Bell inequalities from the incompatibility of experiments:

[Joy Christian]

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There remains much confusion in the physics community regarding the meaning of the so-called experimental "violations" of Bell inequalities. But there shouldn't be.

As is well known, Bell derived his inequalities by considering four incompatible experiments in any possible world, classical or quantum. What I have demonstrated in this paper, however, is that his tacit assumption of compatibility of the manifestly incompatible experiments is the ONLY assumption needed to derive the Bell-CHSH type inequalities. Therefore such inequalities have nothing to do with the issues of locality and realism. Here is a slightly more accurate conclusion from my paper:

To summarize our Corollary, Bell inequalities are usually derived by assuming locality and realism, and therefore
violations of the Bell-CHSH inequality are usually taken to imply violations of either locality or realism, or both. But
we have derived the Bell-CHSH inequality above by assuming only that Bob can measure along the directions b and b′
simultaneously while Alice measures along either a or a′, and likewise Alice can measure along the directions a and a′
simultaneously while Bob measures along either b or b′, without assuming locality. The violations of the Bell-CHSH
inequality therefore simply confirm the impossibility of measuring along b and b′ (or along a and a′ ) simultaneously.

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[Joy Christian]

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Needless to say, the above result is just the first step. The subsequent physical question is: Whether it is possible to understand all quantum correlations in a purely local and realistic manner. Not surprisingly, the answer is in the affirmative, and that is exactly what I have demonstrated recently: viewtopic.php?f=6&t=308#p7666.

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[Joy Christian]

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[Joy Christian]

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Imagine a couple, Jack and Jill, who decide to separate while in Kansas City, and travel to the West and East Coasts separately. Jack decides to travel to Los Angeles, while Jill can't make up her mind and might travel to either New York or Miami. So while Jack reaches Los Angeles, Jill might reach either New York or Miami. Now this is a perfectly good example of Einstein's local realism, understandable by anyone with a sound mind. Anyone can understand that there are two possible destinations for the couple. Either Jack reaches Los Angeles and Jill reaches New York, or Jack reaches Los Angeles and Jill reaches Miami. Only a lunatic would say that, no, no, no, local realism dictates that when Jack reaches Los Angeles, Jill reaches New York AND Miami at the same time. Only a lunatic would claim that. And yet, Bell and his followers assume exactly that to derive the upper bound of 2 on the CHSH inequality. How dumb one has to be to not recognize the silly mistake Bell made in 1964?

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[Joy Christian]

Imagine a couple, Jack and Jill, who decide to separate while in Kansas City, and travel to the West and East Coasts separately. Jack decides to travel to Los Angeles, while Jill can't make up her mind and might travel to either New York or Miami. So while Jack reaches Los Angeles, Jill might reach either New York or Miami. Now this is a perfectly good example of Einstein's local realism, understandable by anyone with a sound mind. Anyone can understand that there are two possible destinations for the couple. Either Jack reaches Los Angeles and Jill reaches New York, or Jack reaches Los Angeles and Jill reaches Miami. Only a lunatic would say that, no, no, no, local realism dictates that when Jack reaches Los Angeles, Jill reaches New York AND Miami at the same time. Only a lunatic would claim that. And yet, Bell and his followers assume exactly that to derive the upper bound of 2 on the CHSH inequality. How dumb one has to be to not recognize the silly mistake Bell made in 1964?

Let me extend above analogy to further illustrate the absurdity of Bell's argument. Suppose, for example, that after reaching New York Jill decides to buy either apple juice or orange juice. And likewise, after reaching Miami Jill decides to buy either apple juice or orange juice. Consequently, there are following four counterfactually possible events that can realistically occur at least in our familiar world:

(1) While Jack reaches Los Angeles and buys apple juice, Jill reaches New York and buys apple juice;

(2) While Jack reaches Los Angeles and buys apple juice, Jill reaches New York and buys orange juice;

(3) While Jack reaches Los Angeles and buys orange juice, Jill reaches Miami and buys apple juice;

(4) While Jack reaches Los Angeles and buys orange juice, Jill reaches Miami and buys orange juice.

So far so good. But now consider the following four "events":

(5) While Jack reaches Los Angeles and buys apple juice, Jill reaches New York and buys apple juice and Jill reaches Miami and buys orange juice at the same time;

(6) While Jack reaches Los Angeles and buys apple juice, Jill reaches New York and buys orange juice and Jill reaches Miami and buys apple juice at the same time;

(7) While Jack reaches Los Angeles and buys orange juice, Jill reaches New York and buys apple juice and Jill reaches Miami and buys orange juice at the same time;

(8) While Jack reaches Los Angeles and buys orange juice, Jill reaches New York and buys orange juice and Jill reaches Miami and buys apple juice at the same time.

Needless to say, no one in their right mind would claim that such events are possible in any possible world, even counterfactually. In particular, Einstein's local realism by no means demands such absurd or impossible "events." But Bell and his followers average over precisely such absurd and impossible events to derive the absolute bound of 2 on the CHSH string of expectation values (cf. my paper) and then blame Einstein for their own silly mistake. It is not at all surprising why the unphysical bound of 2 is not respected in the actual experiments. Duh, it is a bound on impossible events in any possible world. Why would it be respected in the experiments?

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[Joy Christian]

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Even though it is hidden in plain sight, let me bring out the sleight of hand in the proof of Bell's theorem (or in the derivation of the CHSH inequality). The sleight of hand occurs in Bell's proof in the form of a nonsensical application of Einstein's idea of realism. For convenience, I am including a figure below from one of my papers so that nothing is left to the imagination. In the present context of an EPR-Bohm type experiment, Einstein's idea of realism dictates that ALL possible measurement results such as A(a, h) about the measurement directions "a" are real in the sense that they are at least counterfactually realizable. If, for example, Alice chooses to perform her measurement about the direction "a", then she would observe the result A(a, h), where "h" is the hidden variable or an initial state of the singlet system, which is usually denoted by "lambda." Thus all, infinitely many, measurement results A(a, h) are "real" in Einstein's sense. They all exist and are in fact predetermined for Alice, at least counterfactually. Likewise, all measurement results B(b, h) are real for Bob, and they again exist, at least counterfactually, about the freely chosen measurement directions "b". And since all such A(a, h) and B(b, h) exist, so do their products such as A(a, h)B(b, h), A(a, h)B(b', h), A(a', h)B(b, h) and A(a', h)B(b', h).



So far so good. However, the proof of Bell's theorem, or the derivation of the bound of 2 in the CHSH inequality, depends on the evaluation of the following integral:



See eq. (16) of my paper to understand the details. As we can see, the integrand of this expression involves quantities like A(a, h) { B(b, h) + B(b', h) }. The immediate question then is: Are the quantities like A(a, h) { B(b, h) + B(b', h) } "real" in Einstein's sense? Are they at least counterfactually realizable? Note that it would be quite deceitful to claim that these are just intermediate mathematical quantities. They are not. They are THE quantities that are averaged over to obtain the bound of 2 in the CHSH inequality. Thus it is quite pertinent to ask whether these quantities are real in Einstein's sense. If they are real in Einstein's sense, then Bell's theorem goes through. But if they are not real in Einstein's sense, then Bell inequalities are simply mathematical curiosities without any physical significance. Now it is very easy to see that quantities like A(a, h) { B(b, h) + B(b', h) } are entirely fictitious. They are NOT realizable in ANY possible world. Indeed, they are analogous to the events like

While Jack reaches Los Angeles and buys apple juice, Jill reaches New York and buys orange juice and Jill reaches Miami and buys apple juice at exactly the same time.

Thus the fictitious quantities such as A(a, h) { B(b, h) + B(b', h) } appearing in the above integral are NOT real in Einstein's sense, and therefore Bell's theorem fails.

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[Xray]

Thanks Joy,

1. Do you have a source (or sources) for this, please: "In the present context of an EPR-Bohm type experiment, Einstein's idea of realism dictates that ALL possible measurement results such as A(a, h) about the measurement directions "a" are real in the sense that they are at least counterfactually realizable. If, for example, Alice chooses to perform her measurement about the direction "a", then she would observe the result A(a, h), where "h" is the hidden variable or an initial state of the singlet system, which is usually denoted by "lambda." Thus all, infinitely many, measurement results A(a, h) are "real" in Einstein's sense. They all exist and are in fact predetermined for Alice, at least counterfactually."

2. Re this: "They all exist and are in fact predetermined for Alice, at least counterfactually." What does it mean for something to exist and be predetermined? And when you add "at least counterfactually", what meaning are you giving to "counterfactual"?

3. What do you make of this particular Einstein: "I concede that the natural sciences concern the “real,” but I am still not a realist. (EA 22-307, ECP-8-624)".

My thanks again,

Xray

[Joy Christian]

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Xray, the standard source for all the points you have raised is J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881 (1978).

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[Joy Christian]

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Every morning I wake up and scratch my head at the mind-boggling absurdity of Bell's argument. How on earth has the physics community come to accept such a silly argument, with many otherwise intelligent people defending it feverishly as if it were a God-given gospel?

Think about this for a second. The absolute bound of 2 on the CHSH inequality is obtained by Bell and his followers by averaging over many spacetime events like



But no such events can possibly occur in any possible world, classical or quantum, even counterfactually, because a, a', b and b' are mutually exclusive macroscopic directions in the three-dimensional physical space! Indeed, the events being averaged over by Bell and company are analogous to impossible events like the following:

While Jack reaches Los Angeles and buys apple juice, Jill reaches New York and buys orange juice and Jill reaches Miami and buys apple juice at exactly the same time, and simultaneously Jack reaches Seattle and buys orange juice, while Jill reaches New York and buys apple juice and Jill reaches Miami and does not buy orange juice at exactly the same time.

No wonder the bound of 2 obtained by averaging over such impossible events is not respected by Nature. Should we not lock up all Bell-belivers in a lunatic asylum?

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[Joy Christian]

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The bottom line:

All infinity of results A(a, h) "exist" in the sense that they are at least counterfactually realizable in an experiment, in principle, in our world.

All infinity of results B(b, h) "exist" in the sense that they are at least counterfactually realizable in an experiment, in principle, in our world.

But the "results" A(a, h) { B(b, h) + B(b', h) } cannot be realized even counterfactually, in any possible world, in any concivable experiment.

Bell's mistake is to explicitly assume and average over results of the type A(a, h) { B(b, h) + B(b', h) } to obtain the upper bound of 2 on CHSH.

Any precocious schoolchild should be able to see that this is a fatal mistake.

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[Joy Christian]

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Bell inequalities are derived by averaging over events that cannot possibly occur in any possible world.

How incredibly stupid one has to be to passionately believe and defend the so-called theorem of Bell ?

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[Joy Christian]

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I have revised the paper on the arXiv to match the published version: https://arxiv.org/abs/1704.02876

Slightly revised Abstract:

Bell inequalities are usually derived by assuming locality and realism, and therefore experimental violations of Bell inequalities are usually taken to imply violations of either locality or realism, or both. But, after reviewing an oversight by Bell, here we derive the Bell-CHSH inequality by assuming only that Bob can measure along the directions b and b' simultaneously while Alice measures along either a or a', and likewise Alice can measure along the directions a and a' simultaneously while Bob measures along either b or b', without assuming locality. The observed violations of the Bell-CHSH inequality therefore simply verify the manifest impossibility of measuring along the directions b and b' (or along the directions a and a') simultaneously, in any realizable EPR-Bohm type experiment.

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[Jarek]

Bell inequalities from the incompatibility of experiments:

If you are saying that the problem is incompatibility of experiments, what do you think of

Pr(A=B) + Pr(A=C) + Pr(B=C) >= 1

Bell-like inequality: that tossing 3 coins, 2 of them give the same value ... but can be violated by QM (and MERW).

It was rather not tested experimentally, but theoretically QM formalism allows to violate it - do you agree with such possibility? How would you explain it?

[FrediFizzx]

It is NOT mathematically possible for anything to "violate" a Bell inequality. You are mathematically tricking yourself. But you are not alone; this trickery has been going on for over 50 years now.

[Jarek]

FrediFizzx, it is a bit more complicated - derivation of Bell-like inequalities uses some assumptions, hence their violation by QM formalism means only that it does not satisfy these assumptions.

For example in
Pr(A=B) + Pr(A=C) + Pr(B=C) >= 1
probability of the "A=B" scenario for three binary variables is Pr(000) + Pr(001) + Pr(110) + Pr(111).
Adding such three sums for "A=B", "A=C" and "B=C" we get all 8 possibilities: 1 plus 2 Pr(000) + 2 Pr(111)
getting above inequality, which corresponds to obvious "tossing 3 coins, at least 2 are equal".

However, Pr(A=B) = Pr(000) + Pr(001) + Pr(110) + Pr(111)
contains a subtle assumption: probability of alternative of disjoint events is sum of their probabilities.

In contrast, QM (and MERW) formalism uses Born rule instead: probability of alternative of disjoint events is proportional to sum of square of their amplitudes.
It allows to violate above inequality by using psi000 = psi111 = 0, and equal for the remaining 6 - getting violation to 3/5.

[minkwe]

"tossing 3 coins, at least 2 are equal".

Sure, tossing 3 coins once each (ABC), 2 will be the same (A=B or B=C or A=C), but What about tossing two of the three coins each time 3 times (AB, AC, BC)? Will you still expect (A=B or B=C or A=C). If you think so, do the experiment and post your results.

This is the Bell delusion.

[Jarek]

Imagine there is some probability distribution among 2^3 = 8 possibilities, which sum to 1.
Pr(A=B) = Pr(000) + Pr(001) + Pr(110) + Pr(111)
Pr(A=C) = Pr(000) + Pr(010) + Pr(101) + Pr(111)
Pr(B=C) = Pr(000) + Pr(100) + Pr(011) + Pr(111)
Summing all three and using that \sum_ABC Pr(ABC) = 1:
Pr(A=B) + Pr(A=C) + Pr(B=C) = 1 + 2 Pr(000) + 2 Pr(111) >= 1

However, it is violated in QM formalism - see page 9 of Preskill's lectures: http://www.theory.caltech.edu/people/pr ... /chap4.pdf

If we would measure all three "coins", above derivation would apply - we couldn't violate the inequality.
For violation it is crucial that we measure only 2 at once - the third one remains unfixed - so we should use Born rules here: with square allowing for the violation.

For example take the following amplitudes:
psi000 = psi111 = 0,
psi001 = psi010 = psi100 = psi011 = psi101 = psi110 = 1/sqrt(6)

Now using Born rules, for example measuring only A and B:
Pr(A=B) ~ (psi000 + psi001)^2 + (psi110 + psi111)^2
what leads to Pr(A=B) + Pr(A=C) + Pr(B=C) = 3/5

[FrediFizzx]

If we would measure all three "coins", above derivation would apply - we couldn't violate the inequality.
For violation it is crucial that we measure only 2 at once - the third one remains unfixed - so we should use Born rules here: with square allowing for the violation.

Easy to see where it goes wrong. If C is not measured at the same time as A and B, and you are only using A and B, then you have changed the inequality. The original inequality is never violated.

[Jarek]

Sure, but saying that while measuring A and B there is still some unknown value of C - would lead to this inequality, violated e.g. by QM formalism.

It is far nontrivial to get a model violating Pr(A=B) + Pr(A=C) + Pr(B=C) >= 1 ... we need to understand what does falseness of "C has some unknown value" means?
Do you have some self-consistent model allowing for such violation?

Personally, I think this violation comes from Born rule: probability of alternative of disjoint events is proportional to sum of square of their amplitudes.
It is built in quantum formalism, and can be seen in euclidean path ensembles or equivalent MERW ( https://en.wikipedia.org/wiki/Maximal_E ... andom_Walk ) - that we need to consider statistical ensembles of entire paths.
This way we mathematically get amplitudes while considering ensembles of half-paths: toward past or future.
To get probability of randomly getting given value now, we need to get it from both time directions: probability is proportional to square of amplitudes ... like in below prediction of stationary probability for infinite potential well:

[FrediFizzx]

Sure, but saying that while measuring A and B there is still some unknown value of C - would lead to this inequality, violated e.g. by QM formalism.

It is far nontrivial to get a model violating Pr(A=B) + Pr(A=C) + Pr(B=C) >= 1 ... we need to understand what does falseness of "C has some unknown value" means?
Do you have some self-consistent model allowing for such violation?

Of course not since violation is impossible. The string Pr(A=B) + Pr(A=C) + Pr(B=C) is meaningless if C is not measured at the same time as A and B. You have changed the inequality to only including Pr(A=B).

Of course there is a local realistic model that does match the predictions of QM. But it also never "violates" any of the Bell inequalities. It is apples to oranges.