by Joy Christian » Fri Sep 18, 2015 9:31 pm
minkwe wrote:Just a few comments about the above:
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:
None of the following expressions is the CHSH, although in some special circumstances, some of them may be equivalent to the CHSH:
1.
2.
, where
are non-scalar vectors or bivectors.
3.
 - 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
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
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
. You can see this by expanding the terms according to the derivation:
 = \int_{\Lambda} d\lambda \rho(\lambda )A(a,\lambda )B(b,\lambda ))
Therefore CHSH is:
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))
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:
.
However, I am careful to use
two different fonts to stress that
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
)
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
. 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.
[quote="minkwe"]Just a few comments about the above:
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:
[tex]\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\}[/tex]
None of the following expressions is the CHSH, although in some special circumstances, some of them may be equivalent to the CHSH:
1. [tex]\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\}[/tex]
2. [tex]\langle A_1B_1\rangle - \langle A'_1B_1\rangle + \langle A_1B'_1\rangle + \langle A'_1B'_1 \rangle :[/tex], where [tex]A_1, B_1, A'_1, B'_1[/tex] are non-scalar vectors or bivectors.
3. [tex]E(a,b|i) - E(a',b|j) + E(a,b'|k) + E(a',b'|l)[/tex]
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 [tex]2 \sqrt 2[/tex] 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 [tex]A_1 = 1, A_3 = 1, B_1 = 1, B_2 = -1, A'_2 = 1, A'_4 = 1, B'_3 = 1, B'_4 = 1[/tex] 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 [tex]i = j = k = l[/tex]. You can see this by expanding the terms according to the derivation:
[tex]E(a,b|\Lambda) = \int_{\Lambda} d\lambda \rho(\lambda )A(a,\lambda )B(b,\lambda )[/tex]
Therefore CHSH is:
[tex]\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 )[/tex]
Which is equal to
[tex]E(a,b|\Lambda) - E(a',b|\Lambda) + E(a,b'|\Lambda) + E(a',b'|\Lambda)[/tex]
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![/quote]
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 -- [url=http://arxiv.org/pdf/1501.03393.pdf]this paper[/url] 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:
[tex]\langle A_1B_1\rangle - \langle A'_1B_1\rangle + \langle A_1B'_1\rangle + \langle A'_1B'_1 \rangle[/tex].
However, I am careful to use [u]two different fonts[/u] to stress that [tex]A_1, B_1, A'_1, B'_1[/tex] 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]i.e[/i]., 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 [tex]A_i(a)[/tex] are [b][i][u]standardized variables[/u][/i][/b], 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 [url=http://arxiv.org/pdf/1211.0784.pdf]this paper[/url]] 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 [url=http://arxiv.org/pdf/1501.03393.pdf]this paper[/url]]:
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 [url=http://arxiv.org/pdf/1501.03393.pdf]this paper[/url]] 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 [tex]2\sqrt{2}[/tex]. 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.
