The Efficiency Loophole: A Local Hidden Variables Theory?

[DrChinese]

And that is one of the problems - all these experiments try to test local realism but they don't test falsifiable predictions of QM . However they are presented as scientific tests of QM.
And that is just sick.

This is incorrect. They absolutely test a falsifiable prediction of QM as well! That prediction being the cos^2(theta) relationship. The EPR paper contemplated the idea that QM was not complete. Please recall that Bell says that if QM is incorrect, then the Bell Inequality is respected and the cos^2 relationship is wrong. In fact, there are local realistic models in which QM and LR yield different predictions for this relationship. In such, usually the LR model is linear.

 

[zonde]

This is incorrect. They absolutely test a falsifiable prediction of QM as well! That prediction being the cos^2(theta) relationship.

Yes, cos^2(theta) relationship is falsifiable prediction.
So can you give reference to some experiment that does scientific test of this relationship and which you would prefer as an example?

 

[DrChinese]

Yes, cos^2(theta) relationship is falsifiable prediction.
So can you give reference to some experiment that does scientific test of this relationship and which you would prefer as an example?

One of many I could cite:

http://arxiv.org/abs/quant-ph/9810080
Violation of Bell's inequality under strict Einstein locality conditions (1998)
Authors: Gregor Weihs, Thomas Jennewein, Christoph Simon, Harald Weinfurter, Anton Zeilinger

"Quantum theory predicts a sinusoidal dependence for the coincidence rate Cqm++(A, B) ∝ sin2(B − A) on the difference angle of the analyzer directions in Alice’s and Bob’s experiments. ... Thus, because the visibility of the perfect correlations in our experiment was about 97% we expect S to be not higher than 2.74 if alignment of all angles is perfect and all detectors are equally efficient. ... A typical observed value of the function S in such a measurement was S = 2.73±0.02 for 14700 coincidence events collected in 10 s. ... Our results confirm the quantum theoretical predictions..."

I would say the above description is fairly typical, and I did not include the portion in which local realistic predictions are calculated and then falsified. My point being that QM makes a specific prediction different than LR. The QM prediction would be falsified if the LR value was seen - or in fact if any other value than the QM prediction was seen. So QM is tested.

 

[billschnieder]

What does "separable formalism" mean? You have a habit of not answering direct questions I ask you, which is frustrating. In my previous post I asked about the meaning of the similar phrase "parameter separability":

Can you please tell me if by "separable formalism" you just mean this idea that we can find local variables lambda in Alice's region that screen off the correlation between Alice's result with setting c and Bob's result with setting b, i.e. P(c+|b+, lambda) = P(c+|lambda)?

Not answering for ThomasT but just to chime in on what "parameter separability" means. Given an expression such as

ab + bc < ac

Separability allows me to rearrange the terms at will in the expression. I can factor out b on the LHS and treat each of the parameters as a standalone variable.

Note that this can not be done if our parameters are not communtative. In other words, if the value of a when it occurs together with b, is not the same as the value of a when it occurs with c, then we can not factor at will. The parameters will not be separable either, and therefore each term in the inequality (ie "ab", "bc", "ac") is a single indivisible whole which must be treated as such.

What has this got to do with Bell?
Bell derives his inequality by making use of the ability to factorize the terms at will. This introduces a separability requirement. If you are in doubt about this, see his derivation starting at equation 14. He introduces a P(a,c) term which he subtracts from a P(a,b) term, and by factorization and rearagement, he obtains a P(b,c) term. The fact that the P(b,c) term pops out from the P(a,b) and P(b,c) terms affirms this point.

What has this got to do with QM?
P(a,b) from QM does not commute with P(b,c), nor with P(a,c). So off the bat, we have a problem already before we can even do an QM calculations as those terms will not be compatible with Bell's inequality.

What about the experiments?
P(a,b) from one run of the experiment, does not commute with P(b,c) nor with P(a,c) from a different run of the experiment either. That is what QM has been telling us all along! For those whose concept of reality involves ridgit pre-existing properties which are passively revealed in Bell-type experiments it will be difficult to see how this is possible. All you need is for the parameters being measured to be contextual. Which simply means, a pre-existing property of the particles combines with a property of the device to reveal the outcome of an experiment.

Yet some may exclaim that if the value of 'a' in combination with 'b' is different from the value of 'a' in combination with 'c', it means there is spooking action between "setting a" and "setting b". That is certainly the naive interpretation since all that is required is for the process which produces the particle pairs to be non-stationary (http://en.wikipedia.org/wiki/Stationary_process)

Therefore Bell's theorem is mistated in my opinion. It will be better stated as:

Non-commuting expectation values are not compatible with Bell's inequalities
Or
Non-separable expectation values are not compatible with Bell's inequalities
Or
You can not eat your cake and have it

Which would have been stating the obvious if not of all the noise surrounding Bell's theorem.

 

[JesseM]

Not answering for ThomasT but just to chime in on what "parameter separability" means. Given an expression such as

ab + bc < ac

Separability allows me to rearrange the terms at will in the expression. I can factor out b on the LHS and treat each of the parameters as a standalone variable.

Note that this can not be done if our parameters are not communtative. In other words, if the value of a when it occurs together with b, is not the same as the value of a when it occurs with c, then we can not factor at will. The parameters will not be separable either, and therefore each term in the inequality (ie "ab", "bc", "ac") is a single indivisible whole which must be treated as such.

What has this got to do with Bell?
Bell derives his inequality by making use of the ability to factorize the terms at will. This introduces a separability requirement. If you are in doubt about this, see his derivation starting at equation 14. He introduces a P(a,c) term which he subtracts from a P(a,b) term, and by factorization and rearagement, he obtains a P(b,c) term. The fact that the P(b,c) term pops out from the P(a,b) and P(b,c) terms affirms this point.

What has this got to do with QM?
P(a,b) from QM does not commute with P(b,c), nor with P(a,c). So off the bat, we have a problem already before we can even do an QM calculations as those terms will not be compatible with Bell's inequality.

In that derivation Bell is not trying to show what's true in QM, he's showing what would necessarily be true in this experiment under a local realist theory (assuming the local realist theory meets the condition that when the experimenters both choose the same detector setting they are guaranteed to get opposite results, the condition expressed in equation 13), and then showing that this is incompatible with QM's predictions about the same experiment. His derivation in equations 14-15 in this paper is about what would be true in a local realist theory (of the type I discussed in [post=3231977]post 31[/post]).

That said I'm still not really clear on what you mean by "separability"--"Separability allows me to rearrange the terms at will in the expression" is a bit vague, and the relation of this to commuting/non-commuting is also unclear, the notion of commuting or not commuting is usually applied to measurement operators, not expectation values. Position x, momentum p and energy E don't all mutually commute, but if you are interested in the expectation values P(x), P(p) and P(E) for a single state vector, then if you had some expression like P(x)*P(p) + P(p)*P(E) < P(x)*P(E), you could certainly factor out P(p) from the left hand side, for any specific state vector the three expectation values will all just be real numbers with fixed values after all. Non-commuting would imply that if you took a state vector V and then applied the position operator resulting in a collapse to a position eigenvector Vx, then immediately applied the momentum operator to Vx and looked at the expectation value for momentum, this would be different than if you had first applied the momentum operator to V and then immediately applied the position operator.

So, if you had the following:

[P(x)*P(p) for a position measurement followed by a momentum measurement] + [P(p)*P(E) for a momentum measurement followed by an energy measurement] < [P(x)*P(E) for a position measurement followed by an energy measurement]

...then in that case the non-commutativity would mean you could no longer factor P(p) out of the left hand side because the expectation value for momentum would depend if it was measured first as in P(p)*P(E) or second as in P(x)*P(p). Not clear on how this relates to an inequality featuring expectation values for P(a,b), P(b,c) and P(a,c) though, might help if you wrote it out in the same explicit form as I did above.

 

[zonde]

One of many I could cite:

http://arxiv.org/abs/quant-ph/9810080
Violation of Bell's inequality under strict Einstein locality conditions (1998)
Authors: Gregor Weihs, Thomas Jennewein, Christoph Simon, Harald Weinfurter, Anton Zeilinger

Yes, very good experiment.

"Quantum theory predicts a sinusoidal dependence for the coincidence rate Cqm++(A, B) ∝ sin2(B − A) on the difference angle of the analyzer directions in Alice’s and Bob’s experiments. ... Thus, because the visibility of the perfect correlations in our experiment was about 97% we expect S to be not higher than 2.74 if alignment of all angles is perfect and all detectors are equally efficient. ... A typical observed value of the function S in such a measurement was S = 2.73±0.02 for 14700 coincidence events collected in 10 s. ... Our results confirm the quantum theoretical predictions..."

I would say the above description is fairly typical, and I did not include the portion in which local realistic predictions are calculated and then falsified. My point being that QM makes a specific prediction different than LR. The QM prediction would be falsified if the LR value was seen - or in fact if any other value than the QM prediction was seen. So QM is tested.

We were talking about testing of cos^2(theta) relationship. So let's keep to that.
From paper:
"A nonlinear χ2 -fit showed perfect agreement with the sine curve predicted by quantum theory."
That is about as far as it goes in respect of cos^2(theta) testing.

In order to consider this experiment as scientific test of cos^2(theta) relationship there should be some necessary condition for that test. Then if that necessary condition does not hold we can say that test falsified prediction.

If result about cos^2(theta) testing is formulated like: "experimenter's opinion is that fit is good" we don't call this scientific test, do we? Experimenters opinion can't be this necessary condition if we talk about scientific tests.

So I say that only Bell inequalities are tested scientifically in this experiment. But not cos^2(theta) relationship.

 

[DrChinese]

...

Separability allows me to rearrange the terms at will in the expression. I can factor out b on the LHS and treat each of the parameters as a standalone variable.

Note that this can not be done if our parameters are not communtative. In other words, if the value of a when it occurs together with b, is not the same as the value of a when it occurs with c, then we can not factor at will. The parameters will not be separable either, and therefore each term in the inequality (ie "ab", "bc", "ac") is a single indivisible whole which must be treated as such.

What has this got to do with Bell?
Bell derives his inequality by making use of the ability to factorize the terms at will. This introduces a separability requirement. If you are in doubt about this, see his derivation starting at equation 14. He introduces a P(a,c) term which he subtracts from a P(a,b) term, and by factorization and rearagement, he obtains a P(b,c) term. The fact that the P(b,c) term pops out from the P(a,b) and P(b,c) terms affirms this point.

What has this got to do with QM?
P(a,b) from QM does not commute with P(b,c), nor with P(a,c). So off the bat, we have a problem already before we can even do an QM calculations as those terms will not be compatible with Bell's inequality.

What about the experiments?
P(a,b) from one run of the experiment, does not commute with P(b,c) nor with P(a,c) from a different run of the experiment either. That is what QM has been telling us all along! For those whose concept of reality involves ridgit pre-existing properties which are passively revealed in Bell-type experiments it will be difficult to see how this is possible. All you need is for the parameters being measured to be contextual. Which simply means, a pre-existing property of the particles combines with a property of the device to reveal the outcome of an experiment.

...

Wrong, as per usual. I will repeat what I have said numerous times before: your statements represent your personal theories about Bell, which are completely at variance with the scientific community at large. Other readers may not be aware that you are pushing your personal opinions and not good science.

It is the assertion of the Local Realist that there is no dependence of a measurement here on a result there (separability/locality), which essentially denies entanglement exists as a physical state. It really wouldn't matter in that statement whether QM says this or says that. Further the Local Realist says that there exists values for unobserved measurement settings (realism) for a particle, regardless of whether measuring one setting commutes with the measurement of another. That's all you really need to get Bell's Theorem. Of course, you would also want to know the QM expectation value for comparative purposes, tying back to EPR.

If you think that non-commutativity is relevant in EPR setups (with entangled pairs which don't commute), then I would say you reject Local Realism prima facie. And that same conclusion follows from accepting QM as "complete". Like most Local Realists, you want to have your cake (LR) and eat it too (QM). Bell does not allow this.

 

[DrChinese]

So I say that only Bell inequalities are tested scientifically in this experiment. But not cos^2(theta) relationship.

And I would say the authors of the paper would laff their heads off if they read that.

Since Bell Inequalities come from that relationship. As do the perfect correlations that the paper mentions.

 

[harrylin]

And I would say the authors of the paper would laff their heads off if they read that.

Since Bell Inequalities come from that relationship. As do the perfect correlations that the paper mentions.

"Perfect" correlations don't exist in scientific measurements - scientific measurements work with measurement errors.

 

[harrylin]

[...] It is the assertion of the Local Realist that there is no dependence of a measurement here on a result there (separability/locality), which essentially denies entanglement exists as a physical state. [..]
Further the Local Realist says that there exists values for unobserved measurement settings (realism) for a particle, regardless of whether measuring one setting commutes with the measurement of another. That's all you really need to get Bell's Theorem. [..]

Thanks for the summary, but those assertions are a little (too) extreme.
I would think that it is the assertion of the Local Realist that there is no magical dependence of a measurement here on a result there (separability/locality). Influences at a distance according to known or not yet known physical mechanisms are admitted. However I agree that that does essentially deny physical entanglement at a great distance. [..] Further, a Local Realist assumes that already before the measurement one or more unobserved particle variables exist that will affect the values that will be measured. And I suppose that Bell's theorem is meant to apply to such local realism.

 

[JesseM]

Thanks for the summary, but those assertions are a little (too) extreme.
I would think that it is the assertion of the Local Realist that there is no magical dependence of a measurement here on a result there (separability/locality). Influences at a distance according to known or not yet known physical mechanisms are admitted. However I agree that that does essentially deny physical entanglement at a great distance. [..] Further, a Local Realist assumes that already before the measurement one or more unobserved particle variables exist that will affect the values that will be measured. And I suppose that Bell's theorem is meant to apply to such local realism.

The meaning of "locality" specifically has to do with there being no FTL causal influences...if there are such FTL influences this would be a violation of locality, it's irrelevant whether the influences obey some well defined "physical mechanism" or if they appear "magical" to us.

 

[DrChinese]

"Perfect" correlations don't exist in scientific measurements - scientific measurements work with measurement errors.

You got me there!

 

[harrylin]

The meaning of "locality" specifically has to do with there being no FTL causal influences...if there are such FTL influences this would be a violation of locality, it's irrelevant whether the influences obey some well defined "physical mechanism" or if they appear "magical" to us.

FTL would be a violation of some interpretations of SR. Quite some people accept the possibility of FTL influences as long as they cannot be used for signaling. A "FTL influence" even implies the concept of locality: with a non-local concept there is no locality to influence another one.
However, probably this is not really relevant for loopholes.

 

[JesseM]

FTL would be a violation of some interpretations of SR. Quite some people accept the possibility of FTL influences as long as they cannot be used for signaling. A "FTL influence" even implies the concept of locality: with a non-local concept there is no locality to influence another one.
However, probably this is not really relevant for loopholes.

I don't think you're using "locality" the way most physicists use it, there is no possibility of signaling in the Bohmian interpretation but no one would call this a "local" interpretation, and "locality" does not just mean that causal influences are transmitted by discrete localizable particles which might nevertheless be moving FTL as you seem to imply, it specifically is used to refer to the idea that no causal influences move FTL.

 

[harrylin]

I don't think you're using "locality" the way most physicists use it, there is no possibility of signaling in the Bohmian interpretation but no one would call this a "local" interpretation, and "locality" does not just mean that causal influences are transmitted by discrete localizable particles which might nevertheless be moving FTL as you seem to imply, it specifically is used to refer to the idea that no causal influences move FTL.

I kind of agree with that (I see that I didn't formulate it well) but we drifted far away from "local realist" on which we probably already agreed (I wrote: "I agree that that does essentially deny physical entanglement at a great distance"). The meaning of such words as "locality" depend on the context, and here I take it to refer to the physical process of detection which a Local Realist supposes to be of negligible influence far away - thus a "local" process. I think that "FTL" isn't the essential point of "locality" - nor is this little excursion relevant for loopholes.

 

[JesseM]

We drifted far away from "local realist" on which we probably already agreed ("I agree that that does essentially deny physical entanglement at a great distance"). The meaning of such words as "locality" depend on the context

In the specific context of Bell and local realism, I think "locality" is generally used to forbid FTL causal influences. It might be that in some other contexts "locality" simply means that no causal influences are infinitely fast (instantaneous), but this definition would be completely untestable since no experiment can ensure that the time between the two distant measurements is precisely zero. Can you find any examples of physicists using "locality" in a way that allows FTL "local" influences in the context of Bell/local realism?

here I take it to refer to the physical process of detection which a Local Realist supposes to be of negligible influence far away - thus a "local" process.

I don't know what "of negligible influence far away" means, if my detector's interaction with the particle causes a localized FTL influence to be transmitted to the other particle which influences how that particle interacts with the other detector, isn't that a non-negligible influence?

 

[harrylin]

In the specific context of Bell and local realism, I think "locality" is generally used to forbid FTL causal influences. It might be that in some other contexts "locality" simply means that no causal influences are infinitely fast (instantaneous), but this definition would be completely untestable since no experiment can ensure that the time between the two distant measurements is precisely zero. Can you find any examples of physicists using "locality" in a way that allows FTL "local" influences in the context of Bell/local realism?

No, although at one point I poorly formulated it, I stressed that in the context of Bell/local realism ("Bell non-locality"), "locality" is used to indicate a local process or interaction* - thus without measurable effect at a great distance. I'm simply against jargon creep.

* "requiring the value assigned to an operator associated with an individual constitutent to be independent of what is measured on any other constitutent"
- http://plato.stanford.edu/entries/bell-theorem/

I don't know what "of negligible influence far away" means, if my detector's interaction with the particle causes a localized FTL influence to be transmitted to the other particle which influences how that particle interacts with the other detector, isn't that a non-negligible influence?

Certainly that is a non-negligible and non-local process. Again: "locality" in this context is about the influence of detection which is supposed to be limited to the environment of the detection, not about FTL. As I understand it, "Bell non-locality" is the negation of that classical assumption.

Harald

 

[JesseM]

No, although at one point I poorly formulated it, I stressed that in the context of Bell/local realism ("Bell non-locality"), "locality" is used to indicate a local process or interaction* - thus without measurable effect at a great distance. I'm simply against jargon creep.

That seems too vague for a technical term though--what are the precise limits of "great distance"? And surely if enough time has passed then effects at great distances are OK (for example a message sent from Earth to Alpha Centauri today may have a measurable effect there in 4.4 years!), it's only quasi-instantaneous effects at great distances that are a problem?

Also, it occurs to me that if you accept the relativity of simultaneity, then there is really no meaningful distinction between defining locality as "no instantaneous influences" and "no FTL influence"--after all, a particle that travels from location A to location B at a speed even slightly faster than light in one frame will depart A at precisely the same time it arrives at B in some other frame.

 

[harrylin]

That seems too vague for a technical term though--what are the precise limits of "great distance"? And surely if enough time has passed then effects at great distances are OK (for example a message sent from Earth to Alpha Centauri today may have a measurable effect there in 4.4 years!), it's only quasi-instantaneous effects at great distances that are a problem?

Also, it occurs to me that if you accept the relativity of simultaneity, then there is really no meaningful distinction between defining locality as "no instantaneous influences" and "no FTL influence"--after all, a particle that travels from location A to location B at a speed even slightly faster than light in one frame will depart A at precisely the same time it arrives at B in some other frame.

OK, "great" of "great distance" is more for a convincing experiment. But sorry, I can't change the meaning of "local"! One last time: "local" has no direct relationship with speed of transmission. In the context of Bell it simply means that a measurement at one place does not affect the measurement outcome at another place. And the more clearly the particles are separated, the more clearly they are at different locations.

As formulated by Shimony: "Locality is a condition on composite systems with spatially separated constituents, requiring [..] the value assigned to an operator associated with an individual constitutent to be independent of what is measured on any other constitutent."
- http://plato.stanford.edu/entries/bell-theorem/

Harald

PS I suppose that we all agree that the term "local hidden variable theory" points to a theory in which measurement related variables exist before they are measured, and whereby the measurement on one of two spatially separated entities does not affect the measurement outcome on the other entity.

 

[JesseM]

OK, "great" of "great distance" is more for a convincing experiment. But sorry, I can't change the meaning of "local"! One last time: "local" has no direct relationship with speed of transmission. In the context of Bell it simply means that a measurement at one place does not affect the measurement outcome at another place.

But that would suggest that it's irrelevant to Bell's proof whether or not there is a spacelike separation between the two measurements or a timelike one, as long as there is a "great distance" between the two measurements. Do you really think that's the case? What about the locality loophole in Bell experiments?

Also, in Bell's own paper La nouvelle cuisine, much of which can be read on google books starting on p. 216 of this book (it's also available in Speakable and Unspeakable in Quantum Mechanics), he specifically defines his notion of "local causality" in terms of the speed of light limit, and this plays an essential role in his derivation. On p. 217 he starts out connecting locality and the speed of light:

I will be particularly concerned with the idea that effects are near to their causes:

"If the results of experiments on free fall here in Amsterdam would depend appreciably on the temperature of Mont Blanc, on the height of the Seine below Paris and on the position of the planets, one would not get very far.", H.B.G. Casimir.

Now at some very high level of accuracy, all these things would become relevant for free fall in Amsterdam. However even then we would expect their influence to be retarded by at least the time that would be required for the propagation of light. I will be much concerned here with the idea of the velocity of light as a limit.

Then on p. 224 he gives a definition of his "principle of local causality", saying:

The direct causes (and effects) of events are near by, and even indirect causes (and effects) are no further away than permitted by the velocity of light.

Then on p. 225 he makes this more precise with a diagram of the past light cones of two regions 1 and 2, with region 3 being a complete cross-section of the past light cone of region 1 (bounded above and below by spacelike surfaces) which is "above" the region where the two past light cones overlap, so that no point in 3 is part of the overlap region, and any timelike or lightlike worldline which starts from an event in the past light cone of 2 would have to pass through region 3 in order to pass through region 1. Then he defines "local causality" more precisely in terms of this diagram (and also in terms of "local beables", local physical facts taken to be basic elements of any local theory of physics, which he had earlier brought up on p. 219), saying:

A theory will be said to be locally causal if the probabilities attached to the values of local beables in a space-time region 1 are unaltered by specification of values of local beables in a space-like separated region 2, when what happens in the backward light cone of 1 is already sufficiently specified, for example by a full specification of all local beables in a space-time region 3 (figure 6.4).

So you can see that the notion that local facts in region 1 don't depend on facts at a space-like separation in region 2 (and that any correlation can be "screened off" by including facts about another region 3 that any causal influences from the overlap of the two post light cones would have to travel through to get to 1) plays a critical role in his argument, and is the justification for the step on p. 228 where he starts with equation 6.9.2, {A,B|a,b,c,λ}={A|B,a,b,c,λ}{B|a,b,c,λ} (here a and b refer to detector settings in region 1 and 2, while c refers to observable variables in region 3 and λ refers to hidden variables in region 3) and then uses local causality to get the next equation:

Invoking local causality, and the assumed completeness of c and λ in the relevant parts of region 3, we declare redundant certain of the conditional variables in the last expression, because they are at space-like separation from the result in question. Then we have

{A,B|a,b,c,λ}={A|a,c,λ}{B|b,c,λ}

And this step is essential to his proof that QM cannot be explained by a "locally causal" theory of hidden variables.

 

[harrylin]

But that would suggest that it's irrelevant to Bell's proof whether or not there is a spacelike separation between the two measurements or a timelike one, as long as there is a "great distance" between the two measurements. Do you really think that's the case? What about the locality loophole in Bell experiments?

Not at all: for testing theories against each other it is essential to test situations where there can be no doubt that they predict something different; if one fails to do so, one creates a loophole*. For those (like Einstein) who accept Relativity, local causality implies that no influence towards another location can occur faster than light.

However, you do make a good argument here below that Bell defined local causality slightly different from what I am used to; see next.

Also, in Bell's own paper La nouvelle cuisine, much of which can be read on google books starting on p. 216 of this book (it's also available in Speakable and Unspeakable in Quantum Mechanics), he specifically defines his notion of "local causality" in terms of the speed of light limit, and this plays an essential role in his derivation. On p. 217 he starts out connecting locality and the speed of light:

Thanks for the very useful link! There he starts out with the common meaning of local causality:

"the idea that effects are near to their causes",

and next indeed he states that he will be very much concerned with the limit of the speed of light.

Then on p. 224 he gives a definition of his "principle of local causality", saying:

"The direct causes (and effects) of events are near by, and even indirect causes (and effects) are no further away than permitted by the velocity of light."

Indeed. Note that he doesn't suggest that a theory would be necessary "non-local" if events happen close enough to each other to be within the light cone. And what you did not cite:

"Here we have preferred to see it not as a formulation of local causality but as a consequence thereof."
("It" = factorizability due to independence of A on B and vice versa).

[..] And this step is essential to his proof that QM cannot be explained by a "locally causal" theory of hidden variables.

Yes indeed - thanks for your nice summary.

I find Shimony's definition natural and linguistically pure, while I now see that indeed Bell's is subtly different; thanks for pointing that out. Note that it has no consequence for the derivation. Bell admitted that what is "very often" done (differently form his formulation), is to define that A and B do not depend on each other nor on remote polarizers. (p.228/109).Harald

*PS: Shimony calls it the "communication loophole"
- http://plato.stanford.edu/entries/bell-theorem/