′)|≤ min(6/gamma- 4, 4).

Later in the paper we prove that that stronger bound min(6/gamma - 4, 4) can be attained. Meaning, for instance, that if we take gamma = 0.5, so that min(6/gamma - 4, 4) = 4, then a local realist simulation model can "hit" gamma = 0.5 and have all four correlations in question equal to +/-1, three positive and one negative (or three negative and one positive). Get it?

Our paper tells us the exact (large N limit) of what local realistic simuations can achieve through use of the coincidence loophole (the other loopholes excluded).
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Re: The real puzzle

Postby minkwe » Sat May 03, 2014 2:17 pm

gill1109 wrote:
minkwe wrote:With gamma equal 0.5, Your expressions become
δ > 0
δ ≥ -2
| E(AC′|ΛAC′) + E(AD′|ΛAD′)| + |E(BC′|ΛBC′) − E(BD′|ΛBD′)|≤ 8


I see no contradictions either between these three expressions or with anything else.

If I say "pi = 3.14..." and then later prove a theorem saying "pi >= 3" there is no contradiction.


You admit that your theorem is only valid if δ > 0
Therefore your theorem is only valid if γ >= 0.75
Do you deny that?
This means your theorem is not valid if γ < 0.75
Therefore you should not be surprised by the nonsensical results you obtain for values of γ less than 0.75.

A theorem that suggests the two statements S <= 4, and S <= 8 are valid at the same time is inconsistent. The second statement cannot be correct if the first one is. We know that the first one is correct, therefore the second one is wrong, even though 4 <= 8. The first statement says S can be any value less than or equal to 4. The second one says S can be any value less than or equal to 8. Which is wrong because we know for a fact that S can never have a value between 4 and 8.

Your theorem is therefore stuck with either restricting it's domain to γ >= 0.75, or admitting nonsensical upper bounds. In fact, your δ > 0 assumption already forces you to γ >= 0.75. There is no other way. How then can it be true that your theorem rescues Bell's analysis from the coincidence time problem, as you claim in the paper?
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Re: The real puzzle

Postby gill1109 » Sat May 03, 2014 2:33 pm

minkwe wrote:A theorem that suggests the two statements S <= 4, and S <= 8 are valid at the same time is inconsistent. The second statement cannot be correct if the first one is. We know that the first one is correct, therefore the second one is wrong, even though 4 <= 8. The first statement says S can be any value less than or equal to 4. The second one says S can be any value less than or equal to 8. Which is wrong because we know for a fact that S can never have a value between 4 and 8

Hold it. If I say that S <= 4 I am not saying that S can take *any* value <= 4. I am saying that in any particular situation covered by the conditions of the theorem, the value it does take, whatever that might be, is <= 4.

It does occur to me now that maybe we were careless in distinguishing between the two directions of possible interest.

There is a proof of a bound, and a proof that the bound is sharp.

The bound "6/gamma - 4" is a trivially true bound, but a trivially unattainable bound, in the case that 6/gamma - 4 > 4.

So how about we add to the assumptions: "suppose 6/gamma - 4 <= 4" (ie gamma > = 0.75), then ...

If gamma <= 0.75 then the bound is 4 and it is attainable. Happy with that?
Last edited by gill1109 on Sat May 03, 2014 2:41 pm, edited 1 time in total.
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Re: The real puzzle

Postby minkwe » Sat May 03, 2014 2:38 pm

gill1109 wrote:So there is also a trivial bound
| E(AC′|ΛAC′) + E(AD′|ΛAD′)| + |E(BC′|ΛBC′) − E(BD′|ΛBD′)|≤ 4
so we could have restated our theorem, trivially, as
min(| E(AC′|ΛAC′) + E(AD′|ΛAD′)| + |E(BC′|ΛBC′) − E(BD′|ΛBD′)|≤ min(6/gamma- 4, 4).

You could have but, you didn't. Even if you did, you already made the assumption that δ > 0 and since according to you, δ ≥ 4 − 3/γ, it follows that an assumption of δ > 0, is equivalent to an assumption of γ > 3/4. Your theorem is clearly only valid for those values of γ.

Larsson & Gill wrote:Proof. The proof consists of two steps; the first part is similar to the proof of Theorem 1, using the intersection
ΛI = ΛAC′ ∩ ΛAD′ ∩ ΛBC′ ∩ ΛBD′ ,
on which coincidences occur for all relevant settings. This ensemble may be empty, but only when δ = 0 and then the inequality is trivial, so δ > 0 can be assumed in the rest of the proof.
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Re: The real puzzle

Postby gill1109 » Sat May 03, 2014 2:42 pm

Splendid, I agree! You truly did spot a real oversight in our paper! Thanks.

If gamma >= 0.75 then the bound is 6/gamma - 4 and it is attainable

If gamma is <= 0.75 then the bound is 4 and it is attainable.

This is worth a correction note and a word of acknowledgement.

Or a new, short, joint paper.
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Re: The real puzzle

Postby minkwe » Sat May 03, 2014 2:47 pm

gill1109 wrote:Hold it. If I say that S <= 4 I am not saying that S can take *any* value <= 4. I am saying that in any particular situation covered by the conditions of the theorem, the value it does take, whatever that might be, is <= 4.

You forget basic logic and what is on the LHS. S is the possible result of a whole family of theories. S is not just one theory. the CHSH is prescribing what any LHV theory cannot do. So the inequality is clearly any value below the RHS is possible in LHV theories which meet those conditions, but those above the RHS are impossible.

You seem to misunderstand this basic fact, since in the past you have proposed one example of a theory which produces an S around 2, as evidence that all LHV theories should be around 2 (remember your silly R-script). You make the same mistake in your recent paper by mixing up the probability of a specific theory violating a bound, with the probability that any LHV theory can violate a bound, a rather nonsensical concept. In any case, that is fodder for a different discussion.
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Re: The real puzzle

Postby minkwe » Sat May 03, 2014 2:51 pm

gill1109 wrote:Splendid, I agree! You truly did spot a real oversight in our paper! Thanks.

If gamma >= 0.75 then the bound is 6/gamma - 4 and it is attainable

If gamma is <= 0.75 then the bound is 4 and it is attainable.

This is worth a correction note and a word of acknowledgement.

Or a new, short, joint paper.

We made some progress then.
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Re: The real puzzle

Postby gill1109 » Sat May 03, 2014 2:55 pm

minkwe wrote:You make the same mistake in your recent paper by mixing up the probability of a specific theory violating a bound, with the probability that any LHV theory can violate a bound, a rather nonsensical concept.

No. Take a LHV theory. There is some probability it can generate a value of CHSH larger, say, than 2.2. I gave a bound for the probability of violating the bound. Not a bound on the probability of a LUH theory. You are confusing the scope of universal quantifiers again, and you are again confusing "there exists" with "for all".

Anyway, we rescued Larsson and Gill. It makes a rather interesting statement, don't you think, for gamma = 10/11? It tells you something that a local realist computer simulation can't do, but apparently that QM can do. (Something which I believe you didn't know before).

(Though to be sure, no one tried to do that experiment in the quantum optics lab yet, so you would apparently consider that uninteresting, after all).

It makes a very different statement for gamma = 0.5, say. Possibly not so interesting. It tells you something that a local realist computer simulation can do. (Something which I believe you knew already).
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Re: The real puzzle

Postby gill1109 » Sat May 03, 2014 2:57 pm

minkwe wrote:
gill1109 wrote:Splendid, I agree! You truly did spot a real oversight in our paper! Thanks.

If gamma >= 0.75 then the bound is 6/gamma - 4 and it is attainable

If gamma is <= 0.75 then the bound is 4 and it is attainable.

This is worth a correction note and a word of acknowledgement.

Or a new, short, joint paper.

We made some progress then.

Yes, it has been a wonderful day, on many counts. I would say everyone learnt something to their benefit.
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Re: The real puzzle

Postby minkwe » Sat May 03, 2014 8:31 pm

gill1109 wrote:No. Take a LHV theory. There is some probability it can generate a value of CHSH larger, say, than 2.2. I gave a bound for the probability of violating the bound. Not a bound on the probability of a LUH theory. You are confusing the scope of universal quantifiers again, and you are again confusing "there exists" with "for all".

I don't believe I am. But in your opinion is Bell's theorem a "there exist" or a "for all". You'll realize by answering that question that I am not mistaken. When you say something is impossible, you are applying a negation to those. A negation of "there exists" clearly implies "for all ... it is impossible to ...". That's why to disprove the CHSH, you only need to find one counter-example, but to prove it, you need "all". Providing one example which obeys it does not tell you anything about the other possible examples out there which might violate it. And without a clear understanding of the other possibilities, you can not possibly calculate "a probability that the CHSH can be violated". Such a "probability" is only meaningful in the context of a specific LHV theorem.

Anyway, we rescued Larsson and Gill. It makes a rather interesting statement, don't you think, for gamma = 10/11? It tells you something that a local realist computer simulation can't do, but apparently that QM can do. (Something which I believe you didn't know before).

That's stretching it a lot. As I've been explaining since you first contacted me, and throughout on this forum, so-called violations of the CHSH are simply due to mathematical incompatibility of terms used. That has not changed. You say, LG provides a limit to what LR theorem can do but I fail to see how it does that. The original CHSH <= 2 provides that limit, and so those the other CHSH_disjoint <= 4. Those limits continue to be valid for the appropriate terms. For completely disjoint terms like in experiments, that absolute limit is 4. For a single set of particles, the absolute limit is 2. Of course it was obvious that there will be a variation from 2 to 4 as the sets become more and more disjoint. Although you (LG) discuss it in the light of coincidence probability, I believe it can be more generally applied to degree of "disjointedness". A local realistic simulation can be done for a single set just as easily as disjoint sets. The full range of 2 to 4 is therefore available, the appropriate bound therefore (being a mathematical tautology) continues to be impossible to violate either by QM or anything else. Maybe that is interesting to you but to me, it was already to be expected.
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Re: The real puzzle

Postby minkwe » Sat May 03, 2014 8:39 pm

Because of that, I disagree with you as concerns your claim that "QM can do". But you prefix that by saying "apparently" so I will scratch that and agree with you instead. QM appears to violate a certain bound, but such a violation is only apparent, if we fail to realize a few crucial points I've raised elsewhere. The QM correlations are for disjoint sets, therefore the upper bound to be applied should be 4. Now you'd say what about Tsirelson's bound? I have an answer for that if you are interested.

It makes a very different statement for gamma = 0.5, say. Possibly not so interesting. It tells you something that a local realist computer simulation can do. (Something which I believe you knew already).

My interpretation is somewhat different. LG doesn't say anything for gamma <= 0.75, above 0.75, it gives the appropriate bound that should be used to compare with. As I've told you from the beginning, I do not believe those bounds can ever be violated by anything if the correct bound is used. Not even a non-local theory can violate them. Yet it is possible to pick just the right degree of disjointedness, and compare it with an inappropriate bound in order to obtain apparent violations. Therefore I disagree that it is a limit to what a LR computer simulation can do. It is a limit to everything that is possible, experiments, simulations QM etc. Take your pick. The simulations are not privileged outsiders here. It is mathematically impossible to violate those bounds without having made a mathematical error. This is what the "QM can do that" story is all about. If you are interested we can discuss that part , starting with Bell's 1966 paper as well as Adenier's paper on the subject. The 4 QM correlations are disjoint. You have to use the appropriate inequality to compare it with. This is not being done.
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Re: The real puzzle

Postby gill1109 » Sat May 03, 2014 8:51 pm

minkwe wrote:But in your opinion is Bell's theorem a "there exist" or a "for all". You'll realize by answering that question that I am not mistaken. When you say something is impossible, you are applying a negation to those. A negation of "there exists" clearly implies "for all ... it is impossible to ...". That's why to disprove the CHSH, you only need to find one counter-example, but to prove it, you need "all".

In my opinion so-called Bell's theorem is not a theorem. Bell proved no theorem. I agree with his statement of the state of affairs in Chapters 13 and 16 of Speakable and Unspeakable. All we have at present, as far as logic is concerned, are metaphysical alternatives. It's a matter of taste what you like to believe, if you insist on believing something. Things might change in a few years. For instance, if someone did "Larsson and Gill's experiment" and observed gamma = 0.91 (0.01) and S = 2.80 (0.01) (standard errors in brackets) that would be something.

Take a look at http://arxiv.org/abs/1403.2811. Khrennikov is now working *with* the Vienna experimentalists and doing statistics for them.

Now various people have proven theorems. For instance, I proved Theorem 1 in my infamous recent paper. (Today I must submit the final revision). The mathematics in this paper has been studied in excuciating detail by numerous competent people and no-one has identified an error. No-one has given a counter-example. Many hoped it wasn't true, many tried.

Here is another very interesting paper with *theorems*.

Constraints on determinism: Bell versus Conway-Kochen
Eric Cator, Klaas Landsman
[url][url]http://arxiv.org/abs/1402.1972[/url][/url]
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Re: The real puzzle

Postby gill1109 » Sat May 03, 2014 8:54 pm

minkwe wrote:Therefore I disagree that it is a limit to what a LR computer simulation can do. It is a limit to everything that is possible, experiments, simulations QM etc. Take your pick. The simulations are not privileged outsiders here. It is mathematically impossible to violate those bounds without having made a mathematical error. This is what the "QM can do that" story is all about. If you are interested we can discuss that part , starting with Bell's 1966 paper as well as Adenier's paper on the subject. The 4 QM correlations are disjoint. You have to use the appropriate inequality to compare it with. This is not being done.

I disagree. You have to use statistics and give up certainty. Forget inequalities. Or more precisely: study statistics, and use it to build a bridge between theory and experiment.

The question is, are observed correlations close to theoretical QM correlations yes or no? CHSH says: without loopholes, under local realism, and with a large sample size, with large probability, no.

But let's leave this for the time being: nothing new here.
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Re: The real puzzle

Postby gill1109 » Sun May 04, 2014 5:06 am

minkwe wrote:δ > 0 (M1); LG (-)
δ ≥ 4 − 3/γ (M2); LG (17)
| E(AC′|ΛAC′) + E(AD′|ΛAD′)| + |E(BC′|ΛBC′) − E(BD′|ΛBD′)|≤ 6/γ − 4 (M3); LG (20)

My labels M1, M2, M3 (Michel) and LG formula numbers

Michel Fodje thinks that LG (Larsson and Gill (2004); http://arxiv.org/abs/quant-ph/0312035, contains untrue statements.
He determines this by looking at three equations in the paper and then trying out the test value gamma = 0.5.

Let's take a look at the context.

Suppose that we are given a LHV model with coincidence-interval post-selection, no other loopholes, the sample size is infinite - we talk just about probabilities, not about relative frequencies. Population means, not sample averages. Probability theory, not statistics. The statistical bridge between theory and experiment is another matter.

delta is defined in LG formula (10) so Michel's first statement that it is non-negative is a triviality: it is the minimum of four different conditional probabilities. In fact we know it lies between 0 and 1.

LG's Theorem 1 (the only theorem in their paper) states that under clearly defined conditions,

| E(AC′|ΛAC′) + E(AD′|ΛAD′)| + |E(BC′|ΛBC′) − E(BD′|ΛBD′)|≤ 4 - 2 delta (11)

He has not indicated anything wrong with the theorem. Obviously we also know, trivially, that the left hand side is nonnegative (it's the sum of two absolute values of various quantities. As delta varies between 0 and 1 (as obviously, it could, by varying the LHV model), the bound on the right hand side varies (in the opposite direction) between 2 and 4. A rather pretty extension of standard CHSH which has delta = 1 and bound = 2, that's one end of the spectrum. At the other end we have delta = 0 and bound = 4. The trivial bound which holds because the four "E(..|..) quantities lies between -1 and +1.

As a corollary to Theorem 1 which includes (11), LG proceed to derive (M2), which is LG's (20). On the way they of course have to define the new quantity gamma (there was no gamma in Theorem 1, but there is one in (M2) = (20)

gamma is defined in LG's (16). It is obviously a number between 0 and 1 since it is the minimum of four probabilities.

The proof that delta >= 4 - 3 / gamma is pretty trivial, Fodje gave no objections to it. This is LG's (17) and Fodjes (M2).

From this we trivially obtain Fodje's (M3) which is LG's (20).

From the context it is clear that LG claim that (20) is true if the conditions of Theorem 1 are true. No more and no less, at this point.

The fact that LG's (17) = Fodjes (M2) is trivially true if gamma <= 0.75 is not a problem. The corollary to Theorem 1 is trivially true if gamma <= 0.75.

So far, LG have simply said nothing whatsoever of any interest in the case that gamma <= 0.75. They have said something novel in the case that gamma > 0.75 because for any gamma between 0.75 and 1.00 their theorem tells us something which was not known before.

Next, LG say "let us see whether this bound is necessary and sufficient". They should have said, let us see if this bound is sharp.

They do not state any more formal theorems but merely give an example. The example shows that if you pick a number l between 0 and 1, you can arrange that gamma = (3 + l) / 4 and S = 4 * (3 - l) / (3 + l) which is the same as saying that S = 6/gamma - 4. Thus: gamma van be chosen freely between 3/4 and 1, and for each choice there is a LHV model with that value of gamma and with S = 6/gamma - 4.

This takes care of all of the values of gamma between 3/4 and 1. There is no need to consider smaller gamma, since for gamma = 3/4 the bound S <= 4 can be attained. So trivially we can attain the bound S <= 4 with gamma < 3/4.

Thus I conclude that there is no error at all in LG (2005) though to be sure, we had to cut off quite a few corners in order (a) to keep under the four page limit and (b) not burden the physicist reader with "pedantic" mathematical precision and formality.

Still I congratulate Michel on his careful reading of our paper which certainly brought up some points which would need clarification for non-experts in the field. Really, ever result likes this needs to be pulbished in two versions: one for the physicists, one for the mathematicians. However, I would say that a competent and knowledgable mathematician can make the translation from physics language to mathematics language without any difficulty, hence it is not even interesting, as pure mathematics; while the physicist doesn't understand the proof anyway - they just need to get the general idea, understand the main results, and hope that the referees did a good job checking math details. I can assure you, they did - they were very nit-pickish, but they too understood the conflicting constraints on a letter written by mathematicians with a somewhat more mathematical content than usual in europhysics letters... I think we tried PRL first but got completely supid referee reports and there was no point in wasting time arguing with them ... the editor seemed totally uninterested, anyway. Obviously this is more of a European than a US subject. Indeed, the Europeans continue to be the leaders in the field. The US physicists did not catch on till it was a bit late.
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Re: The real puzzle

Postby gill1109 » Sun May 04, 2014 5:14 am

By the way I don't see what Michel's analysis of LG (2005) Europhysics Letters in this thread has got to do with the question he raised and named "the real puzzle". Maybe he can explain why we are where we are now. Wildly off topic, but what the topic really was, was never quite clear to me, so my apologies if I helped us steer further still off course.
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Re: The real puzzle

Postby minkwe » Sun May 04, 2014 7:09 am

gill1109 wrote:I disagree. You have to use statistics and give up certainty. Forget inequalities. Or more precisely: study statistics, and use it to build a bridge between theory an experiment.

The question is, are observed correlations close to theoretical QM correlations yes or no? CHSH says: without loopholes, under local realism, and with a large sample size, with large probability, no.

But let's leave this for the time being: nothing new here.

Unfortunately, you are just wrong. Statistics is irrelevant. I've challenged you in the past to produce an appropriate data set from any source, which violates the appropriate upper bound by even 0.0000000000001 statistically. You have been unable. Keyword is appropriate. Introduce as much error as you like.

Your "silly" R script used an inappropriate upper bound.

This is a very important challenge isn't it:

You claim a certain upper bound applies to only LHV theories and simulations, not QM, and you propose bets to anyone who can violate it.

I call your bluff and in turn claim that your upper bounds are universal, nothing can violate it, not even QM, not even non-realism, not even nonlocalism. Then I challenge you to produce the non-real/non-local dataset which violates the appropriate bound by even 0.00000001.

Since you're in the business of formulating appropriate bounds for different scenarios, I would have expected you will take this seriously.

Interestingly, your LG paper, while attempting to rescue Bell, actually confirms its death. One may ask, what does QM say about delta or gamma? Adeniers paper answers that. There are only two possibilities. Either gamma is 1 (strongly objective) or gamma is zero ( weakly objective).

http://arxiv.org/abs/quant-ph/0006014

Bell's Theorem was developed on the basis of considerations involving a linear combination of spin correlation functions, each of which has a distinct pair of arguments. The simultaneous presence of these different pairs of arguments in the same equation can be understood in two radically different ways: either as `strongly objective,' that is, all correlation functions pertain to the same set of particle pairs, or as `weakly objective,' that is, each correlation function pertains to a different set of particle pairs.
It is demonstrated that once this meaning is determined, no discrepancy appears between local realistic theories and quantum mechanics: the discrepancy in Bell's Theorem is due only to a meaningless comparison between a local realistic inequality written within the strongly objective interpretation (thus relevant to a single set of particle pairs) and a quantum mechanical prediction derived from a weakly objective interpretation (thus relevant to several different sets of particle pairs).


Clearly, all your claims about what QM can do that LHV theories cannot do, rely on comparing apples and oranges. This is the real puzzle.
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Re: The real puzzle

Postby gill1109 » Sun May 04, 2014 7:20 am

minkwe wrote:Statistics is irrelevant.

I rest my case. Read Bell (chapters 13 and 16).
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Re: The real puzzle

Postby gill1109 » Sun May 04, 2014 7:23 am

minkwe wrote:Interestingly, your LG paper, while attempting to rescue Bell, actually confirms its death. One may ask, what does QM say about delta or gamma? Adeniers paper answers that. There are only two possibilities. Either gamma is 1 (strongly objective) or gamma is zero ( weakly objective).

The whole point is that gamma is observable. The right hand side of our inequality is just as experimentally accessible as the left hand side. In real experiments, gamma might be anything between 0 and 1.

The inequality refers to the population.

Take a large sample, and the observed averages and relative frequencies will, with large probability, give you decent approximations of all the terms. Publish a paper in Nature when the number of standard errors between bound and data is large enough. Like the guys who found the Higgs boson. Signal to noise ratio = 6. That was good enough for them. But I forgot ... statistics has no place in science. The only decent thing in science is writing computer programs which simulate the data of past experiments. This sounds to me like stamp collecting. One stamp collection equals past experiments. Another stamp collection equal computer simulations which match nicely. That's all that is real science - the rest is just chit-chat, speculation, pure mathematics (uninteresting tautologies). And a mathematical theorem which says that when gamma = 0.5 then S <= 8 is clearly a false theorem. So all theorems in mathematics are either false or irrelevant.

No wonder you are puzzled and ask puzzling questions.

Andrei Khrennikov, who used to write papers claiming that Bell was wrong, is nowadays working with top experimentalists and doing rather interesting statistics with them.
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Re: The real puzzle

Postby minkwe » Sun May 04, 2014 7:31 am

Things might change in a few years. For instance, if someone did "Larsson and Gill's experiment" and observed gamma = 0.91 (0.01) and S = 2.80 (0.01) (standard errors in brackets) that would be something.

1) No experiment will ever be done which violates the appropriate bound. It would be something as it would invalidate all of mathematics. Keep dreaming. It will never be done. Keyword is "appropriate".

2) The appropriate upper bound must take into account the observed gamma and its uncertainty. It can never be violated.
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