Is Bell's 1971 CHSH the same as Gill's CHSH version?

[FrediFizzx]

If not, what exactly is the difference? I'm on vacation right now so can't really do the detail of Bell's derivation of CHSH but it would be nice if someone could show the extra detail steps for the derivation. And keep in mind that the A and B functions are averages so show them like this, <A(a, lambda)>, etc.

From Michel's thread I gather that Bell's LHV CHSH derivation and Gill's version of CHSH are both "rigged" and nothing can violate them as far as violating a bound of 2. It is also easy to see from that thread that for both QM and quantum experiments the bound is 4 and not 2.

[gill1109]

You are asking the wrong question, Fred, and jumping to conclusions.

Suppose according to a certain mainstream theory, some theoretical constant mu is not larger then 2. Suppose there is an alternative theory which says mu can be as large as 2.8. Suppose we do an experiment to measure mu. Suppose there will be error in our measurement. For instance, the error might be Gaussian with standard deviation 0.1. In other words the experiment concludes with reporting a number for mu, say 2.3, 1.8, or whatever ... with error bars, +/-0.1.

If we got to observe 2.1 or 1.9, that would be consistent with said mainstream theory. Note: seeng a noisy measurement outcome for mu of 2.1 with s.e. 0.1 is not proof that the theoretical bound 2 has been exceeded. On the contrary, it is entirely consistent with mu=2. Or even with mu=1.9.

If we got to observe 2.2, then we would say, that's interesting. Two standard errors. "Statistically significant", good for a publication in psychology, ecology, and similar fields. Only 2.5% chance of seeing such a large value if mainstream theory were true. Maybe we should investigate the alternative theory. It has gained some (weak) experimental support.

If we got to observe, say, 2.7, we'd say, wow, 7 standard deviations above 2. The theory that says mu is not larger than 2 has bern disproved.

Remember the Higgs boson? The experimental proof was a signal 5 standard deviations larger than what it would have been if Higgs didn't exist.

[FrediFizzx]

Richard, we already know that you think they are the same. What you wrote above is completely irrelevant to the question.

[gill1109]

So are you also saying that Joy Christian's proof of the Tsirelson bound 2 sqrt 2 is wrong?

Wow!

Why did no one ever report a violation of the Tsirelson bound, 2.828..., if the only good bound is 4?

[Mikko]

And keep in mind that the A and B functions are averages so show them like this, <A(a, lambda)>, etc.

I would propose <A(a, lambda)> to show that lambda is a random variable, or <A(a, lambda)>, if a is, too. More generally, we should aim to maximal clarity of presentation.

[gill1109]

We need to distinguish between population means and sample averages. We need to be aware of the necessity of calculating error bars and taking account of statistical error. We need to think carefully about the relation between theory and experiment.

According to Fred and Michel, experiment is senseless because the only thing we can say is that the observed "CHSH" (I refer to that particular linear combination of four particular correlations) will not exceed 4.

We know that in advance, it's a logical necessity, it's the only logical necessity. (All this is true!).

So forget experiment. Forget measurement. Forget the Higgs. Close down CERN, put the money into logic.

Close down the Nobel prizes. From now on there will only be Abel prizes.

Because Fred Diether and Michel Fodje are so certain of their logic, they are not going to do a little experiment which I proposed in another thread. They know they know everything. They have nothing to learn. No experiment. No observation. Just logical deduction. The bound is 4.

Shut down all the threads. Close down the forum. Science has been reduced to logic and logic says CHSH <= 4.

[FrediFizzx]

And keep in mind that the A and B functions are averages so show them like this, <A(a, lambda)>, etc.

I would propose <A(a, lambda)> to show that lambda is a random variable, or <A(a, lambda)>, if a is, too. More generally, we should aim to maximal clarity of presentation.

That would be good also. Back from my mini-vacation and I see no one seems to want to tackle the finer details of the Bell derivation. I will give it a stab this up-coming weekend.

[minkwe]

Back from my mini-vacation and I see no one seems to want to tackle the finer details of the Bell derivation. I will give it a stab this up-coming weekend.

Fred, isn't the one on Wikipedia I already discussed the same as Bell's?

[gill1109]

And keep in mind that the A and B functions are averages so show them like this, <A(a, lambda)>, etc.

I would propose <A(a, lambda)> to show that lambda is a random variable, or <A(a, lambda)>, if a is, too. More generally, we should aim to maximal clarity of presentation.

That would be good also. Back from my mini-vacation and I see no one seems to want to tackle the finer details of the Bell derivation. I will give it a stab this up-coming weekend.

Fred, I think the finer details of the Bell derivation are irrelevant. I think the problem is that people don't distinguish between between theory and experiment, between population and sample.

Quantum theory predicts that in a certain situation

rho(0, 45) = - 0.7
rho(0, 135) = + 0.7
rho(90, 45) = - 0.7
rho(90, 135) = - 0.7

A certain classical physical theory predicts, however,

rho(0, 45) = - 0.5
rho(0, 135) = + 0.5
rho(90, 45) = - 0.5
rho(90, 135) = - 0.5

An experiment might result (standard errors in parentheses) in

E(0, 45) = - 0.63 (0.04)
E(0, 135) = + 0.69 (0.08)
E(90, 45) = - 0.83 (0.10)
E(90, 135) = - 0.54 (0.06)

Notice that the numbers E( , ) are just averages of products of outcomes (+/-1) of finitely many measurements, based on different runs. ie N(0, 45) pairs of particles measured at setting 0 and 45 degrees, another N(0, 135) at 0 and 135, and so on.

The numbers rho( , ) are given by mathematical formulas involving in the quantum case Hilbert space projections and in the classical case integration over the possible values of hypothetical infinite population, weighting with respect to their probability density.

If we don't bear these distinctions in mind we are going to stay going in circles for every. Once however we are able to understand these distinctions, we might be able to lift the discussion to a higher level.

In a theory, one can have "proof", "inequality", "bound", "violation". When talking about an experiment, we really can't use those words any more. Or only with some qualification, like "statistical proof", or "statistically significant violation". The qualification has a precise technical meaning though it does depend on some parameters which need to be specified (statistical size, power, ...), not part of everyone's common knowledge. The phrase "statistical proof" is an abbreviation for a quite subtle concept. And a concept which has nothing to do with proof in the sense it is used in mathematics or logic.

You might like to look at the pictures in http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=31&start=190#p1467. Joy and I are planning to use these graphics in a media offensive to drum up interest in our experiment and hence funding... Anyone who wants to help make them better, is welcome. Maybe Mathematica or Python can do this much more easily.

[minkwe]

Fred, I think the finer details of the Bell derivation are irrelevant.

Not true, you just submitted a paper discussing finer details. How else would you introduce the assumption of counterfactual definiteness without finer details? How else would you arrive at Bells theorem and all the talk of nonlocality or non-realism?

The magician's trick is in the finer details. Of course he won't want us to look at the finer details. But that is where we must look.

I think the problem is that people don't distinguish between between theory and experiment

I agree, people do not realise that genuine CHSH type experiments, although imaginable in the abstract theory, are unperformable. And performable experiments are governed by different inequalities which have never been violated, not even by experimental error.

Once they realise it, they will stop believing that LHV theories are impossible and they will stop discouraging others from looking for such theories.

[FrediFizzx]

Back from my mini-vacation and I see no one seems to want to tackle the finer details of the Bell derivation. I will give it a stab this up-coming weekend.

Fred, isn't the one on Wikipedia I already discussed the same as Bell's?

Yes, it is except you left out the averaging so all the A and B outcomes are averages below. I would like to see all the mathematical steps to get from here,

To here,

Additional explanations for the steps would be good also.

[gill1109]

Quick Bell: suppose that the random variables . Make a table with the eight possible values of and observe that . If the joint distribution of exists, then , yielding Bell's inequality .

Note 1: the existence of local hidden variables, and perfect anticorrelation in the case of equal settings, is used by Bell to argue that the joint probability distribution of A, B and C does exist.

Note 2:
|A(B-C)|=1-BC because if B = C, then both sides equal 0, while if B != C, both sides equal +2.

But indeed one can also prove this by writing out a table of all 8 possibilities.

Note 3:
People unfamiliar with probability might be unfamiliar with the rule |E(X)| <= E(|X|)

Proof. Define Xp := max(X, 0) and Xm := max(-X, 0). One can say that Xp and Xm are the positive and negative parts of X (where we gave the negative part a plus sign!). It follows from this definition that X = Xp - Xm. Therefore E(X) = E(Xp) - E(Xm).

On the other hand, |X| = Xp + Xm, so E(|X|) = E(Xp) + E(Xm).

E(Xp) and E(Xm) are two nonnegative numbers, and for two nonnegative numbers a and b, |a-b| <= a + b. QED

If you are unhappy with the statements about random variables like the two definitions and then four relations

Xp := max(X, 0)
Xm := max(-X, 0)

Xp >= 0
Xm >= 0
X = Xp - Xm
|X| = Xp + Xm

just check that for any particular numerical value taken on by X, the two definitions make sense and the four subsequent properties are all true. Probably no-one is unhappy with the claim that if a random variable is >= 0, then its expectation value is >= 0 too.

[gill1109]

Note 3:
People unfamiliar with probability might be unfamiliar with the rule |E(X)| <= E(|X|)

People unfamiliar with probability should not study this subject. They should grab, say, Feller's two volumes and learn it.

And after that, they should grab a book on statistics. I recommend John Rice, introduction to mathematical statistics and data analysis. It should be completely accessible to physicists and engineers. It's not only for mathematicians. Half of that book is an introduction to probability theory. If you don't need to know all the proofs of all the theorems, and don't need to learn about the deeper subtelties, you needn't study Feller's two volumes. They are only required reading for future probabilists. A statistician, especially an applied statistician, need not be a professional (academic) probabilist. (Probabilists are a kind of pure mathematicians. But statisticians are useful people).

[FrediFizzx]

Fred, isn't the one on Wikipedia I already discussed the same as Bell's?

Yes, it is except you left out the averaging so all the A and B outcomes are averages below. I would like to see all the mathematical steps to get from here,

To here,

Additional explanations for the steps would be good also.

Getting back to this and I will just do in plain text and use L for lambda and <> for averages. The only way I see getting from the first expression to the second is if,

<A(a', L)><B(b', L)> = 0 and <A(a', L)><B(b, L)> = 0

But how can that be? What am I missing?

[FrediFizzx]

LOL! Never mind. Those averages for A and B will tend to zero for large enough N. Both the derivation on wikipedia and in the Clauser-Shimony review paper for Bell's 1971 derivation kind of throw you off since they show |<A(a, L)>|=< 1; same for B which is of course true. Well... I'm an old codger and tend to forget things easily. So what we really have is for large enough N,

<A(a, L)> = <A(a', L)> = <B(b, L)> = <B(b', L)> = 0

[minkwe]

Fred, I too do not yet see how Bell went from 1 to 2. The way I have usually derived the CHSH has been similar to how Alain Aspect did it in his paper http://arxiv.org/ftp/quant-ph/papers/0402/0402001.pdf

s = A(a,L).B(b,L) - A(a,L).B(b',L) + A(a',L).B(b,L) + A(a',L).B(b',L)
= A(a,L)[B(b,L) - B(b',L)] + A(a',L)[B(b,L) + B(b',L)]

Since A, B can only take values ±1, when [B(b,L) - B(b',L)] = ±2, [B(b,L) + B(b',L)] must be 0, therefore

s = ±2

which means

−2 ≤ ∫ dLp(L)s(L, a, a',b,b') ≤ 2
−2 ≤ S(a, a',b,b') ≤ 2
and
S(a, a',b,b') = E(a,b) - E(a,b') + E(a',b) + E(a',b')

−2 ≤ E(a,b) - E(a,b') + E(a',b) + E(a',b') ≤ 2

I'll try to find the original paper in which he did that so we can verify his justifications.

[FrediFizzx]

Fred, I too do not yet see how Bell went from 1 to 2. The way I have usually derived the CHSH has been similar to how Alain Aspect did it in his paper http://arxiv.org/ftp/quant-ph/papers/0402/0402001.pdf

s = A(a,L).B(b,L) - A(a,L).B(b',L) + A(a',L).B(b,L) + A(a',L).B(b',L)
= A(a,L)[B(b,L) - B(b',L)] + A(a',L)[B(b,L) + B(b',L)]

Well, that is what Richard is doing so I would say that derivation by Aspect is the same as Gill's version. AFAICT, Bell's derivation only works if we assume that for large N,

<A(a, L)> = <A(a', L)> = <B(b, L)> = <B(b', L)> = 0

Thanks for the link to the Aspect paper; I have never read his papers yet.

[gill1109]

Here is the reference to Bell (1971):

Introduction to the hidden-variable question
JS Bell
Foundations of quantum mechanics, 171-181

Preprint:
http://cds.cern.ch/record/400330/files/CM-P00058691.pdf

The innovation in this proof relative to earlier proofs of CHSH is that he allows hidden variables to reside also inside the instruments as well as just in the source. His assumptions are more general than those of Clauser, Horne, Shimony and Holt.

Fourty years on, we can derive the CHSH inequality under even weaker assumptions. All the assumption sets which people have ever used so far, imply the mathematical existence of a complete set of both outcomes of both of Alice and Bob's possible measurements. And that weakest possible assumption implies CHSH. Conversely, the complete set of 8 CHSH inequalities implies the mathematical existence of a complete set of both outcomes of both of Alice and Bob's possible measurement. This was already known to A. Fine, I. Pitowsky and others thirty years ago.

[minkwe]

Richard, you are badly mixing up the meanings of "existence" and "possibilities". The phrase "mathematical existence" is ill-defined baloney.

In any case thanks for the link to the article.

[gill1109]

Richard, you are badly mixing up the meanings of "existence" and "possibilities". The phrase "mathematical existence" is ill-defined baloney.
In any case thanks for the link to the article.

The phrase "mathematical existence" means a lot to mathematicians. It doesn't mean much to anyone else. Too bad.