Is Bell's Theorem a Valid Solution to the Locality Versus Nonlocality Issue?

[rlduncan]

Theoretically, Bell’s theorem can be not be violated by any experiment when applied to a two-valued variables, such as S(T,F), S(H,T), or S(+,-). Whether the measured values are true/false, heads/tails, or up spin/down spin, etc. Bell’s theorem is a mathematical truth, a tautology. If misapplied by not meeting the conditions of the theorem, then violations may occur. Two examples will demonstrate using a coin tossing experiment where the upper most face is observed and the sequence of heads and tails is recorded. Three coins are tossed simultaneously by three individuals. For simplicity, let's them be a, b, and c and each coin is tossed eight times.

Example 1:
a =HTTTHTHH
b=TTHHTHHH,

b=TTHHTHHH
c=HTHTTTHH,

a=HTTTHTHH
c=HTHTTTHH,
Bell’s Theorem, nab(HH) + nbc(HH) ≥ nac(HH) or 2+3 ≥ 3 (True)

Example 2:
a1=HTTHTHHH
b1=THHTTHTT,

b2=HTHHTHHT
c1=TTTTHHTH,

a2=THHTHTTH
c2=HHHTHTTT,
Bell’s Theorem, nab(HH) + nbc(HH) ≥ nac(HH) or 1+1 ≥ 3 (False)

There is a one-to-one mapping of the three sequences in Example 1 for ab, bc, and ac. In the EPRB experiments only one angle can be measured at a time. As a result there are six sequences necessary which give different runs of photons and a1 sequence is not the same as a2, etc and the one-to-one mapping is lost; and violation of Bell’s theorem may occur as demonstrated in Example 2. This may be the case for the EPRB experiments. Assuming the above analysis is correct. Is there a possibility that Bell’s theorem cannot validly be used to resolve the issue of locality versus nonlocality?

 

[Rap]

Theoretically, Bell’s theorem can be not be violated by any experiment when applied to a two-valued variables, such as S(T,F), S(H,T), or S(+,-). Whether the measured values are true/false, heads/tails, or up spin/down spin, etc. Bell’s theorem is a mathematical truth, a tautology. If misapplied by not meeting the conditions of the theorem, then violations may occur. Two examples will demonstrate using a coin tossing experiment where the upper most face is observed and the sequence of heads and tails is recorded. Three coins are tossed simultaneously by three individuals. For simplicity, let's them be a, b, and c and each coin is tossed eight times.

(Examples)

There is a one-to-one mapping of the three sequences in Example 1 for ab, bc, and ac. In the EPRB experiments only one angle can be measured at a time. As a result there are six sequences necessary which give different runs of photons and a1 sequence is not the same as a2, etc and the one-to-one mapping is lost; and violation of Bell’s theorem may occur as demonstrated in Example 2. This may be the case for the EPRB experiments. Assuming the above analysis is correct. Is there a possibility that Bell’s theorem cannot validly be used to resolve the issue of locality versus nonlocality?

Yes - it is an experimental fact that, for example, if two detectors are oriented in the same direction, they will always record opposite spins. One of the assumptions in the Bell inequalities is that of "counter factual definiteness" (CFD). (see http://en.wikipedia.org/wiki/Counter_factual_definiteness ). CFD means, for example, that if you do not have the detectors oriented in the same direction and measured a1, you may assume that if you had decided to orient them in the same direction, you would have measured a string of opposite spins (a1=a2) - this is not the same thing as knowing that every time you actually measure them, you will get a string of opposite spins. Violations of Bell's inequalities may be explained by the absence of CFD (i.e. a1 may not be a2, etc).

It is important to state CFD explicitly when dealing with Bell's inequalities, if only to affirm it as obvious or axiomatic. I don't know the full implications of rejecting CFD, but I am willing to reject it just to see where things go.

 

[DrChinese]

Theoretically, Bell’s theorem can be not be violated by any experiment when applied to a two-valued variables, such as S(T,F), S(H,T), or S(+,-). Whether the measured values are true/false, heads/tails, or up spin/down spin, etc. Bell’s theorem is a mathematical truth, a tautology. If misapplied by not meeting the conditions of the theorem, then violations may occur. Two examples will demonstrate using a coin tossing experiment where the upper most face is observed and the sequence of heads and tails is recorded. Three coins are tossed simultaneously by three individuals. For simplicity, let's them be a, b, and c and each coin is tossed eight times.

Example 1:
a =HTTTHTHH
b=TTHHTHHH,

b=TTHHTHHH
c=HTHTTTHH,

a=HTTTHTHH
c=HTHTTTHH,
Bell’s Theorem, nab(HH) + nbc(HH) ≥ nac(HH) or 2+3 ≥ 3 (True)

Example 2:
a1=HTTHTHHH
b1=THHTTHTT,

b2=HTHHTHHT
c1=TTTTHHTH,

a2=THHTHTTH
c2=HHHTHTTT,
Bell’s Theorem, nab(HH) + nbc(HH) ≥ nac(HH) or 1+1 ≥ 3 (False)

There is a one-to-one mapping of the three sequences in Example 1 for ab, bc, and ac. In the EPRB experiments only one angle can be measured at a time. As a result there are six sequences necessary which give different runs of photons and a1 sequence is not the same as a2, etc and the one-to-one mapping is lost; and violation of Bell’s theorem may occur as demonstrated in Example 2. This may be the case for the EPRB experiments. Assuming the above analysis is correct. Is there a possibility that Bell’s theorem cannot validly be used to resolve the issue of locality versus nonlocality?

This is a significant misunderstanding of the EPR/Bell situation. It is the REALIST who is asserting something, namely simultaneous a b and c. You have it correct in your Example 1, which shows compliance with Bell. (Of course, these statistics will never match QM.)

Your example 2 exactly follows the QM analysis, namely that only 2 of a b and c have meaning (actually only 1 but you can define a statistical relationship for the 2).

My point is that you have demonstrated Bell precisely, you are simply labeling the resulting conclusion backwards. The purpose of a Bell test is to determine whether nature follows the QM prediction or some other function (such as Product State stats). If Product State stats were seen, for example, then QM would be wrong.

If, on the other hand, you care to put forth a Local Realistic dataset (i.e. like your Example 1) for the angle settings 0, 120 and 240 degrees, you will discover something very quickly. Your match rate will be greater than 33 percent. Now YOUR prediction will not match experiment (which is 25%). Try it, really (psssst you already have proven my point with your example 1)! Suddenly, the assumptions you made about this not being a meaningful test completely falls apart because YOUR predictions will be flat wrong.

 

[rlduncan]

This is a significant misunderstanding of the EPR/Bell situation. It is the REALIST who is asserting something, namely simultaneous a b and c. You have it correct in your Example 1, which shows compliance with Bell. (Of course, these statistics will never match QM.)

Your example 2 exactly follows the QM analysis, namely that only 2 of a b and c have meaning (actually only 1 but you can define a statistical relationship for the 2).

My point is that you have demonstrated Bell precisely, you are simply labeling the resulting conclusion backwards. The purpose of a Bell test is to determine whether nature follows the QM prediction or some other function (such as Product State stats). If Product State stats were seen, for example, then QM would be wrong.

If, on the other hand, you care to put forth a Local Realistic dataset (i.e. like your Example 1) for the angle settings 0, 120 and 240 degrees, you will discover something very quickly. Your match rate will be greater than 33 percent. Now YOUR prediction will not match experiment (which is 25%). Try it, really (psssst you already have proven my point with your example 1)! Suddenly, the assumptions you made about this not being a meaningful test completely falls apart because YOUR predictions will be flat wrong.

I am not questioning the predictions of QM or suggesting whether local realistic view of Enstein, Podolsky, and Rosen vs the Copenhagen interpretation that these correlations are a result of the nonlocality of the measurement process is the correct view.

And, you have made my point, how convenient that the QM analysis requires two of the a,b, and c values, but Bell’s theorem demands that all three be known simultaneously. From the examples given, it is clear that Bell’s theorem may be violated by the EPRB experiments, unless you can explain how measuring just one angle and thus knowing two of the three values somehow meets the conditions of Bell’s theorem.

 

[JesseM]

And, you have made my point, how convenient that the QM analysis requires two of the a,b, and c values, but Bell’s theorem demands that all three be known simultaneously.

No, it doesn't require that they be known, it just shows that in any local realistic theory where we find measurements of the same property always give identical results, that implies each particle must have an identical set of predetermined values for a,b,c prior to a measurement of any two, even if we can never know all three values for a given particle pair.

From the examples given, it is clear that Bell’s theorem may be violated by the EPRB experiments

Of course it can be violated, it's just that such a violation rules out the idea that there is an underlying local realistic theory determining the results.

 

[Rap]

Of course it can be violated, it's just that such a violation rules out the idea that there is an underlying local realistic theory determining the results.

Question - just to be sure I understand the terminology - when you say "local realistic", the "realistic" part means "counter factual definiteness"? Counter factual definiteness is the idea that if you measure spins a and b with equally aligned detectors and always get equal and opposite spins, then if they are not aligned during your measurement, you can assume that if they were aligned, they would have given equal and opposite spins.

Or is there more to "realistic"?

 

[JesseM]

Question - just to be sure I understand the terminology - when you say "local realistic", the "realistic" part means "counter factual definiteness"? Counter factual definiteness is the idea that if you measure spins a and b with equally aligned detectors and always get equal and opposite spins, then if they are not aligned during your measurement, you can assume that if they were aligned, they would have given equal and opposite spins.

To me "local realistic" refers to the idea of a theory that provides a local and objective description of reality at all times, not just the moment of measurement. From another thread, my basic definition:

1. The complete set of physical facts about any region of spacetime can be broken down into a set of local facts about the value of variables at each point in that regions (like the value of the electric and magnetic field vectors at each point in classical electromagnetism)

2. The local facts about any given point P in spacetime are only causally influenced by facts about points in the past light cone of P, meaning if you already know the complete information about all points in some spacelike cross-section of the past light cone, additional knowledge about points at a spacelike separation from P cannot alter your prediction about what happens at P itself (your prediction may be a probabilistic one if the laws of physics are non-deterministic).

And I added this clarification to avoid confusion about 1):

Keep in mind that 1) doesn't forbid you from talking about "facts" that involve an extended region of spacetime, it just says that these facts must be possible to deduce as a function of all the local facts in that region. For example, in classical electromagnetism we can talk about the magnetic flux through an extended 2D surface of arbitrary size, this is not itself a local quantity, but the total flux is simply a function of all the local magnetic vectors at each point on the surface, that's the sort of thing I meant when I said in 1) that all physical facts "can be broken down into a set of local facts". Similarly in certain Bell inequalities one considers the expectation values for the product of the two results (each one represented as either +1 or -1), obviously this product is not itself a local fact, but it's a trivial function of the two local facts about the result each experimenter got.

Also see [post=3248153]this post[/post] where I linked to some specific sections of Bell's "La nouvelle cuisine" paper showing that his notion of "local causality" is the same as the above. Anyway, to go from this notion of local realism to "counterfactual definiteness" you have to add what's called the "no-conspiracy condition" which says that there isn't a correlation between the experimenter's choices of what detector setting to use on each trial and the variables that determine what result the particle gives when it encounters a detector at a given setting (this condition is generally noted in more mathematically rigorous proofs of the theorem, see for example section D on p.6 of this proof). A violation of this condition would require what's called superdeterminism. On this thread I gave an intuitive argument for why I think it's reasonable to say it's very implausible that there could be violations of the no-conspiracy condition:

A violation of this condition is logically possible but would be physically bizarre, it would mean for example that if an experimenter chose on a whim each day whether to have cereal, pancakes or an omelet for breakfast, and on each day used this seemingly random choice to decide which detector setting to use for a particle which had been in flight for exactly a year, then one year earlier the laws of physics must have behaved as if they were "choosing" what hidden variables to assign to the particle based on what the experimenter would decide to have for breakfast one year later.

 

[billschnieder]

Theoretically, Bell’s theorem can be not be violated by any experiment when applied to a two-valued variables, such as S(T,F), S(H,T), or S(+,-). Whether the measured values are true/false, heads/tails, or up spin/down spin, etc. Bell’s theorem is a mathematical truth, a tautology. If misapplied by not meeting the conditions of the theorem, then violations may occur. Two examples will demonstrate using a coin tossing experiment where the upper most face is observed and the sequence of heads and tails is recorded. Three coins are tossed simultaneously by three individuals. For simplicity, let's them be a, b, and c and each coin is tossed eight times.
...

Despite some of the comments above, your argument is valid. You can find more rigorous variations of it in the following articles:

* Hess, K. and Michielsen, K. and De Raedt, H. Possible experience: From Boole to Bell. 2009. EPL (Europhysics Letters), 87:60007. http://arxiv.org/pdf/0907.0767

* Khrennikov, A. Bell-Boole inequality: nonlocality or probabilistic incompatibility of random variables?. 2008. Entropy, 10(2):19--32. http://www.mdpi.com/1099-4300/10/2/19/pdf

* Sica, L. Bell's inequalities:: I: An explanation for their experimental violation. 1999. Optics communications, 170(1-3):55--60. http://arxiv.org/pdf/quant-ph/0101087

* Sica, L. Bell's inequalities:: II: Logical loophole in their interpretation. 1999. Optics communications, 170(1-3):61--66. http://arxiv.org/pdf/quant-ph/0101094

* Kracklauer, AF. Bell’s inequalities and EPR-B experiments: are they disjoint?. 2005. AIP Conf. Proc, 750(1):219--227. http://link.aip.org/link/?APCPCS/750/219/1

 

[JesseM]

Despite some of the comments above, your argument is valid.

Not in the specific case where the experimental setup matches Bell's (and where experimental loopholes are assumed to be closed off, since we are just talking about whether QM is theoretically compatible with local realism), and we assume the results are determined by local realistic underlying laws, along with the no-conspiracy condition. If the papers you quote argue otherwise they are crackpot stuff, and presumably not from any mainstream peer-reviewed journals.

 

[billschnieder]

Not in the specific case where the experimental setup matches Bell's (and where experimental loopholes are assumed to be closed off, since we are just talking about whether QM is theoretically compatible with local realism), and we assume the results are determined by local realistic underlying laws, along with the no-conspiracy condition. If the papers you quote argue otherwise they are crackpot stuff, and presumably not from any mainstream peer-reviewed journals.

As usual you offer no published rebuttals of the above arguments, and your only argument is akin to "if those papers do not agree with me, they are crackpot stuff". I'm not surprised. Note that those articles are squarely focused on the applicability of Bell's inequalities to actual performable Bell-test experiments, and they all come to the conclusion that an a faithful Bell-test experiment can never be performed. It is not about experimental loopholes, or local realism or even realism. It is about logical errors of interpretation and the simple example of the OP illustrates this quite nicely.

Earlier you said :

No, it doesn't require that they be known, it just shows that in any local realistic theory where we find measurements of the same property always give identical results, that implies each particle must have an identical set of predetermined values for a,b,c prior to a measurement ...

Can you show us an experiment in which the experimenters were able to make sure that every single pair of particles possessed an identical predetermined set of values for a,b,c prior to measurement? Failing which, you would have confirmed the arguments made in those articles.

 

[DrChinese]

Despite some of the comments above, your argument is valid. You can find more rigorous variations of it in the following articles:

* Hess, K. and Michielsen, K. and De Raedt, H. Possible experience: From Boole to Bell. 2009. EPL (Europhysics Letters), 87:60007. http://arxiv.org/pdf/0907.0767

* Khrennikov, A. Bell-Boole inequality: nonlocality or probabilistic incompatibility of random variables?. 2008. Entropy, 10(2):19--32. http://www.mdpi.com/1099-4300/10/2/19/pdf

* Sica, L. Bell's inequalities:: I: An explanation for their experimental violation. 1999. Optics communications, 170(1-3):55--60. http://arxiv.org/pdf/quant-ph/0101087

* Sica, L. Bell's inequalities:: II: Logical loophole in their interpretation. 1999. Optics communications, 170(1-3):61--66. http://arxiv.org/pdf/quant-ph/0101094

* Kracklauer, AF. Bell’s inequalities and EPR-B experiments: are they disjoint?. 2005. AIP Conf. Proc, 750(1):219--227. http://link.aip.org/link/?APCPCS/750/219/1

These papers should be ignored as representing scientific consensus. The publishing he is referring to is not considered substantive. Period. He is a diehard local realist who comes here to dupe others who don't know any better.

 

[DrChinese]

Question - just to be sure I understand the terminology - when you say "local realistic", the "realistic" part means "counter factual definiteness"? Counter factual definiteness is the idea that if you measure spins a and b with equally aligned detectors and always get equal and opposite spins, then if they are not aligned during your measurement, you can assume that if they were aligned, they would have given equal and opposite spins.

Or is there more to "realistic"?

Counterfactual definiteness (CD) has nothing to do with the above. The question is quite simply: Does an individual particle have definite values for all possible measurements INDEPENDENT of that measurement? The realist says yes, a chair is red, tall and fluffy even if you don't measure it (or blue, short, stiff, etc.).

QM - because of the Heisenberg Uncertainty Principle - says a particle's observables only match its precision. Reality is defined by the context of a measurement and not otherwise.

Note that entangled pairs have nothing to do with this position. If you are a local realist, you have adopted the first position and believe in CD.

 

[DrChinese]

Can you show us an experiment in which the experimenters were able to make sure that every single pair of particles possessed an identical predetermined set of values for a,b,c prior to measurement?

billschnieder, this is an exceptionally dopey statement from you. According to EPR, every Bell test meets this requirement. Sad to see someone of your caliber on the dark side.

 

[DrChinese]

I am not questioning the predictions of QM or suggesting whether local realistic view of Enstein, Podolsky, and Rosen vs the Copenhagen interpretation that these correlations are a result of the nonlocality of the measurement process is the correct view.

And, you have made my point, how convenient that the QM analysis requires two of the a,b, and c values, but Bell’s theorem demands that all three be known simultaneously. From the examples given, it is clear that Bell’s theorem may be violated by the EPRB experiments, unless you can explain how measuring just one angle and thus knowing two of the three values somehow meets the conditions of Bell’s theorem.

Bell is following the EPR reasoning. There is an element of reality IF the result at Alice can be predicted with certainty. This is not an added requirement from Bell. It is simply continuing the EPR argument where it left off. It is the REALIST who says there exist simultaneous a, b and c. If you don't believe in a, b, and c, then you aren't a realist and Bell doesn't matter.

 

[DrChinese]

And just to make the EPR argument clear:

There is an element of reality IF I can predict Alice's result in advance. I do this by measuring Bob.

Now, by extension: since I can do this for any angle I care to choose, there must be reality to every one of these. That is called REALISM. If you say instead that there is only an element of reality to the ONE angle I choose for an actual measurement and NO others, then you are adopting the position called Contextuality. EPR explicitly denies contextuality as being unreasonable. Read the last 2 paragraphs if you are unsure of this point.

So you CANNOT attack the Bell argument on the a/b/c point if you accept EPR. Of course, Bohr didn't accept EPR, so I would say that is a very reasonable position (i.e. rejecting EPR). And one followed by many even in the day.

 

[JesseM]

As usual you offer no published rebuttals of the above arguments, and your only argument is akin to "if those papers do not agree with me, they are crackpot stuff". I'm not surprised. Note that those articles are squarely focused on the applicability of Bell's inequalities to actual performable Bell-test experiments, and they all come to the conclusion that an a faithful Bell-test experiment can never be performed.

I wasn't talking about experiment, I was talking about Bell's theorem, which is about proving a theoretical claim. Do all those papers agree with Bell's proof that no local realistic theory can possibly match the theoretical predictions of QM? (assuming it doesn't violate the no-conspiracy condition of course) If so then the papers don't disagree with what I said above, in which case I would not necessarily say they are crackpot stuff (note the if-then conditional in my statement). But if they disagree with this theoretical claim, then they are crackpot.

It is of course conceivable that experiments will prove the "theoretical predictions of QM" wrong in the future, but this would not disprove Bell's theorem, it would just make it irrelevant. QM has a perfect track record on all experiments done so far, however.

It is not about experimental loopholes, or local realism or even realism. It is about logical errors of interpretation

Are the "logical errors of interpretation" supposed to be in Bell's analysis? If so it had better be about local realism, because that was the main assumption used to derive Bell's theorem. Showing that an inequality like Bell's can be violated in a scenario that violates the conditions Bell assumed should not be a surprising result to anybody!

Can you show us an experiment in which the experimenters were able to make sure that every single pair of particles possessed an identical predetermined set of values for a,b,c prior to measurement?

Uh, why would I want the experimenters to "make sure" of something that Bell specifically proved must be wrong if QM statistics are respected? Bell was just saying that if you are guaranteed to get perfectly correlated results whenever both experimenters measure the same property in the type of experiment described, and if local realism is true (with the no-conspiracy assumption), then you get the conclusion that the particles must have had predetermined values for all possible properties the experimenters could have chosen (which in turn implies various Bell inequalities should be respected). If either of the "if" claims fails, the "then" claim no longer is expected to hold.

 

[Rap]

Anyway, to go from this notion of local realism to "counterfactual definiteness" you have to add what's called the "no-conspiracy condition" which says that there isn't a correlation between the experimenter's choices of what detector setting to use on each trial and the variables that determine what result the particle gives when it encounters a detector at a given setting (this condition is generally noted in more mathematically rigorous proofs of the theorem, see for example section D on p.6 of this proof). A violation of this condition would require what's called superdeterminism. On this thread I gave an intuitive argument for why I think it's reasonable to say it's very implausible that there could be violations of the no-conspiracy condition:

A violation of this condition is logically possible but would be physically bizarre, it would mean for example that if an experimenter chose on a whim each day whether to have cereal, pancakes or an omelet for breakfast, and on each day used this seemingly random choice to decide which detector setting to use for a particle which had been in flight for exactly a year, then one year earlier the laws of physics must have behaved as if they were "choosing" what hidden variables to assign to the particle based on what the experimenter would decide to have for breakfast one year later.

If you assume that an experimenter can make a truly random choice, then yes, lack of CFD would imply some sort of conspiracy, but if they really have no choice, and the idea that one can make a choice is yet another classical prejudice that needs to be rejected, then there is no conspiracy. I don't know the full implications of this. I think that until we do we should follow the consequences of both accepting and rejecting CFD, just to see where it goes.

 

[JesseM]

If you assume that an experimenter can make a truly random choice, then yes, lack of CFD would imply some sort of conspiracy, but if they really have no choice, and the idea that one can make a choice is yet another classical prejudice that needs to be rejected, then there is no conspiracy.

No, conspiracy refers specifically to the deterministic case where they don't have a choice. It would still be a very strange "conspiracy" in a deterministic universe if seemingly unrelated events like what hidden variables are assigned to a particle by the emitter, and what I choose to have for breakfast exactly one year after the particle was emitted, are found to have a predictable statistical correlation over many different particle emissions and many different breakfasts. In a normal type of deterministic universe you wouldn't expect such predictable statistical correlations between events which you'd expect to be determined by completely different factors like these.

 

[Rap]

Counterfactual definiteness (CD) has nothing to do with the above. The question is quite simply: Does an individual particle have definite values for all possible measurements INDEPENDENT of that measurement? The realist says yes, a chair is red, tall and fluffy even if you don't measure it (or blue, short, stiff, etc.).

QM - because of the Heisenberg Uncertainty Principle - says a particle's observables only match its precision. Reality is defined by the context of a measurement and not otherwise.

Note that entangled pairs have nothing to do with this position. If you are a local realist, you have adopted the first position and believe in CD.

Ok, I am not a "realist" under the above definition. My point is that if you reject CD, you will not need superluminal effects to explain Bell's paradox. I have seen you write "assuming the experimenter has freedom to choose" and I always translated that into "assume CFD".

 

[Rap]

No, conspiracy refers specifically to the deterministic case where they don't have a choice. It would still be a very strange "conspiracy" in a deterministic universe if seemingly unrelated events like what hidden variables are assigned to a particle by the emitter, and what I choose to have for breakfast exactly one year after the particle was emitted, are found to have a predictable statistical correlation over many different particle emissions and many different breakfasts. In a normal type of deterministic universe you wouldn't expect such predictable statistical correlations between events which you'd expect to be determined by completely different factors like these.

But what if the idea that you can choose your breakfast is invalid? This is the essence of the rejection of CFD. Bell's paradox rests on the assumption that e.g. if you chose to align the detectors, they would register equal and opposite spins. This is an acceptance of CFD. This results in superluminal effects. If you reject CFD, then superluminal effects are not needed. But then we live in a universe where the use of the word "if" must be very carefully used, and the illusion of free choice is a classical concept, which fades away as we go from the classical to the quantum realm. Again, I will not defend the rejection of CFD to the death, but it needs to be explicitly stated and its rejection must be done carefully.

 

[JesseM]

But what if the idea that you can choose your breakfast is invalid?

I didn't mean "choose" in the philosophical sense of having non-deterministic free will, I just meant that my brain deterministically comes to a decision based on a host of complex factors that seemingly have nothing to do with the emission of a distant particle 1 year earlier. If you prefer we can reformulate the argument based on something that has nothing to do with human decisions, like what the weather is like today. Even assuming the weather is generated by (chaotic) deterministic processes, why should there be a reliable correlation between the weather on some region of Earth each day and the properties assigned to a particle by a source one year earlier?

If you reject CFD, then superluminal effects are not needed.

Only if you accept the possibility of bizarre correlations between seemingly unrelated events, like the idea that a particle emitter in space would be consistently more likely to assign particles certain properties on days in 2010 where it would be rainy exactly one year later in 2011 here in Rhode Island, and less likely to assign those same properties on days in 2010 where it would be sunny one year later in 2011. This pretty much makes the very concept of "locality" semi-useless, since we could even "explain" a telephone that allows me to talk instantly with someone on Alpha Centauri by saying "there's no actual causal influence between the phone on Alpha Centauri and the phone on Earth, it just happens that there's a perfect statistical correlation between the input received by the Alpha Centauri phone and the output of the phone on Earth (which is being created using a series of 1's and 0's generated by a radioactive decay events, converted into sound) due to some very special initial conditions of all the matter at the big bang."

 

[rlduncan]

JesseM wrote:

Of course it can be violated, it's just that such a violation rules out the idea that there is an underlying local realistic theory determining the results.

I disagree. One may logically concluded that the violations are the result of too many sequences! Not the result of nonlocality. And if this is the case, Bell's theorem can't be used to rule out a local realistic theory and says nothing about locality or nonlocality.

 

[JesseM]

I disagree. One may logically concluded that the violations are the result of too many sequences!

What do you mean "too many sequences"? Do you agree with the logic of Bell and EPR that if we always find perfect correlations when we measure the same property of both particles (and choose what properties to measure at a spacelike separation so under locality neither particle can "know" what property of the other particle is being measured), then that means even before measurement the particles must have had identical predetermined values for all possible properties we could measure? (assuming local realism, of course) If you don't agree with that, probably you just haven't tried to follow the argument carefully enough. If you do agree with that, then do you disagree that for any collection of particle pairs where each member of the pair is predetermined to either have or not have each of three properties A,B,C (so some pairs might have predetermined values [A, not B, C], others might have [not A, B, not C] and so forth), then when considering the total collection of all pairs, it must be true that Number(A, not B) + Number(B, not C) ≥ Number(A, not C)? (see the derivation on this page) As I noted in [post=3290345]this post[/post], the inequality above is not quite the same as Bell's inequality dealing with measured pairs, but it only takes the addition of the no-conspiracy assumption (no correlation between the predetermined values of each particle pair and the experimenter's later choice of what properties to measure for each pair) to derive Bell's from the above.

As for your example involving coin flips I don't understand how it is supposed to relate to Bell's inequality--what is supposed to correspond to each of two experimenters randomly picking a possible property out of the three possible ones? And what corresponds to the fact that whenever they both pick the same property, they are guaranteed to get the same result? Are we supposed to imagine for example that each time a,b,and c flip their coins, two experimenters then have a choice of asking any of the three what their flip was, so for example on one trial I might ask b his result and get the answer "H", while you might ask c her result and get the answer "T"? But that doesn't seem to work in the second example where b1 doesn't match b2, so if I asked b1 and got the answer "H", on the same trial you might ask b2 and get the answer "T", contradicting the EPR/Bell scenario where each time both experimenters measure the same property they are guaranteed to get the same (or opposite, depending on the specific experiment) result. That assumption, which comes directly from the predictions of QM, is critical to deriving the inequality P(A, not B) + P(B, not C) ≥ P(A, not C), although there are other Bell inequalities that don't require the assumption of perfect correlations.

 

[rlduncan]

Bell is following the EPR reasoning. There is an element of reality IF the result at Alice can be predicted with certainty. This is not an added requirement from Bell. It is simply continuing the EPR argument where it left off. It is the REALIST who says there exist simultaneous a, b and c. If you don't believe in a, b, and c, then you aren't a realist and Bell doesn't matter.

Let me restate my concern: If the polarizer angle is set to ab and a string of photons are measured, then a sequence of spin detections result for a1. Now change the polarizer to orientation c and measure the angle ac, experimentally this is a different run and presumably a different string of photons is being measured resulting in a2, that is, a different sequence. If a1≠a2, then a violation of Bell’s theorem can occur. As demonstrated in Example 2.

Please clarify your position. Are you saying that a1=a2 and no violation can occur (for this cause), or a1≠a2 and is irrelevant to the conditions of Bell’s theorem.

Thanks in advance.

 

[billschnieder]

I wasn't talking about experiment, I was talking about Bell's theorem, which is about proving a theoretical claim.

So let's get this straight. The theoretical claim is only valid if all particles have the exact same variables for "a", "b", "c". Yet it is fine to compare the theoretical claim with experiments in which the experimenters have no control over detailed physical parameters to draw a far reaching conclusion about nature? I'm ashamed that such arguments are still made in the 21st century.

Do all those papers agree with Bell's proof that no local realistic theory can possibly match the theoretical predictions of QM?

They all disagree with that claim. They show that Bell's inequalities are not applicable to QM or any experiment that can ever be performed, whether or not the operating physical principles are non-local or local.

But if they disagree with this theoretical claim, then they are crackpot.

We now have here the new definition for crackpot -- "any one who disagrees with Bell and his proponents". Note the absence of any specific rebuttal to the issues raised in those papers. It's been 12 years since the first one.

It is of course conceivable that experiments will prove the "theoretical predictions of QM" wrong in the future

You are baiting and switching here. The issue is not whether QM is correct or not. All of those papers acknowledge the correctness of QM. The issue is whether the data gathering requirements of Bell's inequalities can every be realized in any experiment. You have just confirmed that they can't, by your statement that "... implies each particle must have an identical set of predetermined values for a,b,c prior to a measurement". As a result, you can never be sure that failure to meet this requirements is not solely responsible for any violations. Unless you can demonstrate that this requirement has been met in an experiment, any other conclusion you make is fallacious and sloppy.

Are the "logical errors of interpretation" supposed to be in Bell's analysis? If so it had better be about local realism,

Evidently you haven't read any of the articles, yet you call them crackpot. The logical error is the following which the OP nicely illustrates.

You take three different independent terms P0(a0,b0), P1(b1,c1), P2(a2,c2) and substitute it into the inequality, and then triumphantly proclaim "non-locality! non-reality!" when a violation is obtained, forgetting that Bell's analysis (according to you) requires "... each particle must have an identical set of predetermined values for a,b,c prior to a measurement"!

By naively representing the three independent terms as P0(a,b), P1(b,c), P2(a,c) (without qualifying subscripts), you fool yourself into believing the requirement "... each particle must have an identical set of predetermined values for a,b,c prior to a measurement" has been met.

QM gives you three separate terms P0, P1, P2 for three separate experiments. Actual experiments produce 3 separate terms for three different experiments. The issue addressed in those articles which you haven't responded to is the simple question: Is it mathematically and logically correct to plug those terms from QM and the experiments into Bell's inequality for purposes of comparison. And the answer is unanimously no! So unless you can demonstrate that such an operation is logical and mathematically correct, you can not be sure that the incompatibility is not responsible of the violation.

Showing that an inequality like Bell's can be violated in a scenario that violates the conditions Bell assumed should not be a surprising result to anybody!

Of course. If you have 6 assumptions, you have to make sure you have eliminated 5 of them before you proclaim the failure of the 6th as the cause of the violation.

In case the simple explanation above is still not clear let me rephrase:

Bell's proof relies on the idea that there exists a single probability distribution p(a,b,c) from which three distributions p(a,b), p(b,c), and p(a,c) are then extracted. Experiments measure three arbitrary distributions p(a0,b0), p(b1,c1), P(a2,c3). It is a logical error to just assume that those terms from the experiment originate from a single probability distribution. Similarly, QM gives you three independent terms from three different incompatible experiments and it is a logical error to plug those terms into Bell's inequality.

In the experiment actually performed for which QM provides the predictions, a pair of particles is measured at only two settings so that the outcomes can be expressed as follows:

a,b - outcome after measure particle 1 using setting (a) and particle 2 using setting (b)
b,c - outcome after measure particle 1 using setting (b) and particle 2 using setting (c)
a,c - outcome after measure particle 1 using setting (a) and particle 2 using setting (c)

Clearly, it is impossible in any experiment for all three measurement to be performed on the very same pair of particles. It is therefore impossible in any experiment to obtain p(a,b), p(b,c), and p(a,c) all drawn from a single probability distribution. I'm sure you understood this, that is probably why you said "... each particle must have an identical set of predetermined values for a,b,c prior to a measurement", because then, you could just use another pair of particles and avoid this problem. So unless this requirement of yours is also met, you have no justification for using the terms from the experiments in Bell's inequality in the first place.

Uh, why would I want the experimenters to "make sure" of something that Bell specifically proved must be wrong if QM statistics are respected?

Because they have no other grounds for comparing what they will get with Bell's inequalities, hence the claims in the above articles that a genuine Bell-test experiment is impossible to perform.

Bell was just saying that if you are guaranteed to get perfectly correlated results whenever both experimenters measure the same property in the type of experiment described, and if local realism is true (with the no-conspiracy assumption), then you get the conclusion that the particles must have had predetermined values for all possible properties the experimenters could have chosen (which in turn implies various Bell inequalities should be respected). If either of the "if" claims fails, the "then" claim no longer is expected to hold.

Bell's argument recast in the light of the above goes as follows:

IF there exists a single probability distribution p(a,b,c) from which three distributions p(a,b), p(b,c), and p(a,c) can be extracted, they must necessarily obey the following inequality:

|P(a,b) - P(a,c)| <= P(b,c) + 1

If we find a situation for which the inequality is not obeyed, then a single probability distribution p(a,b,c) does not exist in that situation.

For the QM case, the three terms are independent and incompatible, so it is not surprising that the inequality is obeyed. For the experiment case, I have already explained above why all three terms can not possibly originate from a single distribution as well, not because of any failure of locality or realism but simply because we can only measure two specific particles at two settings in any experiment that can ever be performed.

So I ask, what has this all got to do with realism and non-locality? Is it your argument that if locality is true, and realism is true, then a single probability distribution p(a,b,c) must exist for the experimental results? Is it your argument that the impossibility of ever measuring a specific pair of two particles at three angles implies, the results of the measurements actually performed are not predetermined?

 

[JesseM]

Let me restate my concern: If the polarizer angle is set to ab and a string of photons are measured, then a sequence of spin detections result for a1. Now change the polarizer to orientation c and measure the angle ac, experimentally this is a different run and presumably a different string of photons is being measured resulting in a2, that is, a different sequence. If a1≠a2, then a violation of Bell’s theorem can occur. As demonstrated in Example 2.

Violations can occur with a small number of measurements, but if it's true that both particles had identical predetermined results for all three polarizer settings, and if it's assumed that the probabilities of various predetermined results are uncorrelated with the choice of polarizer settings (the no-conspiracy assumption which says the source cannot "anticipate") what the experimenters will choose on each measurement, then the probability of a violation would be expected to become vanishingly small with a large number of measurements (the law of large numbers).

Suppose it is your job to prepare a series of values for a,b,c on each trial, but you don't know in advance which two Alice and Bob will choose to "measure". In this case, it's not hard to see that since you lack such foreknowledge, while it's possible that for a small number of trials the inequality would be violated, in the long run there is no winning strategy for you and the probability it still gets violated over a long series of trials becomes astronomically small.

 

[rlduncan]

What do you mean "too many sequences"? Do you agree with the logic of Bell and EPR that if we always find perfect correlations when we measure the same property of both particles (and choose what properties to measure at a spacelike separation so under locality neither particle can "know" what property of the other particle is being measured), then that means even before measurement the particles must have had identical predetermined values for all possible properties we could measure? (assuming local realism, of course) If you don't agree with that, probably you just haven't tried to follow the argument carefully enough. If you do agree with that, then do you disagree that for any collection of particle pairs where each member of the pair is predetermined to either have or not have each of three properties A,B,C (so some pairs might have predetermined values [A, not B, C], others might have [not A, B, not C] and so forth), then when considering the total collection of all pairs, it must be true that Number(A, not B) + Number(B, not C) ≥ Number(A, not C)? (see the derivation on this page) As I noted in [post=3290345]this post[/post], the inequality above is not quite the same as Bell's inequality dealing with measured pairs, but it only takes the addition of the no-conspiracy assumption (no correlation between the predetermined values of each particle pair and the experimenter's later choice of what properties to measure for each pair) to derive Bell's from the above.

As for your example involving coin flips I don't understand how it is supposed to relate to Bell's inequality--what is supposed to correspond to each of two experimenters randomly picking a possible property out of the three possible ones? And what corresponds to the fact that whenever they both pick the same property, they are guaranteed to get the same result? Are we supposed to imagine for example that each time a,b,and c flip their coins, two experimenters then have a choice of asking any of the three what their flip was, so for example on one trial I might ask b his result and get the answer "H", while you might ask c her result and get the answer "T"? But that doesn't seem to work in the second example where b1 doesn't match b2, so if I asked b1 and got the answer "H", on the same trial you might ask b2 and get the answer "T", contradicting the EPR/Bell scenario where each time both experimenters measure the same property they are guaranteed to get the same (or opposite, depending on the specific experiment) result. That assumption, which comes directly from the predictions of QM, is critical to deriving the inequality P(A, not B) + P(B, not C) ≥ P(A, not C), although there are other Bell inequalities that don't require the assumption of perfect correlations.

Bell's theorem requires three sequences. More than three may result in a violation of the theorem that should be obvious given the two examples.

Coin tossing certainly relates to Bell's inequality. Read the post again. Bell's theorem pertains to any two-valued variables. Do you disagree?

Coin tossing also compares to the ERPB experiment. Imagine a glass top where Alice’s views from the top and Bob from the bottom. When viewing the same coin they always get opposite results. Now flip coins ab to generate a sequence of flips. Repeat a second time using coins bc and record the sequence. Finally do the same for ac, where the first coin is viewed by Alice and the second coin by Bob. Try this yourself and you will find that a violation of Bell’s inequality can occur.

 

[rlduncan]

Violations can occur with a small number of measurements, but if it's true that both particles had identical predetermined results for all three polarizer settings, and if it's assumed that the probabilities of various predetermined results are uncorrelated with the choice of polarizer settings (the no-conspiracy assumption which says the source cannot "anticipate") what the experimenters will choose on each measurement, then the probability of a violation would be expected to become vanishingly small with a large number of measurements (the law of large numbers).

Suppose it is your job to prepare a series of values for a,b,c on each trial, but you don't know in advance which two Alice and Bob will choose to "measure". In this case, it's not hard to see that since you lack such foreknowledge, while it's possible that for a small number of trials the inequality would be violated, in the long run there is no winning strategy for you and the probability it still gets violated over a long series of trials becomes astronomically small.

The law of large numbers will not change the fact that the sequences a1 and a2 are different, and a violation will still occur. (IMHO)The only way a violation would not occur is if a1 equals a2 and are identical.

 

[JesseM]

So let's get this straight. The theoretical claim is only valid if all particles have the exact same variables for "a", "b", "c".

"All particles"? No, just that each member of a particle pair must have same predetermined values as the other one, different pairs may have different predetermined values of course (for example one pair might share predetermined values [a=+1, b=+1, c=-1] while a different pair might share predetermined values [a=-1, b=+1, c=+1]). And this is not itself a starting assumption, rather it's a conclusion that must be true if there is a perfect correlation between measurements with the same setting (which is the prediction of QM) and if we assume local realism.

If you disagree with this, you'd have to claim it's possible that you could have a local realist model that predicts a perfect correlation between measurements whenever the same setting is used in a Bell type experiment, but somehow doesn't involve predetermined values for each of the three settings. Are you in fact making this claim?

They all disagree with that claim. They show that Bell's inequalities are not applicable to QM or any experiment that can ever be performed, whether or not the operating physical principles are non-local or local.

OK, assuming you've read them correctly (I don't know if you have), then they are crackpots and you shouldn't be posting such nonsense here.

We now have here the new definition for crackpot -- "any one who disagrees with Bell and his proponents".

Not Bell specifically, rather anyone who disagrees with a theoretical proof that thousands of physicists have concluded is airtight is almost certainly a crackpot. For example, if someone disagrees that according to GR it is possible to pass through the event horizon of a black hole and reach the singularity in finite proper time, they are a crackpot.

Note the absence of any specific rebuttal to the issues raised in those papers. It's been 12 years since the first one.

What, you expect people to read through them all and post detailed rebuttals? Why don't you describe a particular argument you found convincing. Incidentally, in case you've forgotten I did discuss the "Possible Experience: from Boole to Bell" paper with you in the past, see [post=2780659]this post[/post] and [post=2781956]this one[/post].

It is of course conceivable that experiments will prove the "theoretical predictions of QM" wrong in the future

You are baiting and switching here.

No I'm not, I was responding to your comment which seemed to suggest the papers might have something to do with experimental loopholes: "Note that those articles are squarely focused on the applicability of Bell's inequalities to actual performable Bell-test experiments, and they all come to the conclusion that an a faithful Bell-test experiment can never be performed." Of course it is in theory possible to perform a "faithful Bell-test experiment", if you disagree this is probably related to your delusions that it is somehow necessary to "control" the hidden variables in order to perform such an experiment. In fact all that needs to be assumed about an ideal theoretical experiment is that the experimenters are choosing randomly between 3 detector settings, the choice of detector settings and the measurements with those settings are made at a spacelike separation (no locality loophole), all emitted particles are detected (no detector efficiency loophole), and for some versions of the proof we also assume it is verified that whenever both experimenters pick the same setting they are guaranteed to get the same result (a theoretical prediction of QM). I don't think there are any other experimental conditions needed, if you combine these assumptions about the experiment with the theoretical assumption of local realism (and the no-conspiracy assumption), then any other conclusions (like predetermined results for each particle pair) are derived, not assumed.

The issue is not whether QM is correct or not. All of those papers acknowledge the correctness of QM. The issue is whether the data gathering requirements of Bell's inequalities can every be realized in any experiment. You have just confirmed that they can't, by your statement that "... implies each particle must have an identical set of predetermined values for a,b,c prior to a measurement".

But that doesn't need to be verified experimentally, it's an inescapable logical conclusion given the previous assumptions I mentioned about 1) the experimental setup, 2) the perfect correlation whenever the same detector setting is chosen, and 3) local realist laws with the no-conspiracy assumption. There's no theoretical way to satisfy 1), 2) and 3) without such predetermined values for each pair, I'm sure none of your papers provide a model which satisfy 1-3 but don't involve predetermined values.

You take three different independent terms P0(a0,b0), P1(b1,c1), P2(a2,c2) and substitute it into the inequality, and then triumphantly proclaim "non-locality! non-reality!" when a violation is obtained

Only if conditions 1-3 are satisfied would any such triumphant proclamation be made. As I pointed out in my last two comments to rlduncan it's unclear how his lists are supposed to relate to a Bell-type experimental setup or how they are supposed to guarantee that if the experimenters randomly choose the same setting, they will always get the same result.

Bell's proof relies on the idea that there exists a single probability distribution p(a,b,c) from which three distributions p(a,b), p(b,c), and p(a,c) are then extracted.

Yes, that assumption is called the "no-conspiracy assumption", and a violation under local realism would imply that the properties of the particles which they had at some time prior to the choice of detector settings were statistically correlated with the later choice of detector setting of the experimenters, a weird sort of foresight. See my comments to Rap about why I think it's reasonable to consider this very implausible.

Clearly, it is impossible in any experiment for all three measurement to be performed on the very same pair of particles. It is therefore impossible in any experiment to obtain p(a,b), p(b,c), and p(a,c) all drawn from a single probability distribution.

Not exactly possible, but the the probability of a significant difference gets tinier and tinier the more trials you do (so in the limit of an infinite number of trials, which is what frequentist probabilities are supposed to mean, the probability distributions become identical). To see this, suppose you and your buddy are playing a game where you can confer, then you have to go in separate rooms where you will be asked either question a, question b, or question c (analogous to the particles being measured at a spacelike separation), to which you answer "yes" or "no". This is repeated many times, and the one rule is that the two of you must always make sure that on any trial where you are both asked the same question, you must give the same answer (analogous to the particles always giving the same result when measured with the same setting)--if two get this wrong even once over thousands of trials, you lose out on your chance to win a FABULOUS PRIZE! If you two have no way to communicate directly once you are taken into separate rooms (analogous to no nonlocal influences between particles), and you have no way to anticipate which question you will be asked (analogous to the no-conspiracy assumption), do you doubt that the only winning strategy for the two of you is to always agree on all three answers on each trial? (if the answer is no, please outline your alternate strategy) And if you agree to this, is it not obvious that whatever the fraction of all trials where your predetermined answers include a given pair like "yes to question b, no to question c", then if we look at the subset of trials where you are asked questions b and c, the fraction of those trials where you answered "yes to b, no to c" is likely to be very close to the fraction on all trials if the number of trials is very large? (to make it more technical, let's say that the probability p that the first fraction differs from the second fraction by more than some small amount ε can be made as small as you like by picking a sufficiently large number of trials N--this is basically just a restatement of the law of large numbers)

I'm sure you understood this, that is probably why you said "... each particle must have an identical set of predetermined values for a,b,c prior to a measurement"

That "must" was not meant to be a condition, rather it was a logical deduction (implied by conditions 1-3 above), kind of like saying "if you have a prime number larger than 2, it must be odd".

So I ask, what has this all got to do with realism and non-locality? Is it your argument that if locality is true, and realism is true, then a single probability distribution p(a,b,c) must exist for the experimental results?

All conditions 1-3 are needed (including the fact of being guaranteed to get the same result on any trial where the same setting is used by both experimenters), but if they are all satisfied, then on every trial there must have been predetermined results for all three settings. If you grant this, then we don't even need to talk about "probability distributions" if you want to avoid getting into discussions of the definition of probability (as you know I define probabilities in terms of frequencies in the limit as the number of trials goes to infinity), we can just talk about the fraction of trials where each of the eight possible triplets of predetermined results occurred. Then if the number of trials is large, the probability of a significant difference between (fraction of trials where triplet of predetermined results included +1 for b, -1 for c) and (fraction of trials where b and c were measured, measurement of b gave +1 and measurement of c gave -1) becomes very small. And if you are willing to use a "limit frequentist" definition of probability for the sake of the argument, this should show why P(triplet of predetermined results included +1 for b, -1 for c) = P(b gave +1, c gave -1 | b and c chosen as detector settings).

 

[JesseM]

The law of large numbers will not change the fact that the sequences a1 and a2 are different, and a violation will still occur.

If the sequences are different, then I don't see how you can meet one of the conditions required for deriving that Bell inequality, namely the one that says anytime the experimenters pick the same setting they are guaranteed to get the same answer. From my recent post to billschnieder, the assumptions necessary to derive that Bell inequality:

In fact all that needs to be assumed about an ideal theoretical experiment is that the experimenters are choosing randomly between 3 detector settings, the choice of detector settings and the measurements with those settings are made at a spacelike separation (no locality loophole), all emitted particles are detected (no detector efficiency loophole), and for some versions of the proof we also assume it is verified that whenever both experimenters pick the same setting they are guaranteed to get the same result (a theoretical prediction of QM). I don't think there are any other experimental conditions needed, if you combine these assumptions about the experiment with the theoretical assumption of local realism (and the no-conspiracy assumption), then any other conclusions (like predetermined results for each particle pair) are derived, not assumed.

If your example of coin sequences doesn't match these conditions, it's no surprise that you can get a Bell inequality violation! But if you do assume conditions equivalent to these--for example if after each set of three flips, you and a buddy memorize the results, then travel to two different "experimenters" who can ask you about a single one of the three flips, with you and the buddy being questioned at a spacelike separation--then the probability that your answers will violate the Bell inequality become vanishingly small as the number of trials becomes very large. Try to think in terms of the "game" I suggested to billschnieder:

suppose you and your buddy are playing a game where you can confer, then you have to go in separate rooms where you will be asked either question a, question b, or question c (analogous to the particles being measured at a spacelike separation), to which you answer "yes" or "no". This is repeated many times, and the one rule is that the two of you must always make sure that on any trial where you are both asked the same question, you must give the same answer (analogous to the particles always giving the same result when measured with the same setting)--if two get this wrong even once over thousands of trials, you lose out on your chance to win a FABULOUS PRIZE! If you two have no way to communicate directly once you are taken into separate rooms (analogous to no nonlocal influences between particles), and you have no way to anticipate which question you will be asked (analogous to the no-conspiracy assumption), do you doubt that the only winning strategy for the two of you is to always agree on all three answers on each trial? (if the answer is no, please outline your alternate strategy)

Suppose that besides the condition that you automatically lose if even once you give different answers when asked the same question, the other condition to win the FABULOUS PRIZE is that your answers must violate this inequality:

(Number of trials where you were asked question a and you answered "yes" and your buddy was asked question b and he answered "no")
+
(Number of trials where you were asked question b and you answered "yes" and your buddy was asked question c and he answered "no")
≥
(Number of trials where you were asked question a and you answered "yes" and your buddy was asked question c and he answered "no")

If the number of trials is large, say 10,000, do you think there is any strategy that you and your buddy can adopt that will prevent the probability of your winning the FABULOUS PRIZE from being astronomically small?

Coin tossing certainly relates to Bell's inequality. Read the post again. Bell's theorem pertains to any two-valued variables. Do you disagree?

Yes, of course I disagree! Bell's theorem is only meant to pertain to situations that match the conditions he assumed, any violation of these conditions makes it trivial to violate them. For example, in the "game" above, if you were told what two questions you were going to be asked in advance, winning would be as easy as pie!