Understanding Bell's mathematics

[Gordon Watson]

Bell locality just says it should be possible to find a set of information about hidden variables λ that's sufficiently complete such that the two sides of the above equation will be equal. It doesn't say the opposite, that you can't find a more restricted λ such that the two sides are unequal--of course you can! For example, λ could be defined in such a way that it contains no information about anything in the past light cone of either measurement, in which case it should be completely irrelevant and P(G|X,a,λ) = P(G|X,a) and so forth. In that case the above inequality may be correct, despite the fact that X is a local realistic theory. Again, Bell is just saying that under local realism it should be possible to define some λ so the two sides of the equation become equal.

It's not a supposition, any physicist will agree that under a local realist theory, as long as enough information about local variables (hidden or otherwise) in the past light cones of G and G' is contained in λ, your equation (3) should be satisfied. If this doesn't make sense to you, perhaps you could address my post #29?

Also, consider the following:

--In a local realist theory, all physical facts--including macro-facts about "events" spread out over a finite swatch of space-time--ultimately reduce to some collection of local physical facts defined at individual points in spacetime (or individual 'bits' if spacetime is not infinitely divisible). See my first few comments in this post from another thread. So, any fact of the matter about the result of a measurement can be reduced to a set of local facts about events associated with the smallest possible units of spacetime. Without loss of generality, then, let G and G' be two possibilities for what happens at some single point in spacetime P.

--In a deterministic local realist theory, if λ represents the complete set of local physical facts that lie in the past light cone of P at some time t prior to P, then this allows us to determine whether G or G' occurs with probability one, so any additional information would not cause us to alter our probability estimate. Thus in this case it should be clear that your equation (3) above is satisfied.

--An intrinsically probabilistic local realist theory is a somewhat more subtle case, but for any probabilistic local realist theory it should be possible to break it up into two parts: a deterministic mathematical rule that gives the most precise possible probability of a given event happening at point P based on information in the past light cone of P (if information outside that past light cone of P was required to get the most precise possible probability estimate, the theory would not be a local one), and a random "seed" number whose value is combined with the probability to determine what event actually happened. This "most precise possible probability" does not represent a subjective probability estimate made by any observer, but is the probability function that nature itself is using, the most accurate possible formulation of the "laws of physics" in a universe with intrinsically probabilistic laws.

For example, if the mathematical rule determines the probability of G is 70% and the probability of G' is 30%, then the random seed number could be a randomly-selected real number on the interval from 0 to 1, with a uniform probability distribution on that interval, so that if the number picked was somewhere between 0 and 0.7 that would mean G occurred, and if it was 0.7 or greater than G' occurred. The value of the random seed number associated with each probabilistic choice (like the choice between G and G') can be taken as truly random, uncorrelated with any other event in the universe, while the precise probability of different events could be generated deterministically from a λ which contained information about all local physical facts at all points in spacetime in the past light cone of P. In this case, it would again be true that your equation (3) above would be satisfied.

Do you think it is possible to imagine a "local realist" theory where one of the above would not be true?

Dear Jesse,

Thank you for this detail. You provide much to study. Maybe some is beyond me.

Certainly there are some distracting generalities --

"It's not a supposition, any physicist will agree that under a local realist theory, as long as enough information about local variables (hidden or otherwise) in the past light cones of G and G' is contained in λ, your equation (3) should be satisfied."

For the moment -- to minimize distractions and focus on Bell's mathematics --

I have shown the realistic locality assumption that reduces (1) to (2). With Einstein and Bell as supporters. [Edit: I also give the assumptions to pass from (2) to (4) and from (2) to (5).]

1. Could you show me the assumption that you use to reduce (2) to (3) please? With some support.

2. With your assumption, would all probabilities be zero or one only?

Again thank you,

Jenni

 

[billschnieder]

Suppose we are given candidate theory X for which we can calculate the various probabilities associated with the possible outcomes of Alice and Bob. Let the possible outcomes for Alice be A = {G, R}, for Bob be B = {G', R'}. Upon calculating with X we find for some λ, A, B, a, and b that

P(G|X,a,λ) P(G'|X,b,λ) ≠ P(G,G'|X,a,b,λ) .

Then X [in your view] does NOT satisfy "Bell Locality".

For example [in your view], this is the case for QM:

P(G|X,a,λ) P(G'|X,b,λ) = ½ ∙ ½ = ¼ ≠ P(G,G'|X,a,b,λ) .
__________________________

MEANING of "Bell Locality" VIOLATION and the error in X.

If G and G' are correlated, probability theory teaches [equation (1)] that

(1) P(G,G'|X,a,b,λ) = P(G|X,a,b,λ).P(G'|X,a,b,λ,G) =

(2) P(G|X,a,λ).P(G'|X,a,b,λ,G);

(2) following from (1) because (with Einstein, Bell, and many others) we agree that setting b can have no relevance for outcome G. A realistic locality condition.

But Bell goes further. Bell supposes [Bertlmann's Socks, page 13] that

(3) P(G'|X,a,b,λ,G) = P(G'|X,b,λ),

a result [you say] known as Bell locality.

Bell supposes that the condition aG has no relevance for λ. So Bell locality is a restraint on the λ-s under consideration in candidate theory X. This explains why theory X fails to be realistic.

"Bell locality" might be less confusing if known as "Bell's supposition"?

The condition aG has relevance for λ because it indicates how the λ-s in P(G'|X,a,b,λ,G) respond when subject to a measurement interaction with a measuring device oriented a -- they yield outcome G -- that is the relevance and physical significance of condition aG = |a,G.

[See the one emphasized phrase in Bohr's response to EPR; a response which Bell did not understand.]

Thus condition aG eliminates, from consideration in P(G'|X,a,b,λ,G), just those λ-s that are irrelevant.

Or, better, clearer:

Condition aG identifies, for consideration in P(G'|X,a,b,λ,G), just those λ-s that are relevant. That is, the λ-s that would respond, if subject to a measurement interaction with a measurement device oriented a, to yield outcome G -- that being the relevance and physical significance of aG.

Bell's supposition is a restriction on candidate λ-s.

Bell's supposition should not be associated with locality, nor with realistic constraints on locality.

Note 1:
A realistic constraint on locality was exercised in reducing (1) to (2).

Note 2:
Parameter independence is allowed --
(4) P(G'|X,a,b,λ,G) = P(G'|X,b,λ,G).

Note 3:
Outcome independence is allowed --
(5) P(G'|X,a,b,λ,G) = P(G'|X,a,b,λ).

Note 4:
The one thing not allowed, when X relates to EPR-Bell settings, is Bell's supposition
(3) P(G'|X,a,b,λ,G) = P(G'|X,b,λ).

IMHO.

I completely agree with this. As I mentioned already, I believe Bell's misunderstanding originates from his admitted belief that any form of logical dependence implies a causal link. So he mistakenly thinks by removing aG, he is being true to the "locality condition".

Using my earlier example to illustrate

1: abs-blue, brs-blue
2: abs-blue, bbs-red
3: abs-red, bbs-blue
4: abs-red, brs-red
5: ars-blue, bbs-blue
6: ars-blue, brs-red
7: ars-red, brs-blue
8: ars-red, bbs-red

In calculating P(B|abs, bbs, z, A) as a term while calculating P(AB|a,b,z), including absA on the RHS does not imply that A or bbs has any causal influence on B. It is just a straight forward application of the chain rule of probability theory. It simply means, as JenniT points out, that in calculating the probablity we consider only the relevant cases , and ignore the irrelevant cases. In other words, we should not ignore the context of the calculation while doing it.

Also, according to the chain rule of probability theory, it is possible under certain conditions some of which are outlined by JenniT to reduce as follows:
1) P(AB|CD) = P(A|CD)P(B|ACD)
2) P(AB|CD) = P(A|CD)P(B|CD)


However, each of those expressions must give the same result under those conditions
If ever we come across a situation in which the results are different, it simply tells us that in such a situation, we are not justified to reduce (1) to (2) . If Bell is using the chain rule, then all those expressions must be equivalent for him to be able to reduce (1) to (2). If he is not using the chain rule, there is no other rule in existence apart from the chain rule that permits writing down equation (2). Indeed according to his Bertlmann socks paper, it appears he is relying on the chain rule.

It does not say anything about non-locality as proven by my example. Therefore indeed Bell's so called "locality condition" is not about locality at all but a "no logical dependence condition".

 

[Gordon Watson]

,,,

Bell's misunderstanding originates from his admitted belief that any form of logical dependence implies a causal link. So he mistakenly thinks by removing aG, he is being true to the "locality condition".
,,,

Bill, quick question --

What sources do we have for the bold above?

The generality here ["any form"] is a no-no where I come from.

Thank you

 

[ThomasT]

Say H_a represents the full set of information about all local variables in the past light cone of measurement A at some time t prior to A, and H_b represents the full set of information about all local variables in the past light cone of B at the same time t which is also prior to B, with t chosen so that it happens after the last moment the two light cones overlap.

In this case, if we simply define H as the sum of all the information contained in both H_a and H_b, then the equation is equivalent to this:

F(AB|abH_aH_b) = F(A|aH_aH_b)*F(B|bH_aH_b)

Do you still have a problem with this equation? Some of those terms may represent unecessary information--for example, I could give you an argument that F(A|aH_aH_b) = F(A|aH_a), i.e. H_b would give no additional information about the probability/frequency of A given knowledge of a and H_a--but that shouldn't affect their validity.

Ok, first, can we switch to Bell's notation? Let's say that λ_a and λ_b represent the polarization vectors of sinusoidal disturbances incident on the polarizer settings a and b.
Lets also say that λ_a = λ_b, so that λ represents a single, continuous polarization vector extending between a and b, following Bell. So, we write, P(AB|Habλ) = P(A|Haλ_a) P(B|Hbλ_b), where H denotes a local common cause of λ, λ_a, and λ_b, and which says that, given H, a, b, and λ determining the union of the data sets A and B, then the data sets A and B are statistically independent. Is this correct? If not, what does it say?

 

[JesseM]

No, I provided the full universe for my hidden "variable" z, which were clearly defined.

But the way you defined z was not sufficient to give any reason for Bell or any of his advocates to expect that the equation P(AB|abs, bbs, z) = P(A|abs, z) P(B|bbs, z) would be valid, thus you are just engaging in a silly strawman. Bell only says that in a local realist universe it is possible to find a set of variables to define z such that this equation will hold, not that it should hold for any arbitrary definition of z. And given your argument that the response to pushing the switches depended only on a simple internal mechanism in each ball which either activated the same-color light or the opposite-color light, it is indeed possible to define such a z that guarantees the equation will hold, as I showed.

The term "variable" is actually not appropriate since it carries a connotation of something with multiple values. Maybe that is what is confusing you. That is why t'Hooft prefers to use the terms "beables" as different from "changeables".

t'Hooft did not mean to imply anything about whether the values can "change", either in the sense of changing with time for a single particle, or in terms of being different for different particles in different trials.

No it must not! "z" was clearly defined and my full universe includes all possibilities given that "z" is true.

There's nothing "wrong" with your definition of z in itself, but as I said, if you think Bell would have said P(AB|abs, bbs, z) = P(A|abs, z) P(B|bbs, z) for your definition of z, then this is either a strawman or a complete misunderstanding of the physical meaning behind equation (2) in his paper.

P(B|bbs, aObN) is a nonsensical expression with no meaning. It is similar to saying. What is the probability of Bob getting a blue light if Bob gets a blue light.

No, it's "what is the probability of Bob getting a blue light given that he pushed the blue switch, and given that his ball contains a mechanism that causes the ball to light up blue if the blue switch is pressed and red if the red switch is pressed, while Alice's ball contains a mechanism that causes the ball to light up red if the blue switch is pressed and blue if the red switch is pressed".

Do you think any case where P(Y|X)=1, i.e. X implies Y with probability 1, is "a nonsensical expression with no meaning"? If you don't have a problem with conditional probabilities where the condition implies the result with probability 1, what is your special objection to P(B|bbs, aObN)?

No surprise that you will never get any result other than 1 or 0 (certainties) because your full universe will consist of exactly one case and no more, in which case it makes no sense to compare such a result with QM where values other than 1 and 0 are obtained.

You really don't understand local hidden variables theories then! In a case where the experimenters always obtain identical results with probability 1 whenever they choose identical detector settings, a local hidden variables theory says hidden variables associated with each particle on a given trial must completely determine their responses to each detector setting, and they are always created with identical predetermined responses. However, these identical predetermined responses may be different on different trials, which is what accounts for the fact that any given detector setting can yield different results on different trials, and thus the experimenter (who doesn't know the hidden variables on each trial) has a probability estimate for the result on a given detector setting which is not 0 or 1.

Note that all this is directly analogous to the scratch lotto card analogy I gave you on post #18 of the other thread, which I repeatedly asked you to address but you kept ignoring. Can you tell me whether you agree that both the following would be true?

1. On each trial, if you know the full set of "hidden fruit" behind each box on the cards given to Alice and Bob, then the probability Alice gets a cherry if she scratches a given box is also known with probability 1.

2. If the source generating the cards picks the "hidden fruits" randomly on each trial, then if you don't know the hidden fruits, the probability of cherry vs. lemon when a given box is scratched would not be 0 or 1.

I'm afraid it is you who is badly misunderstanding probability theory. The conditioning variable must be specific, not just a vague concept of "everything in past light cone".

Nothing vague about it, in a local realist universe. Do you agree that in a local realist universe, the laws of physics associate some well-defined set of basic physical facts with each point in spacetime? Do you agree that for some event A, the set of points in spacetime which are in the past light cone of A at some specific time t is also completely well-defined and unambiguous?

Besides Bell was not dealing with certainties but with probabilities. Your approach MUST NECESSARILY result in a certainties (0, or 1) rather than probabilities.

0 and 1 are allowable probabilities, surely you don't disagree? Would you like some examples appearing in textbooks where the probability of something is 0 or 1?

Again, "z" does not represent all different possible hidden variable states. You are confusing functional notation with Probability notation. When I write P(A|ab) in a generic equation, a and b are not variables but place-holders for "beables" not variables.

"beables" is a physical term, nothing to do with a particular type of notation in probability theory. If you disagree, find me an example of a probability text not relating to discussions of fundamental physics which uses the term "beables".

It is true that there is sometimes a distinction between uppercase and lowercase in probability theory notation, so X could represent a random variable and x would then represent some particular possible value of that random variable (so X might have possible values x₁, x₂, etc.). See here. Of course if we want to use this convention, it'd be better not to have capital A and B represent particular measurement outcomes as we have done so far!

I don't need to multiply by P(z) because my full universe is already defined by z, ie within my full universe, P(z) is already 1. I don't need to add anything from different hidden variables because I am dealing with a specific hidden variable. I don't need to consider all possible hidden variables because being omniscient about the workings of my machine, I know for sure that it only operates as I described. Yet, according to Bell's choice of equations, my machine is non-local.

No, this is the strawman and/or misunderstanding. Bell's equation is not meant to apply to any arbitrary definition of the extra variable (z in your equation), it's based on the idea that in a local realist universe it's possible to define the variable in such a way as to make the equation hold. And this was indeed true in your example when I defined z to give the information about which mechanism was in Alice's ball and which was in Bob's.

P(A|abs, z) means the Probability that A is observed, given that abs is True and z is true. So when you talk about summing over all values of z,what does the term P(A|abs, z) mean to you.

It means the probability for any single value z of the random variable Z. When I talked about summing over all the different values of z, that was for the purposes of eliminating it from the equation to get P(A|abs). Suppose for example the random variable Z has only two possible values z₁ or z₂, so on a large set of N trials, we'd expect the number of trials with z₁ to be N*P(z₁), and the number of trials with z₂ to be N*P(z₂), with P(z₁) + P(z₂) = 1. Then if we want to know P(A|abs), do you disagree that the following equation would hold? P(A|abs) = P(A|abs, z₁)*P(z₁) + P(A|abs, z₂)*P(z₂)

If you are thinking of adding up the results from say z1 with those of z2, ..., zn then clearly the results from z1 are due to a completely different context from z2 etc Therefore you can not add them up legitimately.What is the rule of probability theory that permits you to do that addition?

I would have to look a bit for a formal statement of the rule, but first tell me, do you disagree that if each trial in the sample space has some specific value of the random variable Z drawn from the set {z₁, z₂, ... z_n}, then it must be true that P(A) = (sum over all values of i from 1 to n) P(A|z_i)*P(z_i) ? If you don't disagree with that, then do you disagree that for some other fact b which is true on some trials but false on others, it would be true that P(A|b) = (sum over all values of i from 1 to n) P(A|b, z_i)*P(z_i) ?

The value of the hidden variable z in my example is the description of the mechanism of the machine I provided.

But your z did not explicitly detail which mechanism was in the ball given to Alice and which was in the ball given to Bob. You defined z like this:

Here is the so-called 'full universe' of possibilities defined by z: where abs means Alice presses blue switch and brs means Bob presses red switch. The color after the hyphen, is what Alice and Bob see as a result of their pressing their switches.
Alice, Bob
1: abs-blue, brs-blue
2: abs-blue, bbs-red
3: abs-red, bbs-blue
4: abs-red, brs-red
5: ars-blue, bbs-blue
6: ars-blue, brs-red
7: ars-red, brs-blue
8: ars-red, bbs-red

If each of these supposed to be a distinct z, so z₁ would represent possibility 1, z₂ would represent possibility 2, then of course this would allow us to infer which mechanism was in Alice's ball on each trial. But from your later statements it seems you didn't mean z to have distinct values, but just to represent the knowledge that one of these possibilities will hold on each trial, without knowing which one will hold on any given trial.

The concept of multiple values for the hidden variable is a misunderstanding carried over from functional notation and probably over-reading into the term "variable".

This comment reveals a complete lack of understanding of Bell's paper. His λ was of course supposed to represent a variable that could take different values on different trials; if that wasn't true, equation (2) in which he integrates over all possible values of λ would make no sense at all!

 

[billschnieder]

Bill, quick question --

What sources do we have for the bold above?

The generality here ["any form"] is a no-no where I come from.

Thank you

In Bell's Bertlmann's socks paper (http://cdsweb.cern.ch/record/142461/files/198009299.pdf), page 15, second paragraph, he says:

To avoid the inequality, we could allow P1 in (11) to depend on b or P2 to depend on a. That is to say we could admit the signal at one end as a causal influence at the other end.

By saying this he is clearly saying that in equation (11)

P(AB|abz) = P1(A|az)P2(B|bz)

if we include a as a condition for P2, ie write P2 as P(B|abz) it means a is a causal influence on B. However, in probability theory P(B|abz) only means B and a are logically dependent not that a is a causal factor in B.

Further in his original paper, in going from (1) to (2) he states that

The vital assuption [2] is that the result B for particle 2 does not depend on the setting a, of the magnet for particle 1, nor A on b

In reading this statement, you may think it is reasonable since Bell's equation (1) is in functional notation not probabilty notation. It is reasonable to say the result at station 1 is a function of only the hidden variable and settings at station 1. However, this does not translate to a generic statement about logical dependence in calculating probabilities, since the above statement will also eliminate all logical dependence, as illustrated by my previous example.

Finally still within his original paper, third page (page 405) just before section IV, Bell says

Thirdly, and finally, there is no difficulty in reproducing the QM correlation (3) if the results A and B in (2) are allowed to depend on b and a respectively as well as on a and b. ...

... However, for given values of hidden variables, the results of measurement with one magnet now depend on the setting of the distant magnet, which is just what we would wish to avoid.

Either Bell intends to avoid all dependence, including logical dependence, or he is completely unaware that there is any difference between logical dependence and causal dependence since he lumps them up as causal dependence.

That is why I say Bell's so called "locality condition" is really a "no-logical dependence condition". In the example I gave above, in calculating P(AB|abz), there is logical dependence between Alice's outcome and the switch pressed by Bob, mandated by the use of the chain rule, even though according to the situation as described, there is clearly no causal link between the two.

 

[ThomasT]

P(G|azbG') = cos^2(a-z)cos^2(b-z), P(G|HazbG') = ?

What are you illustrating here? It seems to me that the addition of H (the EPR-Bell experiment condition) then conditions {cos^2(a-z)cos^2(b-z)} to only apply to certain settings (ie., any values of z, but not all values of a and b).

How can the joint probability, cos^2 (a-b), be gotten to from the product of the individual probabilities, which involves all values of a and b, but, apparently, no values of z.

 

[billschnieder]

But the way you defined z was not sufficient to give any reason for Bell or any of his advocates to expect that the equation P(AB|abs, bbs, z) = P(A|abs, z) P(B|bbs, z)

On the contrary, I defined a probability space with all it's possibilities and went on to calculate probabilities within that space according to the chain rule. Easily verifiable, do you deny that my full universe is consistent with z? No. Do you claim that the calculated results are inconsistent with probability theory? No. So what is your issue. You apparently will prefer a situation in which the probability space was ill defined as the one you are trying to force in, the strawman is on your side.

And given your argument that the response to pushing the switches depended only on a simple internal mechanism in each ball which either activated the same-color light or the opposite-color light, it is indeed possible to define such a z that guarantees the equation will hold, as I showed.

I do not need to guarantee that it will hold, it is already expressed directly in my probability space where z is true. The generation of the full universe takes into account the fact that the results at Alice is determined only by the inner working of the ball, and the switch selected by Alice and same for Bob. It is silly to try to add extra conditions to "make sure" it is so. However, this does not translate to a condition that in calculating the joint conditional probabilities of the outcomes at both stations, logical independence must be dropped.

No, it's "what is the probability of Bob getting a blue light given that he pushed the blue switch, and given that his ball contains a mechanism that causes the ball to light up blue if the blue switch is pressed and red if the red switch is pressed, while Alice's ball contains a mechanism that causes the ball to light up red if the blue switch is pressed and blue if the red switch is pressed".

P(It will rain tomorrow| tomorrow is Friday, it rains every Friday) is not any different from calculating
P(it will rain tomorrow|it will rain tomorrow) This is deduction and you do not need any probabilities for this. Why would somebody attempt to use probability theory to answer such a question unless they were trying to pull a fast one. No doubt you always get 1 or 0.

Oh By the way, in my example, when they press the same switch their results are also always perfectly anti-correlated. Yet I did not use a ridiculous probability space as the one you are trying to push. I'd rather not go down another scratch card rabbit trail.

Nothing vague about it, in a local realist universe. Do you agree that in a local realist universe, the laws of physics associate some well-defined set of basic physical facts with each point in spacetime? Do you agree that for some event A, the set of points in spacetime which are in the past light cone of A at some specific time t is also completely well-defined and unambiguous?

While it makes sense to calculate the probability of an event at a space-time point given a specific set of well defined physical facts, I do not agree that it makes sense to calculate the probability of an event at a given space-time point conditioned on the vague concept of all possible values of all possible physical facts that could be realized at that position. The former clearly defines a probability space, the latter does not.

0 and 1 are allowable probabilities, surely you don't disagree? Would you like some examples appearing in textbooks where the probability of something is 0 or 1?

Since you are pushing a definition of hidden variables that includes all possible information in the past light cone of an event, I asked you for an example in which you knew everything in the past light cone and yet still obtained a probability other than 1 or 0. Did you find it yet?

No, this is the strawman and/or misunderstanding. Bell's equation is not meant to apply to any arbitrary definition of the extra variable (z in your equation), it's based on the idea that in a local realist universe it's possible to define the variable in such a way as to make the equation hold. And this was indeed true in your example when I defined z to give the information about which mechanism was in Alice's ball and which was in Bob's.

If you redefined z, you are no longer talking about my example. z was completely and sufficiently defined in my example.

It means the probability for any single value z of the random variable Z.

Again, z is not variable, and is not random either. It does not have "values". z is a collection of all elements of reality that together with the local settings at Alice, produce the results (similarly for Bob).

I would have to look a bit for a formal statement of the rule, but first tell me, do you disagree that if each trial in the sample space has some specific value of the random variable Z drawn from the set {z₁, z₂, ... z_n}, then it must be true that P(A) = (sum over all values of i from 1 to n) P(A|z_i)*P(z_i) ? If you don't disagree with that, then do you disagree that for some other fact b which is true on some trials but false on others, it would be true that P(A|b) = (sum over all values of i from 1 to n) P(A|b, z_i)*P(z_i) ?

But your z did not explicitly detail which mechanism was in the ball given to Alice and which was in the ball given to Bob.

Again, z is not a variable and does not have "values". My full universe of possibilities exhaustively considers all possible realizations so it doesn't make sense to require me to further restrict the space by stating which of the two balls was sent to Alice and which to Bob. If you want to do that, you may as well also require that I further restrict the space to exactly the result obtained by Alice and Bob, because that is exactly what your tactic results in. Hence your 1 or 0 probability results.

If each of these supposed to be a distinct z, so z₁ would represent possibility 1, z₂ would represent possibility 2, then of course this would allow us to infer which mechanism was in Alice's ball on each trial. But from your later statements it seems you didn't mean z to have distinct values, but just to represent the knowledge that one of these possibilities will hold on each trial, without knowing which one will hold on any given trial.

Again, z is not variable and does not have "values".

This comment reveals a complete lack of understanding of Bell's paper. His λ was of course supposed to represent a variable that could take different values on different trials; if that wasn't true, equation (2) in which he integrates over all possible values of λ would make no sense at all!

You can only integrate or add probabilities defined in the same probability space. If as you claim, Bell is adding probabilities defined for different contexts, it is no surprise that his equations do not work for contextual elements of reality.

 

[zonde]

To use P(A) [as you do] and to then specify some conditions B [which you do] is to invoke (P(A|B).

Your B = These following expressions do not describe EPR-Bell experiments but simple experiments with single photon beam and single analyzer and are covered by Malus law. wow.

Does your wiki reference give anywhere a value for P(A) without a B?

I would like to see that.

Thank you again,

Jenni

The way you describe B it is context. But in P(A|B) B signifies condition not context. Condition is some event B that signifies when we should count particular event A and when not.
We should describe context of course but why do you want to include it in mathematical formalism in some arbitrary way? I see no need to put common formalism on it's head.

 

[JesseM]

But the way you defined z was not sufficient to give any reason for Bell or any of his advocates to expect that the equation P(AB|abs, bbs, z) = P(A|abs, z) P(B|bbs, z)

On the contrary, I defined a probability space with all it's possibilities and went on to calculate probabilities within that space according to the chain rule. Easily verifiable, do you deny that my full universe is consistent with z? No. Do you claim that the calculated results are inconsistent with probability theory? No. So what is your issue.

My issue is pretty simple, "Bell or any of his advocates" would not interpret the λ in equation (2) in his paper to be a mere statement of the "probability space" (by which I guess you actually mean the sample space edit: sorry, I see probability space is just a sample space plus a probability distribution on events in this space) of measurement settings, anyone reading the paper with a modicum of understanding of the physical ideas involved would realize λ refers to the hidden variables that are imagined to influence the measured outcomes.

I agree that given your definition of z, your calculations are correct and it is true that P(AB|abs, bbs, z) is not equal to P(A|abs, z)*P(B|bbs, z). The point is that your definition of z has nothing in common with Bell's implicit definition of λ (which was not defined more explicitly in his very short paper, but you can find more detailed derivations of Bell's theorem which would give more careful definitions of the equivalent symbol), so it has no relevance to invalidating the claim that in Bell's own notation, P(AB|a, b, λ) = P(A|a, λ)*P(B|b, λ). Is this really so difficult a point to grasp, that different equations involving probabilities can define their terms differently so that an equation that was true for one set of definitions of the symbols would not be true for a different set of definitions of the symbols? And that therefore you can't prove a given equation "wrong" by picking a completely different set of definitions and showing that the same equation doesn't hold for your definitions? Please answer yes or no whether you understand and agree with this point.

If you do agree with the point in general, are you claiming that the criticism doesn't apply to you because you think Bell was defining his λ in a way analogous to your own definition of z, as nothing but a statement of all possible combinations of detector settings and outcomes?

You apparently will prefer a situation in which the probability space was ill defined

The probability space was perfectly well defined in the various examples I gave, like the modification of your example where we introduced the symbols aObN and aNbO (analogous to different values of λ in Bell's equation) or like the example involving scratch lotto cards from post 18 on the other thread. In Bell's paper we don't know the set of possible values that λ can take, but that's OK, as long as we accept the basic supposition that any given value of λ will completely predetermine the result for a given measurement setting, his equations would hold for all possible sample spaces that λ could range over, and this is perfectly rigorous. Similarly, there are plenty of general equations in probability theory that would hold regardless of how you define your sample space, like P(A and B) = P(A|B)*P(B).

And given your argument that the response to pushing the switches depended only on a simple internal mechanism in each ball which either activated the same-color light or the opposite-color light, it is indeed possible to define such a z that guarantees the equation will hold, as I showed.

I do not need to guarantee that it will hold, it is already expressed directly in my probability space where z is true.

I said "guarantees the equation will hold"--I was talking about the equation P(AB|abs, bbs, z) = P(A|abs, z)*P(B|bbs, z), which does not actually hold given your definition of z, as you yourself showed. My point was that this doesn't somehow prove Bell wrong, because the derivation of the Bell inequality only requires that it is possible to define the equivalent variable in such a way that the equation would always hold, in any situation obeying local realist laws. Your example involving the two balls with internal mechanisms was one compatible with local realism, and I showed that in your example it is indeed possible to define z in such a way that P(AB|abs, bbs, z) = P(A|abs, z)*P(B|bbs, z) is valid, even if the equation is not valid given your definition of z.

P(It will rain tomorrow| tomorrow is Friday, it rains every Friday) is not any different from calculating
P(it will rain tomorrow|it will rain tomorrow) This is deduction and you do not need any probabilities for this. Why would somebody attempt to use probability theory to answer such a question unless they were trying to pull a fast one. No doubt you always get 1 or 0.

And in fact, if you look at equation (2) in Bell's own paper, since he is assuming that the outcome is completely determined by the measurement setting a and the hidden variables λ, he doesn't write it as a probability at all, he just writes A(a, λ). Nevertheless it is technically valid in probability theory to write something like P(A|a, λ) even if the probability is 1 or 0, and I wrote it that way since the original post was asking about whether the equation P(AB|H) = P(A|H)P(B|H) was valid, based apparently on the same equation in your original thread, even though this equation does not appear in Bell's paper and in the paper itself Bell just writes A(a, λ) and B(b, λ) (because his assumption is that the measurement result is completely determined by the measurement setting and the hidden variables, an assumption that a little argument can show must be true in a local realist universe if the settings are chosen at random at a spacelike separation and they always get identical results when they choose the same setting). If you want to blame someone for introducing probability notation in a case that's supposed to have determinate outcomes, blame yourself! But as I said, there is nothing mathematically invalid about such notation even if it may be seen as a bit weird or unnecessary aesthetically, and I think it can actually be useful if we don't want to assume anything at the outset about whether the hidden variables actually completely determine the outcome for a given setting or if they just influence the outcome in a probabilistic way.

While it makes sense to calculate the probability of an event at a space-time point given a specific set of well defined physical facts, I do not agree that it makes sense to calculate the probability of an event at a given space-time point conditioned on the vague concept of all possible values of all possible physical facts that could be realized at that position.

Why not? In any well-defined local realist fundamental theory, the complete set of possible physical facts that obtain at a given point in spacetime should be well-defined, no? If your fundamental theory involves M different fields and N different particles and nothing else, then by specifying the value of all M fields at a given point along with which (if any) of the N particles occupies that point, then you have specified every possible physical fact at that spacetime point. As long as there is some fundamental theory of physics and it is a local realist one, then the theory itself gives a precise definition of the sample space of distinct physical possibilities that can obtain at any given point in spacetime--do you disagree?

Since you are pushing a definition of hidden variables that includes all possible information in the past light cone of an event, I asked you for an example in which you knew everything in the past light cone and yet still obtained a probability other than 1 or 0. Did you find it yet?

In a deterministic theory there would be no such example, but it's conceivable that the fundamental theory of physics could be probabilistic at a fundamental level, yet still be local realist. In this case you could break the theory down into some deterministic equation that gives the most accurate possible probabilities for a given event at a point in spacetime given the past light cone of that point, plus a random "seed" number for the event whose value is uncorrelated with anything else in the universe (it's truly 'random') which can be combined with the probabilities given by the deterministic equation to decide what actually happened at that point in spacetime. See my comment in post 63.

If you redefined z, you are no longer talking about my example. z was completely and sufficiently defined in my example.

As I said I don't disagree with your example, but since the way you define z is not analogous to the way Bell defines λ in his paper, the fact that P(AB|abs, bbs, z) is not equal to P(A|abs, z)*P(B|bbs, z) given your definitions does not show there is anything wrong with his equation (2). And if we redefine your z so it is analogous to how he defined λ, then that would make it so P(AB|abs, bbs, z) is equal to P(A|abs, z)*P(B|bbs, z).

Again, z is not variable, and is not random either. It does not have "values".

But λ in Bell's paper was defined as a variable that can take different values on different trials, do you disagree?

You can only integrate or add probabilities defined in the same probability space. If as you claim, Bell is adding probabilities defined for different contexts, it is no surprise that his equations do not work for contextual elements of reality.

The experimental context is the same every time, but the value of different variables can vary from one trial to another--and he does assume the probability distribution p(λ), which defines the probability of getting a given value of λ on a randomly-selected trial, is known. The idea that a variable can take different values on different trials is the very meaning of a random variable in probability theory, you're not really saying you have a problem with this, are you?

 

[JesseM]

Say H_a represents the full set of information about all local variables in the past light cone of measurement A at some time t prior to A, and H_b represents the full set of information about all local variables in the past light cone of B at the same time t which is also prior to B, with t chosen so that it happens after the last moment the two light cones overlap.

In this case, if we simply define H as the sum of all the information contained in both H_a and H_b, then the equation is equivalent to this:

F(AB|abH_aH_b) = F(A|aH_aH_b)*F(B|bH_aH_b)

Do you still have a problem with this equation? Some of those terms may represent unecessary information--for example, I could give you an argument that F(A|aH_aH_b) = F(A|aH_a), i.e. H_b would give no additional information about the probability/frequency of A given knowledge of a and H_a--but that shouldn't affect their validity.

Ok, first, can we switch to Bell's notation? Let's say that λ_a and λ_b represent the polarization vectors of sinusoidal disturbances incident on the polarizer settings a and b.

But under Bell's notation λ is supposed to represent whatever variables (hidden or otherwise) determine the particle's response to a given measurement setting. Are you suggesting treating the "polarization vectors" as this type of variable? If so, how does it work? Would each particle have the same polarization angle, and then the probability a particle gives spin-up would be determined by the cosine of the angle between the detector and the polarization angle? But in this case you'd have a situation where the particles would not be guaranteed to give identical results in instances where both detectors were set to the same angle--if each particle had a polarization angle of 30 and the detector was set to 90 degrees, then the probability each particle would give spin-up would be cosine(90 - 30) = 0.5, which means there's also an 0.5 chance each particle gives spin-down, so there's a (0.5)*(0.5)=0.25 chance the first particle gives spin-up but the second particle gives spin-down, and likewise an 0.25 chance the first particle gives spin-down but the second particle gives spin-up. So if this is the type of rule you were thinking of, it won't satisfy the quantum condition that identical detector settings = identical measured outcomes, as Bell was assuming in his original paper (though later Bell inequalities did not require such perfect correlations). And if this isn't the type of rule you were thinking of, please explain in quantitative detail how the polarization vector is supposed to combine with the angle of the detector setting to determine whether the outcome of a given measurement is spin-up or spin-down.

Lets also say that λ_a = λ_b, so that λ represents a single, continuous polarization vector extending between a and b, following Bell. So, we write, P(AB|Habλ) = P(A|Haλ_a) P(B|Hbλ_b), where H denotes a local common cause of λ, λ_a, and λ_b, and which says that, given H, a, b, and λ determining the union of the data sets A and B, then the data sets A and B are statistically independent. Is this correct? If not, what does it say?

Yes, given your definitions that's what the equation would be saying, but you haven't really given any reasoning that explains why you think that equation would hold (i.e. why A and B should be statistically independent) given your own definitions. My own way of defining λ and/or H in terms of past light cones was specifically intended to prove that a similar equation would have to hold in a universe obeying local realist laws, so if you doubt the validity of the equation you need to address the specific definitions I gave (and perhaps also address the argument about what these definitions imply from post 63)

 

[JesseM]

Dear Jesse,

Thank you for this detail. You provide much to study. Maybe some is beyond me.

Certainly there are some distracting generalities --

"It's not a supposition, any physicist will agree that under a local realist theory, as long as enough information about local variables (hidden or otherwise) in the past light cones of G and G' is contained in λ, your equation (3) should be satisfied."

For the moment -- to minimize distractions and focus on Bell's mathematics --

I have shown the realistic locality assumption that reduces (1) to (2). With Einstein and Bell as supporters. [Edit: I also give the assumptions to pass from (2) to (4) and from (2) to (5).]

1. Could you show me the assumption that you use to reduce (2) to (3) please? With some support.

OK, in post #59 you wrote (2) and (3) like so:

Suppose we are given candidate theory X for which we can calculate the various probabilities associated with the possible outcomes of Alice and Bob. Let the possible outcomes for Alice be A = {G, R}, for Bob be B = {G', R'}. Upon calculating with X we find for some λ, A, B, a, and b that

P(G|X,a,λ) P(G'|X,b,λ) ≠ P(G,G'|X,a,b,λ) .

Then X [in your view] does NOT satisfy "Bell Locality".

For example [in your view], this is the case for QM:

P(G|X,a,λ) P(G'|X,b,λ) = ½ ∙ ½ = ¼ ≠ P(G,G'|X,a,b,λ) .
__________________________

MEANING of "Bell Locality" VIOLATION and the error in X.

If G and G' are correlated, probability theory teaches [equation (1)] that

(1) P(G,G'|X,a,b,λ) = P(G|X,a,b,λ).P(G'|X,a,b,λ,G) =

(2) P(G|X,a,λ).P(G'|X,a,b,λ,G);

(2) following from (1) because (with Einstein, Bell, and many others) we agree that setting b can have no relevance for outcome G. A realistic locality condition.

But Bell goes further. Bell supposes [Bertlmann's Socks, page 13] that

(3) P(G'|X,a,b,λ,G) = P(G'|X,b,λ),

a result [you say] known as Bell locality.

Assume that there is a space-like separation between the point where Alice makes her measurement (which yields either G or R) and the point where Bob makes his measurement (which yields either G' or R'), meaning that neither measurement lies in the past or future light cone of the other. My general argument is that in a universe with local realist laws, if λ includes the complete information about all local physical facts in the past light cone of a measurement B, then whatever your probability estimate is that B will yield result G' given knowledge of λ, this probability estimate should not be altered by information about some other event G which lies neither in the past or future light cone of the measurement B. And to see why this should be true, just consider the two possible cases I brought up in post 63, the first dealing with the case of deterministic local realist laws and the second dealing with the case of probabilistic local realist laws. In the first case, λ will perfectly determine whether or not G' happens so the probability will be 1 or 0, and obviously if you already know the probability of G' is 1 or 0 then any further information about G won't change this. In the second case, λ will give the "most precise possible probability" that I discussed in post 63, with the outcome of every event being determined by a combination of this "most precise possible probability" and a random "seed number", the latter being truly random and uncorrelated with anything else in the universe (outside of its future light cone which may contain records of the outcome of the event determined by the seed number). So if you already know the most precise possible probability of G' that can be obtained without knowing the value of the random seed number, and the value of the random seed number is uncorrelated with G, then learning G also will not influence your estimate of the probability of G'. So, regardless of whether the local realistic theory X is fundamentally deterministic or fundamentally probabilistic, it should be true that P(G'|X,a,b,λ,G) = P(G'|X,b,λ)

2. With your assumption, would all probabilities be zero or one only?

Only if the local realist theory is a deterministic one, if it's fundamentally probabilistic (second case discussed in post #63) then not necessarily.

 

[ThomasT]

Yes, given your definitions that's what the equation would be saying, but you haven't really given any reasoning that explains why you think that equation would hold (i.e. why A and B should be statistically independent) given your own definitions.

Given those definitions, does that form correctly describe the experimental situation?

 

[JesseM]

Given those definitions, does that form correctly describe the experimental situation?

Are you asking whether your equations are correct as a theoretical description of the experimental situation (which can include variables that are hidden from actual measurement), or are you asking if the equations can be verified to hold empirically? Your equations involved the terms λ_a and λ_b which were supposed to represent "polarization vectors", would there be any way to empirically measure the value of these variables on each trial?

 

[ThomasT]

Are you asking whether your equations are correct as a theoretical description of the experimental situation (which can include variables that are hidden from actual measurement), or are you asking if the equations can be verified to hold empirically?

If the form of the equation doesn't hold empirically (ie., if it's at odds with the experimental situation), then it can't be taken as a correct theoretical description of the experimental situation.

If you agree that the equation is analogous to Bell's (2), then, since we know that it doesn't hold empirically, then it can't be taken as a correct theretical description of the experimental situation. What possibilities does this suggest? (Note: we're going to be walking through my line of thinking on this, so that if there is something wrong with the way I'm thinking about this, then it will become clear to me.)

Your equations involved the terms λ_a and λ_b which were supposed to represent "polarization vectors", would there be any way to empirically measure the value of these variables on each trial?

No.

Yes, given your definitions that's what the equation would be saying, but you haven't really given any reasoning that explains why you think that equation would hold (i.e. why A and B should be statistically independent) given your own definitions.

The equation was presented as an analogy to Bell's (2), with the definitions of the notations therein being compatible with the definitions of the notations in Bell's (2). Is it, and, are they?

My own way of defining λ and/or H in terms of past light cones was specifically intended to prove that a similar equation would have to hold in a universe obeying local realist laws, so if you doubt the validity of the equation you need to address the specific definitions I gave (and perhaps also address the argument about what these definitions imply from post 63).

The equation, if it's analogous to the form of Bell's (2), has been demonstrated, experimentally, to be invalid. We're investigating why it's invalid. I'm not sure that such a form would have to hold in a universe obeying local realistic laws. In fact, I think that such a form would be invalid in a universe obeying local realistic laws if the form misrepresents the experimental situation.

It will, I'm thinking, come down to this consideration:

Are data sets that are logically and statistically dependent necessarily causally dependent? In other words, does a statistical dependence between spacelike separated events imply a causal connection between those events? If the answer to that question is no, then violation of Bell inequalities doesn't imply anything about nature.

Now, since I'm not really sure if you think that violation of BIs implies anything about nature, then, if you think that it doesn't, then we agree. On the other hand, if you think that violation of BIs does imply something about nature (other than that BIs are violated due to an inadeqate representation of the experimental situation), then it will be necessary to resume a detailed analysis (which just might, as I've said, change the way I'm currently thinking about this).

In connection with the above, consider the following statement by DrChinese (which seems to characterize much of the popular thinking regarding Bell's theorem, Bell tests, and violation of BIs:

You shouldn't be able to have this level of correlation if locality and realism apply.

Well, why not? If you take, say, an optical Bell setup where there is a source emitting counter-propagating optical disturbances during the same atomic transition, two polarizers, and two photon detectors, then why wouldn't the coincidence rate be expected to be, in the ideal, cos² (a-b)?

If you can answer why the observed result, approximating cos² (a-b), wouldn't be expected in a universe obeying local realistic laws, then this might help me to understand why violations of BIs are interpreted, by some, to mean that our universe doesn't obey local realistic laws.

And, by the way, also recalling some discussions we've had in the relativity forum, I'm very impressed by and admire your ability to analyze this stuff and make such long and detailed posts. I'm not so gifted. So, I'm counting on you to either confirm that I'm thinking along the right lines, or correct my incorrect thinking in a way that I can understand. No small task. But you seem to be up to it. And besides, it's you're job here at PF, right?

 

[DrChinese]

Well, why not? If you take, say, an optical Bell setup where there is a source emitting counter-propagating optical disturbances during the same atomic transition, two polarizers, and two photon detectors, then why wouldn't the coincidence rate be expected to be, in the ideal, cos² (a-b)?

If you can answer why the observed result, approximating cos² (a-b), wouldn't be expected in a universe obeying local realistic laws, then this might help me to understand why violations of BIs are interpreted, by some, to mean that our universe doesn't obey local realistic laws.

The reason why I say that "You shouldn't be able to have this level of correlation if locality and realism apply" is because you end up with a different formula using those assumptions. Generally, that being the product state result .25+.5*(cos^2(theta)) when Malus is applied rather than the entangled result cos^2(theta). The product state result being one which follows Bell (2) which is a generic statement.

Of course: if you use something other than Malus then you get different results and potentially, they could be closer to experiment (or farther away).

On the other hand: if you accept Malus - combined with the assumption that there is a specific but unknown polarization for entangled photons - then probably you would conclude that Bell (2) is false.

 

[JesseM]

If the form of the equation doesn't hold empirically (ie., if it's at odds with the experimental situation), then it can't be taken as a correct theoretical description of the experimental situation.

If you agree that the equation is analogous to Bell's (2), then, since we know that it doesn't hold empirically, then it can't be taken as a correct theretical description of the experimental situation. What possibilities does this suggest? (Note: we're going to be walking through my line of thinking on this, so that if there is something wrong with the way I'm thinking about this, then it will become clear to me.)

If λ_a and λ_b are not empirically measurable, then we cannot directly measure the probabilities in P(AB|Habλ) = P(A|Haλ_a) P(B|Hbλ_b), but as Bell did we can use this equation to derive other equations dealing only with P(AB|ab) for different values of A,B,a,b, and if these equations don't hold empirically that shows the original equation doesn't hold empirically. That can be taken as a falsification of the original assumption that there exist hidden "polarization vectors" associated with each particle such that if you knew the value of the polarization vector associated with one particle, then any further information about the behavior of the other particle would not help you to refine your estimate of the probability that your particle will behave a certain way.

The equation was presented as an analogy to Bell's (2), with the definitions of the notations therein being compatible with the definitions of the notations in Bell's (2). Is it, and, are they?

Assuming the polarization vectors are local variables (associated solely with the particle's current location at any given time, say), then yes, your equation would just be a special case of the general class of local hidden variable theories that Bell's equation (2) was meant to cover.

The equation, if it's analogous to the form of Bell's (2), has been demonstrated, experimentally, to be invalid. We're investigating why it's invalid. I'm not sure that such a form would have to hold in a universe obeying local realistic laws.

But if you're "not sure", it would help if you would directly address my point about past light cones in post 63, which is intended to show why such a form would necessarily hold in any universe obeying local realistic laws.

In fact, I think that such a form would be invalid in a universe obeying local realistic laws if the form misrepresents the experimental situation.

It will, I'm thinking, come down to this consideration:

Are data sets that are logically and statistically dependent necessarily causally dependent? In other words, does a statistical dependence between spacelike separated events imply a causal connection between those events? If the answer to that question is no, then violation of Bell inequalities doesn't imply anything about nature.

Again, the whole point is that there can be a dependence in the marginal probabilities for spacelike separated events, but in a local realist universe you can always find a set of information λ about the past light cones of these events such that there is no statistical dependence between the events when conditioned on λ. In other words, if A and B are spacelike-separated events, P(A|B) may be different than P(A), but if λ represents the state of every local variable in the past light cone of A, then it must be true that P(A|λ,B) is equal to P(A|λ).

As a simple example, if I periodically send red or green numbered cards to my friends Alice and Bob, and I always arrange it so Alice's card #N is the opposite color of Bob's card #N, then if Alice and Bob lie at equal distances in opposite directions from me and I send the cards simultaneously at the same speed, there should be a spacelike separation between the event of Alice receiving her card #N and the event of Bob receiving his card #N. So here there is a statistical dependence between the spacelike-separated events of Alice seeing the color of her card #N and Bob seeing the color of his card #N; if you don't know what color Alice's card #3 was, but then you are told that Bob's card #3 was green, then you now know Alice's card #3 must have been red with probability 1. On the other hand, if you know everything about what happened in the past light cone of Alice seeing her card #3, which would include the event of me sending out a red card #3 towards Alice and a green card #3 towards Bob, then you can use that to predict with probability 1 that Alice saw a red card #3, and learning that Bob saw a green card #3 won't change that.

Now, since I'm not really sure if you think that violation of BIs implies anything about nature, then, if you think that it doesn't, then we agree. On the other hand, if you think that violation of BIs does imply something about nature (other than that BIs are violated due to an inadeqate representation of the experimental situation), then it will be necessary to resume a detailed analysis (which just might, as I've said, change the way I'm currently thinking about this).

Yes, I think that a violation of BIs shows that no local hidden variables theory can account for the behavior of entangled particles.

If you take, say, an optical Bell setup where there is a source emitting counter-propagating optical disturbances during the same atomic transition, two polarizers, and two photon detectors, then why wouldn't the coincidence rate be expected to be, in the ideal, cos² (a-b)?

"Expected to be" in real life, or "expected to be" in some local hidden variables theory? If the latter, can you explain the nature of the local hidden variables, and how they interact with the angle of the polarizer to give the probabilities of different outcomes? For example, maybe you're suggesting that each particle has an identical hidden variable giving the angle v of its polarization vector, and that to determine the probability a particle is detected we just take the angle of the polarizer it goes through (a or b) and the angle of the particle's polarization vector (which has the same value v for both particles) and calculate cos² of the angle between them (i.e. cos²(a-v) for the first particle going through polarizer a, and cos²(b-v) for the second particle going through polarizer b). If so, this would not give a coincidence rate of cos²(a-b), as you can see if you set a=b while making v different from a and b; in that case cos²(a-v)=cos²(b-v)=some number between 0 and 1, so there is some nonzero probability the two particles will give opposite results, despite the fact that cos²(a-b)=1 (this is basically the same argument I was making in the first paragraph of post 81, except I forgot to take cosine squared rather than just the cosine of the angles).

If you can answer why the observed result, approximating cos² (a-b), wouldn't be expected in a universe obeying local realistic laws

Because the cos² (a-b) coincidence count can be shown to violate various Bell inequalities if each detector can be set to three possible angles: 0 degrees, 60 degrees, and 120 degrees. For example, one Bell inequality says that if the experimenters are choosing randomly between three possible settings, then if they always get the same result when they pick the same setting, on trials where they happen to pick different settings they must get the same result at least 1/3 of the time. But you can see that with these angles, if they picked different angles the probability of identical results would always be 0.25 (since both cos²(120) and cos²(60) equal 0.25), violating the Bell inequality.

For a simple conceptual picture of where this particular Bell inequality comes from, consider my scratch lotto card analogy in post #2 of this thread (just the text in the quote box, you can ignore the later derivation of the CHSH inequality).

And, by the way, also recalling some discussions we've had in the relativity forum, I'm very impressed by and admire your ability to analyze this stuff and make such long and detailed posts. I'm not so gifted. So, I'm counting on you to either confirm that I'm thinking along the right lines, or correct my incorrect thinking in a way that I can understand. No small task. But you seem to be up to it. And besides, it's you're job here at PF, right?

Thanks! And yes, if you think it would be possible to have a local hidden variables theory which violated the Bell inequalities (or which didn't obey equation 2 in his original paper), I will do my best to correct that! ;)