## Why the upper bound on CHSH is 2\/2 and not 4 ?

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

### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

Joy Christian wrote:
Xray wrote:
jreed wrote:If you want to find out where CHSH comes from, you should read the article by Bell, "Bertlmann's socks and the nature of reality". It's in his book. This explains it all very well. I've read it and it makes sense, but I won't try to give all the details here. Basically it says that for certain measurements the joint probability P(A,B|a,b,lambda) can be written as:
P1(A|a,lambda)*P2(B|b,lambda) where lambda is a hidden variable. The rest follows from this.

Dear jreed (who, with due respect and imho, does not write here -- in this instance -- like Richard Gill at all.

Nonsense. "Richard Gill" is written all over John Reed's post above. All John Reed has to do to dispel this accusation is to confirm that his post above was written by him alone, without any interference from Gill. He has to simply state that Gill did not approach him by private email, reffered to this discussion on this forum, and dictated the above post, after John Reed made a real boo-boo in his previous attempt to derive the bound of 2 on CHSH: viewtopic.php?f=6&t=199&p=5572#p5537.

Just to clear this up, yes I realize I made a big mistake in that previous posting. I made these postings without a thorough understanding of Bell's ideas. After that I decided, by myself with no coaching from Richard, that I needed to understand CHSH better. Richard had told me several months ago that Bell's paper on Bertlmann's socks had a good explanation of it. I have Bell's book with that paper and decided to read it. Now that I understand it, it makes sense and I'm happy to say I finally really understand what Bell was writing about, unlike many on this forum. My discussion looks like what Richard would have replied with because we have read the same paper and have reached similar conclusions. However, Richard never told me how to reach those conclusions or what to post. Do you require me to swear an oath to this statement?
jreed

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### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

jreed wrote:Do you require me to swear an oath to this statement?

Whether you are telling the truth or not is on your conscience.

I am also unconcerned that you wish to remain as deluded and misguided as Gill in the face of the incontrovertible evidence I have presented in the following papers:

http://arxiv.org/abs/1501.03393

http://arxiv.org/abs/1405.2355

http://arxiv.org/abs/1301.1653

http://arxiv.org/abs/1211.0784

http://arxiv.org/abs/1203.2529

http://arxiv.org/abs/1106.0748

http://arxiv.org/abs/1103.1879

http://arxiv.org/abs/0904.4259

http://arxiv.org/abs/0806.3078

http://arxiv.org/abs/quant-ph/0703179

But you are of course free to remain as deluded as you wish.

All I request is that you stop cluttering every thread I start with your misguided and discredited arguments. Bell's theorem is finished. We are moving on in this forum.
Joy Christian
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### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

jreed wrote:Just to clear this up, yes I realize I made a big mistake in that previous posting. I made these postings without a thorough understanding of Bell's ideas. After that I decided, by myself with no coaching from Richard, that I needed to understand CHSH better. Richard had told me several months ago that Bell's paper on Bertlmann's socks had a good explanation of it. I have Bell's book with that paper and decided to read it. Now that I understand it it makes sense and I'm happy to say I finally really understand what Bell was writing about, unlike many on this forum.

You did a Ph.D. in physics and a degree from MIT and yet you are just now "understanding" what Bell was talking about this past week, despite having been a member of this forum for more than a year? I bet you thought you understood it also back when you wrote the completely out-of-this-planet post about factoring out zero. It is quite evident that you still have a lot of reading to do before you can be qualified to judge who else on this forum does not understand what Bell was writing about.

I would suggest the following articles:

http://arxiv.org/abs/0901.2546
http://vixra.org/pdf/1305.0129v1.pdf
http://arxiv.org/pdf/quant-ph/0006014.pdf
http://arxiv.org/pdf/1202.0841v3
minkwe

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### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

minkwe wrote:
jreed wrote:Just to clear this up, yes I realize I made a big mistake in that previous posting. I made these postings without a thorough understanding of Bell's ideas. After that I decided, by myself with no coaching from Richard, that I needed to understand CHSH better. Richard had told me several months ago that Bell's paper on Bertlmann's socks had a good explanation of it. I have Bell's book with that paper and decided to read it. Now that I understand it it makes sense and I'm happy to say I finally really understand what Bell was writing about, unlike many on this forum.

You did a Ph.D. in physics and a degree from MIT and yet you are just now "understanding" what Bell was talking about this past week, despite having been a member of this forum for more than a year? I bet you thought you understood it also back when you wrote the completely out-of-this-planet post about factoring out zero. It is quite evident that you still have a lot of reading to do before you can be qualified to judge who else on this forum does not understand what Bell was writing about.

I would suggest the following articles:

http://arxiv.org/abs/0901.2546
http://vixra.org/pdf/1305.0129v1.pdf
http://arxiv.org/pdf/quant-ph/0006014.pdf
http://arxiv.org/pdf/1202.0841v3

Yes, I have a degree from MIT and a Ph.D. in physics, but that was many years ago and Bell inequalities were never taught, and I never heard of them. I used physics in my work, but that was the mathematical variety, using the solution of elastic wave equations in the earth and programming on supercomputers in Fortran. Now retired, I'm getting back into the theoretical aspects, which I always wanted to do. This forum is great! I'm learning about new physics that I never studied, and programming in Mathematica. It's a great learning experience for me. Also, when I make a mistake, I don't receive a failing grade, just a lot of criticism. This gives me the motivation to try to figure out what is correct, as I did with Bertlmann's socks. I probably won't have time to study all your references, but I do appreciate them, and of course your ever present criticism.
jreed

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### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

OK guys enough, let's get this thread back on topic or it will be locked.
FrediFizzx
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### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

Joy Christian wrote:***
After Fred's recent post about my derivation of the upper bound on CHSH, we have been trying to better understand why the upper bound on CHSH is 2\/2 and not 4 ?

In fact I have already answered this question much more rigorously in this paper, but perhaps a simplified explanation is called for, especially because the underlying geometrical reason for the bound 2\/2 instead of 4 is quite straightforward. In particular, it has nothing to do with any mystical notion such as "quantum non-locality."

As in Fred's post linked above, we begin with the square of the CHSH-type string of the (geometric) products of the standardized variables (or standard scores):

$F = (A1B1 + A1B2 + A2B1 - A2B2) (A1B1 + A1B2 + A2B1 - A2B2) = A1B1A1B1 + A1B2A1B1 + A2B1A1B1 - A2B2A1B1 + A1B1A1B2 + A1B2A1B2 + A2B1A1B2 - A2B2A1B2 + A1B1A2B1 + A1B2A2B1 + A2B1A2B1 - A2B2A2B1 - A1B1A2B2 - A1B2A2B2 - A2B1A2B2 + A2B2A2B2$ ,

where $A1 = {\rm I}\cdot{\bf a}_1$, $A2 = {\rm I}\cdot{\bf a}_2$, $B1 = {\rm I}\cdot{\bf b}_1$, and $B2 = {\rm I}\cdot{\bf b}_2$ are unit bivectors about the directions ${\bf a}_1$, ${\bf a}_2$, ${\bf b}_1$, and ${\bf b}_2$, with ${\rm I}$ being the volume form.

Now in Fred's post as well as in my original derivation, the upper bound of 2\/2 on CHSH is derived by using the assumption $\left[ Ai, Bj \right] = 0$, justified on the physical grounds that $Ai$ and $Bj$ are space-like separated, and there does not exist a "third" particle in the EPR-B type experiment that can be detected along the exclusive direction ${\bf a}_i \times {\bf b}_j$ in addition to the detections of the pair of particles along the directions ${\bf a}_i$ and ${\bf b}_j$. This permits some rearrangement in the 16 terms, leading to the maximum value of $F = 8.$ Consequently, the upper bound on the CHSH in this case turns out to be $\sqrt{F}=2\sqrt{2}\,,$ for the four Bell test directions (or angles).

So far so good. But can we do better? Is it possible to increase the upper bound on CHSH to its apparent full potential of 4 ? And if not, then why not? Evidently, this seems to be possible if we can arrange a maximum value of $F$ to be 16. Then we would have $\sqrt{F} = 4$ as desired. Now from the above expansion of $F$ it is clear that we can have $F$ = 16 if each of its 16 terms equals to +1. Is that possible? A little reflection will convince you that it is possible provided we can have all four of the directions ${\bf a}_1$, ${\bf a}_2$, ${\bf b}_1$, and ${\bf b}_2$ orthogonal to each other. But one cannot have four directions orthogonal to each other in a three dimensional space such as R^3.

So there. One cannot have all four directions ${\bf a}_1$, ${\bf a}_2$, ${\bf b}_1$, and ${\bf b}_2$ orthogonal to each other in R^3. And therefore we cannot extend the upper bound on CHSH to 4.

The maximum value of the upper bound on CHSH within R^3 is 2\/2. Therefore the observed upper bound on CHSH is 2\/2. It has nothing to do with any mystical idieas like quantum entanglement or non-locality. It is simply a numerical constraint arising form the geometrical and topological properties of the physical space we live in.

In fact, |CHSH| $\leqslant$ 4 is physically not possible even within S^3, which is locally (or tangentially) just R^3. The only way to get the bound |CHSH| $\leqslant$ 4 is to have most of the observation directions orthogonal to each other, i.e., have a topology stronger than S^3, as shown in the last plot of this simulation: http://rpubs.com/jjc/84238.

Joy Christian

Above is the main topic of this thread. Any more comments on the content above?
FrediFizzx
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### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

By definition, as implied by the derivation by the original authors, the CHSH is a very specific expression with a very specific meaning. Part of the confusion that has occurred is due to misnaming several similar looking but very different expressions "CHSH".

The CHSH is the expression:

$\langle A_1B_1\rangle - \langle A'_1B_1\rangle + \langle A_1B'_1\rangle + \langle A'_1B'_1 \rangle : A_1, B_1, A'_1, B'_1\in \{\pm1\}$

None of the following expressions is the CHSH, although in some special circumstances, some of them may be equivalent to the CHSH:
1. $\langle A_1B_1\rangle - \langle A'_2B_2\rangle + \langle A_3B'_3\rangle + \langle A'_4B'_4 \rangle : A_1, A_3, B_1, B_2, A'_2, A'_4, B'_3, B'_4 \in \{\pm1\}$
2. $\langle A_1B_1\rangle - \langle A'_1B_1\rangle + \langle A_1B'_1\rangle + \langle A'_1B'_1 \rangle :$, where $A_1, B_1, A'_1, B'_1$ are non-scalar vectors or bivectors.
3. $E(a,b|i) - E(a',b|j) + E(a,b'|k) + E(a',b'|l)$

Secondly, the upper bound of the CHSH expression is 2. Not an iota higher. This is exactly what their proofs show, and the proofs are valid. However, the upper bound of some of the other expressions can be higher than 2, and can even be as high as 4. Again, just to be clear, what gives an upper bound above 2 may look like the CHSH but it is not the CHSH. This includes all the expressions calculated from QM, Joy's model and simulations, my simulations, and experiments. None of them is calculating the CHSH expression. If they were, they would never produce a value above 2 because the CHSH expression can never be violated.

The key then is to understand how each of those expressions is different from the CHSH, and what their respective upper bounds would be. From my understanding, the calculations by Fred and Joy above correspond to expression (2) and in that case, the maximum possible value is $2 \sqrt 2$ as has been shown. However, it is not true that such a limit applies to expressions of the type (1). It is very easy to produce a value of 4 for expressions of the type (1). One example, a system which always produces $A_1 = 1, A_3 = 1, B_1 = 1, B_2 = -1, A'_2 = 1, A'_4 = 1, B'_3 = 1, B'_4 = 1$ will give you a value of 4 for the expression.

Expression (3) deserves special mention: It will be equivalent to the CHSH expression, if and only if $i = j = k = l$. You can see this by expanding the terms according to the derivation:

$E(a,b|\Lambda) = \int_{\Lambda} d\lambda \rho(\lambda )A(a,\lambda )B(b,\lambda )$
Therefore CHSH is:
$\int_{\Lambda} d\lambda \rho(\lambda )A(a,\lambda )B(b,\lambda ) \\- \int_{\Lambda} d\lambda \rho(\lambda )A(a',\lambda )B(b,\lambda ) \\+ \int_{\Lambda} d\lambda \rho(\lambda )A(a,\lambda )B(b',\lambda ) \\+ \int_{\Lambda} d\lambda \rho(\lambda )A(a',\lambda )B(b',\lambda )$
Which is equal to
$E(a,b|\Lambda) - E(a',b|\Lambda) + E(a,b'|\Lambda) + E(a',b'|\Lambda)$

Contrary to popular belief, NO EXPERIMENT has ever violated the CHSH nor is it possible for any experiment to ever violate it. Not even by the smallest statistical experimental error. It is mathematically impossible!
minkwe

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### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

By definition, as implied by the derivation by the original authors, the CHSH is a very specific expression with a very specific meaning. Part of the confusion that has occurred is due to misnaming several similar looking but very different expressions "CHSH".

The CHSH is the expression:

$\langle A_1B_1\rangle - \langle A'_1B_1\rangle + \langle A_1B'_1\rangle + \langle A'_1B'_1 \rangle : A_1, B_1, A'_1, B'_1\in \{\pm1\}$

I would add here that the above CHSH expression is actually impossible to achieve in an experiment. It takes at minimum at least four iterations (trials) to obtain all the necessary elements for the expression. So there has to be another CHSH expression that the experiments use with +/- 1 outcomes. Maybe something like;

$\langle A_1B_1\rangle - \langle A'_2B_1\rangle + \langle A_1B'_3\rangle + \langle A'_2B'_3 \rangle : A_1, B_1, A'_2, B'_3\in \{\pm1\}$

For the bivector valued Joy_CHSH we simply use 2;

"2. $\langle A_1B_1\rangle - \langle A'_1B_1\rangle + \langle A_1B'_1\rangle + \langle A'_1B'_1 \rangle :$, where $A_1, B_1, A'_1, B'_1$ are non-scalar vectors or bivectors."
Is this not also what the QM prediction uses?
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### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

FrediFizzx wrote:$\langle A_1B_1\rangle - \langle A'_1B_1\rangle + \langle A_1B'_1\rangle + \langle A'_1B'_1 \rangle : A_1, B_1, A'_1, B'_1\in \{\pm1\}$
I would add here that the above CHSH expression is actually impossible to achieve in an experiment. It takes at minimum at least four iterations (trials) to obtain all the necessary elements for the expression. So there has to be another CHSH expression that the experiments use with +/- 1 outcomes.

Experiments use expressions of the form (1) in my previous post, which I insist should not be called a CHSH expression.

Maybe something like;
$\langle A_1B_1\rangle - \langle A'_2B_1\rangle + \langle A_1B'_3\rangle + \langle A'_2B'_3 \rangle : A_1, B_1, A'_1, B'_1\in \{\pm1\}$
For the bivector valued Joy_CHSH we simply use;

$\langle A_1B_1\rangle - \langle A_1B_1\rangle + \langle A_1B_1\rangle + \langle A_1B_1\rangle$

I think we are just using different notation. The subscript numbers in my expressions do not correspond to settings. Perhaps you are using ((1,2), (1, 2)) as the settings. To me, the settings are $a, a', b, b'$, The numbers $A_1, A_2, A_3$, are outcomes from three different particles all measured at setting $a$, the numbers $A'_1, A'_2, A'_3$ are the exact same three particles as before but now measured at setting $a'$, etc. Within the averages, the numbers represent the set of outcomes that is averaged. Thus $\langle A'_1B_1\rangle$ means the average of the product of results at setting pair $(a', b)$ for a given set of outcome pairs, and the number $\langle A'_2B_2\rangle$ means the average of the product of results results at setting pair $(a', b)$ on a second disjoint set of outcome pairs different from the first, but measured using the same angle settings.

In any case, anything bivector valued is not the CHSH. The CHSH uses scalars +1, -1 by definition.

In this case then the QM expression is simply
$-a\cdot b + a'\cdot b - a\cdot b' - a'\cdot b'$
Which also is should NOT be called CHSH. Note the lack of averages, and no reference to averages. Plus, the addition of those terms implies that they commute with each other, and therefore necessarily apply to independent sets particle pairs, unlike the CHSH expression. If they did apply to the same set of particle pairs, they would be equivalent to the CHSH expression but, they would not commute and therefore their values would not be the same when combined and therefore violation could not be established as we've explained elsewhere.
minkwe

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### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

Sorry, I edited my post after you grabbed it for a quote.

minkwe wrote:In any case, anything bivector valued is not the CHSH. The CHSH uses scalars +1, -1 by definition.

In this case then the QM expression is simply
$-a\cdot b + a'\cdot b - a\cdot b' - a'\cdot b'$

OK, that is what I was looking for. Joy's CHSH calculation basically does the same thing prediction-wise so it matches what QM does for "CHSH".
FrediFizzx
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### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

FrediFizzx wrote:
minkwe wrote:In any case, anything bivector valued is not the CHSH. The CHSH uses scalars +1, -1 by definition.

In this case then the QM expression is simply
$-a\cdot b + a'\cdot b - a\cdot b' - a'\cdot b'$

OK, that is what I was looking for. Joy's CHSH calculation basically does the same thing prediction-wise so it matches what QM does for "CHSH".

Yes, Joy's averages of bivector products gives the same results as QM.

One other thing I like to stress is that the 4 terms in QM expression are not independent since there is a cyclic pair of settings and the dot product depends only on the settings and nothing else. This independence should not be confused with the statistical dependence that arises in the CHSH expression, in which all pairs contribute outcomes to all terms. The setting dependence is present in the CHSH but in addition, you have further statistical dependence due to the fact that all the averages are calculated on the same set of 4xN outcomes.
minkwe

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### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

By definition, as implied by the derivation by the original authors, the CHSH is a very specific expression with a very specific meaning. Part of the confusion that has occurred is due to misnaming several similar looking but very different expressions "CHSH".

The CHSH is the expression:

$\langle A_1B_1\rangle - \langle A'_1B_1\rangle + \langle A_1B'_1\rangle + \langle A'_1B'_1 \rangle : A_1, B_1, A'_1, B'_1\in \{\pm1\}$

None of the following expressions is the CHSH, although in some special circumstances, some of them may be equivalent to the CHSH:
1. $\langle A_1B_1\rangle - \langle A'_2B_2\rangle + \langle A_3B'_3\rangle + \langle A'_4B'_4 \rangle : A_1, A_3, B_1, B_2, A'_2, A'_4, B'_3, B'_4 \in \{\pm1\}$
2. $\langle A_1B_1\rangle - \langle A'_1B_1\rangle + \langle A_1B'_1\rangle + \langle A'_1B'_1 \rangle :$, where $A_1, B_1, A'_1, B'_1$ are non-scalar vectors or bivectors.
3. $E(a,b|i) - E(a',b|j) + E(a,b'|k) + E(a',b'|l)$

Secondly, the upper bound of the CHSH expression is 2. Not an iota higher. This is exactly what their proofs show, and the proofs are valid. However, the upper bound of some of the other expressions can be higher than 2, and can even be as high as 4. Again, just to be clear, what gives an upper bound above 2 may look like the CHSH but it is not the CHSH. This includes all the expressions calculated from QM, Joy's model and simulations, my simulations, and experiments. None of them is calculating the CHSH expression. If they were, they would never produce a value above 2 because the CHSH expression can never be violated.

The key then is to understand how each of those expressions is different from the CHSH, and what their respective upper bounds would be. From my understanding, the calculations by Fred and Joy above correspond to expression (2) and in that case, the maximum possible value is $2 \sqrt 2$ as has been shown. However, it is not true that such a limit applies to expressions of the type (1). It is very easy to produce a value of 4 for expressions of the type (1). One example, a system which always produces $A_1 = 1, A_3 = 1, B_1 = 1, B_2 = -1, A'_2 = 1, A'_4 = 1, B'_3 = 1, B'_4 = 1$ will give you a value of 4 for the expression.

Expression (3) deserves special mention: It will be equivalent to the CHSH expression, if and only if $i = j = k = l$. You can see this by expanding the terms according to the derivation:

$E(a,b|\Lambda) = \int_{\Lambda} d\lambda \rho(\lambda )A(a,\lambda )B(b,\lambda )$
Therefore CHSH is:
$\int_{\Lambda} d\lambda \rho(\lambda )A(a,\lambda )B(b,\lambda ) \\- \int_{\Lambda} d\lambda \rho(\lambda )A(a',\lambda )B(b,\lambda ) \\+ \int_{\Lambda} d\lambda \rho(\lambda )A(a,\lambda )B(b',\lambda ) \\+ \int_{\Lambda} d\lambda \rho(\lambda )A(a',\lambda )B(b',\lambda )$
Which is equal to
$E(a,b|\Lambda) - E(a',b|\Lambda) + E(a,b'|\Lambda) + E(a',b'|\Lambda)$

Contrary to popular belief, NO EXPERIMENT has ever violated the CHSH nor is it possible for any experiment to ever violate it. Not even by the smallest statistical experimental error. It is mathematically impossible!

This is very beautifully clear and agreeable. I am guilty of using the notation CHSH in my post as a generic shortcut for a CHSH-type string of four expectation values.

More importantly, what I am doing in -- for example -- this paper is something that does not quite fit into the above classification of various CHSH-type expressions.

As I have mentioned before, I have always been able to have my cake and eat it too. What I mean is that I use the following expression in two different ways:

$\langle A_1B_1\rangle - \langle A'_1B_1\rangle + \langle A_1B'_1\rangle + \langle A'_1B'_1 \rangle$.

However, I am careful to use two different fonts to stress that $A_1, B_1, A'_1, B'_1$ are bivectors or scalars in a given argument. For scalars I use scrip A's and B's, and then the above string of numbers is precisely and strictly what CHSH meant for CHSH in their paper. But for bivectors I use unscripted A's and B's (i.e., I use mathrm font). So in that case the CHSH string is not strictly the CHSH string used in the CHSH paper. I do this in my papers not to confuse people but to do the physics right.

In my calculations the unscripted $A_i(a)$ are standardized variables, or standard scores, not the experimentally observed raw scores. I have explained the important distinction in many papers [see, for example, Eqs. (105) to (111) of this paper] so let me not repeat all that, but here is the relationship between the two, where I am now using the notation A_i to represent ordinary scalar numbers (+/-1) [see Eqs. (7) and (8) of this paper]:

A_i(a, u) = d(a) h_i(a, u) = +/-1 (= commuting scalar number),

where u is the hidden variable, a is the measurement direction, d(a) is a detector bivector, h_i(a, u) is the non-commuting standard score (a bivector), and A_i(a, u) = +/-1 is the actually observed raw score. Since A_i(a, u) are commuting scalar numbers, when naively calculated [as Gill is shown to do in Eqs. (12) and (14) of this paper] the bound appears to be 2. However, when correctly calculated using the standardized variables h_i(a, u), which are non-commuting numbers, the bound is $2\sqrt{2}$. So here we have a situation where "totally dependent" raw scores A_i(a, u) (which are the same in both E11 and E12 of CHSH) leads to a bound greater than 2.

Well, I have said all this before. The bottom line is that what the above procedure does is calculate the correlation among the usual scalar (+/-1) points within S^3.
Joy Christian
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### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

Joy Christian wrote:Well, I have said all this before. The bottom line is that what the above procedure does is calculate the correlation among the usual scalar (+/-1) points within S^3.

Yes, and since the detector bivectors "drop out" of the calculation, the way I did Joy_CHSH with GAViewer was essentially a short cut. I guess I could try it with the full A_i(a, u) = d(a) h_i(a, u) type expressions.
FrediFizzx
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### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

FrediFizzx wrote:
Joy Christian wrote:Well, I have said all this before. The bottom line is that what the above procedure does is calculate the correlation among the usual scalar (+/-1) points within S^3.

Yes, and since the detector bivectors "drop out" of the calculation, the way I did Joy_CHSH with GAViewer was essentially a short cut. I guess I could try it with the full A_i(a, u) = d(a) h_i(a, u) type expressions.

Ok, here it is.
Code: Select all
//Adaptation of Albert Jan Wonnink's original code
//http://challengingbell.blogspot.com/2015/03/numerical-validation-of-vanishing-of.html
function getRandomLambda()
{
if( rand()>0.5) {return 1;} else {return -1;}
}

batch test()
{
set_window_title("Test of Joy Christian's CHSH derivation");
N=20000; //number of iterations (trials)
I=e1^e2^e3;
s=0;
a1=sin(0)*e1 + cos(0)*e2 + 0.000*e3;
b1=sin(pi/4)*e1 + cos(pi/4)*e2 + 0.000*e3;
a2=sin(pi/2)*e1 + cos(pi/2)*e2 + 0.000*e3;
b2=sin(3*pi/4)*e1 + cos(3*pi/4)*e2 + 0.000*e3;

for(nn=0;nn<N;nn=nn+1) //perform the experiment N times
{
lambda=getRandomLambda(); //lambda is a fair coin
//resulting in +1 or -1
mu=lambda * I;  //calculate the lambda dependent mu
C1=-I.a1;  //C = {-a_j B_j}
D1=I.b1;   //D = {b_k B_k}
C2=-I.a2;  //C = {-a_j B_j}
D2=I.b2;   //D = {b_k B_k}
E1=mu.a1;  //E = {a_k B_k(L)}
F1=mu.b1;  //F = {b_j B_j(L)}
A1=C1 E1;  //eq. (1) of arXiv:1103.1879, A(a, L) = {-a_j B_j}{a_k B_k(L)}
B1=F1 D1;  //eq. (2) of arXiv:1103.1879, B(b, L) = {b_j B_j(L)}{b_k B_k}
E2=mu.a2;  //E = {a_k B_k(L)}
F2=mu.b2;  //F = {b_j B_j(L)}
A2=C2 E2;  //eq. (1) of arXiv:1103.1879, A(a, L) = {-a_j B_j}{a_k B_k(L)}
B2=F2 D2;  //eq. (2) of arXiv:1103.1879, B(b, L) = {b_j B_j(L)}{b_k B_k}
q=0;
if(lambda==1) {q=((-C1) (A1 B1) (-D1))-((-C1) (A1 B2) (-D2))+((-C2) (A2 B1) (-D1))+((-C2) (A2 B2) (-D2));}
else {q=((-D1) (B1 A1) (-C1))-((-D2) (B2 A1) (-C1))+((-D1) (B1 A2) (-C2))+((-D2) (B2 A2) (-C2));}
s=s+q;
}
Joy_CHSH=abs(s/N);
print(Joy_CHSH, "f");
prompt();
}

And the result is,

Joy_CHSH = 2.828427

So it comes out the same as taking the shortcut.
FrediFizzx
Independent Physics Researcher

Posts: 2905
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Location: N. California, USA

### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

minkwe wrote:By definition, as implied by the derivation by the original authors, the CHSH is a very specific expression with a very specific meaning.

This doesn't sound correct. Definitions are stated explicitly, not implied. However, this indeed is the only sensible meaning of the term.
Secondly, the upper bound of the CHSH expression is 2. Not an iota higher. This is exactly what their proofs show, and the proofs are valid. However, the upper bound of some of the other expressions can be higher than 2, and can even be as high as 4. Again, just to be clear, what gives an upper bound above 2 may look like the CHSH but it is not the CHSH. This includes all the expressions calculated from QM, Joy's model and simulations, my simulations, and experiments. None of them is calculating the CHSH expression. If they were, they would never produce a value above 2 because the CHSH expression can never be violated.

This is obviously not the meaning in the article. The intent of "Proposed experiment to test local hidden-variable theories" is to test experimentally whether the inequality is violated. Therefore it does not make sense to define CHSH so that the primary topic of the article is excluded, nor to claim that it would be excluded by the implied definition.

That the proofs about CHSH are valid only means that their conclusions are true if their assumptions are. Otherwise the conclusion need not be true but the definition of CHSH need not be similarly constrained.
Mikko

Posts: 163
Joined: Mon Feb 17, 2014 2:53 am

### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

FrediFizzx wrote:Ok, here it is.
Code: Select all
//Adaptation of Albert Jan Wonnink's original code
//http://challengingbell.blogspot.com/2015/03/numerical-validation-of-vanishing-of.html
function getRandomLambda()
{
if( rand()>0.5) {return 1;} else {return -1;}
}

batch test()
{
set_window_title("Test of Joy Christian's CHSH derivation");
N=20000; //number of iterations (trials)
I=e1^e2^e3;
s=0;
a1=sin(0)*e1 + cos(0)*e2 + 0.000*e3;
b1=sin(pi/4)*e1 + cos(pi/4)*e2 + 0.000*e3;
a2=sin(pi/2)*e1 + cos(pi/2)*e2 + 0.000*e3;
b2=sin(3*pi/4)*e1 + cos(3*pi/4)*e2 + 0.000*e3;

for(nn=0;nn<N;nn=nn+1) //perform the experiment N times
{
lambda=getRandomLambda(); //lambda is a fair coin
//resulting in +1 or -1
mu=lambda * I;  //calculate the lambda dependent mu
C1=-I.a1;  //C = {-a_j B_j}
D1=I.b1;   //D = {b_k B_k}
C2=-I.a2;  //C = {-a_j B_j}
D2=I.b2;   //D = {b_k B_k}
E1=mu.a1;  //E = {a_k B_k(L)}
F1=mu.b1;  //F = {b_j B_j(L)}
A1=C1 E1;  //eq. (1) of arXiv:1103.1879, A(a, L) = {-a_j B_j}{a_k B_k(L)}
B1=F1 D1;  //eq. (2) of arXiv:1103.1879, B(b, L) = {b_j B_j(L)}{b_k B_k}
E2=mu.a2;  //E = {a_k B_k(L)}
F2=mu.b2;  //F = {b_j B_j(L)}
A2=C2 E2;  //eq. (1) of arXiv:1103.1879, A(a, L) = {-a_j B_j}{a_k B_k(L)}
B2=F2 D2;  //eq. (2) of arXiv:1103.1879, B(b, L) = {b_j B_j(L)}{b_k B_k}
q=0;
if(lambda==1) {q=((-C1) (A1 B1) (-D1))-((-C1) (A1 B2) (-D2))+((-C2) (A2 B1) (-D1))+((-C2) (A2 B2) (-D2));}
else {q=((-D1) (B1 A1) (-C1))-((-D2) (B2 A1) (-C1))+((-D1) (B1 A2) (-C2))+((-D2) (B2 A2) (-C2));}
s=s+q;
}
Joy_CHSH=abs(s/N);
print(Joy_CHSH, "f");
prompt();
}

And the result is,

Joy_CHSH = 2.828427

So it comes out the same as taking the shortcut.

Thanks, Fred. This looks great. I wonder what excuses are left for the Bell believers to continue their delusions?
Joy Christian
Research Physicist

Posts: 2793
Joined: Wed Feb 05, 2014 4:49 am
Location: Oxford, United Kingdom

### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

Mikko wrote:This doesn't sound correct. Definitions are stated explicitly, not implied.

That is just a semantic quibble. Stated or implied, the point is that by definition, the expression has a specific meaning.

This is obviously not the meaning in the article. The intent of "Proposed experiment to test local hidden-variable theories" is to test experimentally whether the inequality is violated.

You should understand the difference between intended meaning and actual meaning. There is no doubt that the intended meaning is what you say. But it is also evident that the actual meaning is what I say.

Therefore it does not make sense to define CHSH so that the primary topic of the article is excluded, nor to claim that it would be excluded by the implied definition.

It doesn't make sense to use two different definitions of the CHSH, in order to give the false impression that their intended meaning matches their actual meaning. That is the Bell error which they inherited, and many still do -- the error of not recognising that the different types of expressions have different upper bounds.

That the proofs about CHSH are valid only means that their conclusions are true if their assumptions are.

And all those proofs require as part of the derivation, that the terms in the expression represent counterfactual outcomes from a single set of particle pairs. A requirement that is completely unfulfilled in the other expression you now want to call CHSH.

Otherwise the conclusion need not be true but the definition of CHSH need not be similarly constrained.

The definitions stated in the article, and the requirements implied by the steps of the derivation constrain the meaning of the final expression. It is not up to you or anyone else ( even the authors) to relax these constraints willynilly. If you do that as has been done recklessly for 50 years, be prepared for paradoxes. Perhaps those paradoxes tickle the fancies of the mystically inclined, but its not mathematics.
minkwe

Posts: 1441
Joined: Sat Feb 08, 2014 10:22 am

### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

So the bound on the real CHSH is 2. So I guess this thread is concerning the bound on the "mainstream" CHSH?
FrediFizzx
Independent Physics Researcher

Posts: 2905
Joined: Tue Mar 19, 2013 7:12 pm
Location: N. California, USA

### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

FrediFizzx wrote:So the bound on the real CHSH is 2. So I guess this thread is concerning the bound on the "mainstream" CHSH?

When I first saw the heading for this thread I was going to suggest (as I now do) that it be changed to: Why the EXPERIMENTAL upper bound on CHSH is 2√2 and not 4? *

Note that it was a proposed correction to the primacy of CHSH (and its 2) as a paper and a better representation of the 2√2 -- for better search results in the future. For that would facilitate good educational discussion re the source of the ORIGINAL CHSH and its 2, the relevance of 4, and the how-and-why of the experiments that deliver the well-known intermediate results.

Thus: In short (as I read him) I support minkwe in accurately honouring CHSH with the facts.

* I now also add what I thought of adding back then:

Bell (1981) -- in Bertlmann, reference # 19 -- has the authors wrongly ordered as CHHS (though that's where I would put the S). A bit like David Bohm in his famous textbook (1951) referred to the paradox of ERP [sic]!
.
Gordon Watson

Posts: 403
Joined: Wed Apr 30, 2014 4:39 am

### Re: Why the upper bound on CHSH is 2\/2 and not 4 ?

Gordon Watson wrote:
FrediFizzx wrote:So the bound on the real CHSH is 2. So I guess this thread is concerning the bound on the "mainstream" CHSH?

When I first saw the heading for this thread I was going to suggest (as I now do) that it be changed to: Why the EXPERIMENTAL upper bound on CHSH is 2√2 and not 4?

It is quite clear from what Michel (minkwe) wrote above that the experimental bound is 4 so there is no question about that. Unless you define the exact type of experiment I suppose. I do believe that what Joy was referring to for the title was the "mainstream" bound in the same vein as quantum theory. But perhaps Joy can clarify that for us if something other.
FrediFizzx
Independent Physics Researcher

Posts: 2905
Joined: Tue Mar 19, 2013 7:12 pm
Location: N. California, USA

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