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

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

Joy Christian wrote:
FrediFizzx wrote:And the result is,

Joy_CHSH = 2.828427

So it comes out the same as taking the shortcut.

Fred, your revised correlation plot on the other thread does not seem to be using scalar outcomes --- I mean this one: viewtopic.php?f=6&t=200&p=5550#p5514.

Is it possible to use scalar outcomes as above for the correlation plot? It should be possible, it seems.

Albert Jan used the shortcut for that simulation also. It should work also with the full formulation.

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

Mikko wrote:Experiments seek to find out facts. Meanings are irrelevant.
... [blah blah blah ] ...
The relevant meanings come from experiment

You should review Adenier's paper: http://arxiv.org/abs/quant-ph/0006014
Adenier wrote: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).

And my previous discussion with a fellow named Richard Gill, in which he stumbled on the same question:
viewtopic.php?f=6&t=63&hilit=Strongly+objective#p2632

### 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.htmlfunction 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.

Fred, your revised correlation plot on the other thread does not seem to be using scalar outcomes --- I mean this one: viewtopic.php?f=6&t=200&p=5550#p5514.

Is it possible to use scalar outcomes as above for the correlation plot? It should be possible, it seems.

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

minkwe wrote:And you can do experiments strictly adherring to the meaning of each expression,

That is not how experiments are performed. Experiments seek to find out facts. Meanings are irrelevant. First something is done and some results are recorded. Then some numbers are computed from the observations. These observations and computed numbers are facts. In order to compare with a theory, you must describe the experiment as actually performed in a way that the theory can be applied. The relevant meanings come from experiment, not from definitions.

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

Gordon, Joy,
Please the distinction between experimental and theoretical bounds is a red herring. All the bounds are theoretical. And you can do experiments strictly adherring to the meaning of each expression, and even then, you will never obtain a violation. The idea that an experiment would have a different upper bound than a theory is wrongheaded. You look at the experimental expression and decide what it's upper bound should be, or you look at the theoretical expression and decide precisely what experiment to do. If your results don't agree, then an error has been made. You can't look at an experimental expression and compare it to a theoretical expression with a different meaning.

Fred,
I don't think you can say "mainstream" CHSH. The problem is that according to the mainstream, all those expressions are one and the same expression. That is why they are still confused when they derive a limit of 2 and measure 2 root 2. They do not yet appreciate that they measured something different. They like mysticism so they project their ignorance to spookiness in nature.

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

Mikko wrote:The intended meaning is the only meaning there is.

That may be true in politics or gossip but not in mathematics, where the actual meaning is the only meaning. That many have not yet discerned the difference between the two meanings does not change this fact. And that you continue to ignore the gulf between the two meanings does not make it disappear, it just shows that you too are making the subtle error Bell and the authors of CHSH made.

If a definition says something else then there is an error in the definition, which can be detected by comparison to the actual use of the term.

There is no error in the definitions. The error is simply failing to discern the difference between, the actual meaning implied by the algebra, and the meanings of the other similar looking expressions. Too bad you don't see it yet. I've given a few articles above to help you see it, there are whole threads here dedicated to helping people see it, for example this one.viewtopic.php?f=6&t=181#p4923

Use of different meanings can be confusing, but you cant help that with introduction of one more.

I haven't introduced anything. Bell did by calculating a different expression from QM than the expression he had just derived. Almost everyone since sheepishly repeats the error. Now it is rich of you to accuse me of introducing a new definition when all I've done is identify the different expressions the authors are confused about. Are you denying that the expressions have different meanings, or are you denying that the authors use them interchangeably as if they had the same meaning? You have to pick just one, though both positions are demonstrably false. What's your pleasure?

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

minkwe wrote: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.

The intended meaning is the only meaning there is. If a definition says something else then there is an error in the definition, which can be detected by comparison to the actual use of the term.

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.

Use of different meanings can be confusing, but you cant help that with introduction of one more.

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

Gordon Watson wrote: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?

Although I see where you are coming from, and although I agree with Michel's classification above, I disagree with your suggestion. The title is fine just as it is. By CHSH I simply mean the string of four expectation values. I have a theoretical model that analytically gives the bound of $2\sqrt{2}$ on the string of four expectation values. See, for example, Eq. (5) of this paper. This theoretical bound is more generally referred to as Tsirel’son's bound, and that is what I have in mind in the title of this thread. In short, as Fred also indicates, $2\sqrt{2}$ is not just an experimental bound. It is also a theoretical bound (more precisely, it is a physical bound).

### 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.

### 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]!
.

### 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?

### 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.

### 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.htmlfunction 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?

### 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.

### 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.htmlfunction 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.

### 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.

### 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.

### 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.

### 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".

### 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.

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