QM Assumptions Regarding Entanglement Properties

[Johan0001]

Hi This might be off topic , but I have a question which relates to the "Reality" aspect, after reading this post.
Hopefully someone shed some light on the topic.

1. Regarding the correlation between Alice and Bobs results , AFTER comparing them.
2. If the order of making the observations does not matter i.e. Alice could have made the measurements first OR even light years after BOB did ,
in another light cone for example.
3. Whenever the results are brought together , and compared in the DISTANT future, a correlation IS FOUND.My question:

Can one not deduce that this correlation existed between the entangled particles due to their common history.
Whether measurements were taken or not.
Thus making it a form of realism.

If this is NOT the case, then at what point in Space time , does the correlation manifest , since the sequence of MEASUREMENT events play no part in the results, as per 2 above??

And further , what makes THAT particular point in Space time different from any other, in the sense that, all of a sudden a correlation has been realized by comparing them?

Johan

 

[jerromyjon]

Can one not deduce that this correlation existed between the entangled particles due to their common history.
Whether measurements were taken or not.
Thus making it a form of realism.

That's the part that has been proven. A simple correlation between particles (they stay in sync.) is not possible. The results of experiment (as predicted by quantum theory) cannot be reproduced if the particles each have spacelike separated values. The unrealistic part is that information has to be shared between the results to make the probabilities work, no matter how separated the measurements are made, an instant correlation exists that no local variables can account for.

 

[zonde]

Can one not deduce that this correlation existed between the entangled particles due to their common history.
Whether measurements were taken or not.

I like this simple "proof" as an explanation why this can't be the case.

If this is NOT the case, then at what point in Space time , does the correlation manifest , since the sequence of MEASUREMENT events play no part in the results, as per 2 above??

There is no answer to this question.
If you accept non-locality then you have to assume there is preferred reference frame in which ordering of measurements matters.
Btw ordering of measurements does not affect statistical properties of measurements. About individual measurements the theory is silent and there are no experimental means to test this (in-)dependence on ordering at the level of individual measurements.

 

[DrChinese]

EPR goes on after that sentence and explain their position what it means to insist on that point: "On this point of view, since either one or the other, but not both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this."
So what is this un-reasonable definition of reality that could permit this? It seems to me that this is superdeterministic reality.
And we can step away from philosophical arguments about reality and could ask: what scientific model (of reality) could allow this? And it seems that there is none at all because superdeterministic models are not scientific.

This does not have anything to do with superdeterminism, which I agree does not meet any reasonable standard as a scientific theory.

EPR says: all elements of reality need not be simultaneously predictable to be simultaneously real. A very reasonable position to be sure, and one that they have as an explicit assumption. If (with certainty) you can predict A in advance, and you can predict (non-commuting) B in advance, and you can predict (non-commuting) C in advance, then A, B and C should be simultaneously real. But in QM, they are not - that would violate the Heisenberg Uncertainty Principle (HUP). EPR did not say that the A/B/C values were predetermined *prior to interactions, nor did they say that there was any kind of "conspiracy" to follow the predictions of QM (which is an essential element of superdeterministic concepts).

But Bell took their viewpoint and developed a mathematical argument around their assumption. Bell's separability condition (2) was

$$P(A,B|a,b,λ)=P(A|a,λ)P(B|b,λ)$$
Which also means:
$$P(A,C|a,c,λ)=P(A|a,λ)P(C|c,λ)$$
$$P(B,C|b,c,λ)=P(B|b,λ)P(C|c,λ)$$
This became the foundation of development of his inequality. We know that for certain A/B/C, the QM predicted values for the left side of these equations cannot be simultaneously true. Ergo you might readily conclude that EPR's premise - "all elements of reality need not be simultaneously predictable to be simultaneously real" - cannot hold. Of course that same premise is also diametrically in opposition to the usual interpretation of the HUP. Non-commuting A/B/C do not simultaneously have well-defined values.

 

[zonde]

This does not have anything to do with superdeterminism, which I agree does not meet any reasonable standard as a scientific theory.

EPR says: all elements of reality need not be simultaneously predictable to be simultaneously real. A very reasonable position to be sure, and one that they have as an explicit assumption. If (with certainty) you can predict A in advance, and you can predict (non-commuting) B in advance, and you can predict (non-commuting) C in advance, then A, B and C should be simultaneously real. But in QM, they are not - that would violate the Heisenberg Uncertainty Principle (HUP). EPR did not say that the A/B/C values were predetermined *prior to interactions, nor did they say that there was any kind of "conspiracy" to follow the predictions of QM (which is an essential element of superdeterministic concepts).

I mean this sentence: "This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way." I understood it as saying that reality of P and Q have to "know" in advance what measurement will be carried out on remote system.

Ok, I don't think that philosophical arguments are very good in discussion. I believe that the role of philosophy is to establish common ground for discussion and not much more.
So my position rather relies on conclusion that there are no scientific models that can describe entanglement correlation resulting from two local measurements. So if any theory that goes beyond QM has to be non-local, does it really matter if we can make certain statement about reality itself (assuming we can make such statements in principle).

 

[Johan0001]

#73

That's the part that has been proven. A simple correlation between particles (they stay in sync.) is not possible. The results of experiment (as predicted by quantum theory) cannot be reproduced if the particles each have spacelike separated values. The unrealistic part is that information has to be shared between the results to make the probabilities work, no matter how separated the measurements are made, an instant correlation exists that no local variables can account for.

This is the grey area for me??

If I interpret the paragraph above. It's equivalent to saying.

Some comparison that MAY be done in the FUTURE at some point in space time , will produce a correlation, that could not exist NOW.

Further, this FUTURE correlation is dependent on what I decide to set the angle of my measuring device RIGHT NOW .
Irrespective of whether my college has set His measuring angle before or after me.

But what will ultimately hold is that the angle between these measuring devices SET ( which were space like separated) in the past will produce a correlation in the future predicted by QM ??

If we assume entanglement exists between 2 particles (until it is broken, by measurement).
Why would the correlation not exist before measurement, since the particles had to interact one way or another to become entangled in the first place.

I can't even decide if what iv'e just said makes sense to me or not?? Am I misunderstanding entanglement?​

Regards
Johan

 

[DrChinese]

I mean this sentence: "This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way."

Yes, this is a description of Subjective Reality or Observer-dependent Reality (as it applies to entangled systems). Generally, the HUP (as it applies to entangled systems) is considered to be Observer-dependent. I can choose to measure Alice's P or Alice's Q. That has the apparent effect of casting Bob's into a state where his P or Q is now consistent with the result of the measurement on Alice. EPR thought that was crazy. Now that idea is demonstrated in experiments every day, but they had no way to even dream of it back then.

Please note the last phrase of the EPR sentence "which does not disturb the second system in any way". This is in direct contradiction to a phrase you agreed with above: "an entangled system of 2 spacelike separated particles does not consist of 2 independent particles". A measurement on one is like an identical measurement on the other. You and I agree on this, so naturally we disagree with EPR's assumption. Please note that I believe the following is equally true:

An entangled system of 2 space-time separated particles does not consist of 2 independent particles.

I.e. the separation can be in space, in time, or both. Particles that have never interacted can also be entangled. Non-local theories do not really provide a reasonable explanation for that. So you can see that simply rejecting locality alone does not quite answer the foundational questions we all have. In fact, I would say that a single particle can, in a sense, be said to occupy no space (pointlike) or all of space. So when we talk about what is "real" in the quantum world, there isn't the usual language to fully convey the picture reasonably. Perhaps a density matrix is the best we can do...

 

[jerromyjon]

Why would the correlation not exist before measurement, since the particles had to interact one way or another to become entangled in the first place

The correlation exists, but the results of observations that indicate the correlation can't have predetermined values. The results are always random for anyone measurement, could be spin up, could be spin down. It's entangled partner will always have the opposite result on the same axis. When you offset the angle between the measurement axes is when it becomes clear that the odds of match/no match don't add up like they do in a realistic classical sense.
This youtube video explains it quite thoroughly with three polarizing filters:

 

[DrChinese]

Why would the correlation not exist before measurement, since the particles had to interact one way or another to become entangled in the first place.

That's not accurate. Particles do not need to have interacted to be entangled. They do not need to have existed in any common light cone. And they don't even need to have both existed at any common time. For example:

https://arxiv.org/abs/0911.1314
https://arxiv.org/abs/1209.4191

These blow away any remaining conventional explanations for correlations. Please keep in mind that when these papers refer to non-local correlations, they are referring to the same quantum non-locality I mention in my previous post (#78). And they are not referring to Bohmian non-locality in particular. Any accepted quantum interpretation will feature quantum non-locality, even where c is respected.

 

[Johan0001]

great video jerromyjon

#79
The correlation exists, but the results of observations that indicate the correlation can't have predetermined values

Agreed fully

However

Could the values of the photons properties( hidden or not) , have changed ever so slightly(that we cannot measure this change as of yet) , when it came into contact with the respective filters , 1st then second or third in the case of 3 filters.
.
I.E. they are never predetermined values but evolving values as it passes/ or blocked by the filter.

So in essence if the angle between the filters is small ( say 1 degree) the resultant photon's property changes ever so slightly, and passes through
or not.

And when the angle is 90 degrees , the photons property changes so much , on contact, that it is not able to pass through.

So the emerging photon does not have the same property as the incident photon when it entered the system.

Regards
Johan

 

[morrobay]

The upshot is that the conditional joint probability can now (with this locality assumption) be written as $$P(A,B | a,b, \lambda ) = P(A | a, \lambda ) P(B | b, \lambda )$$The last piece is the 'realism' bit - this gets used later on in the derivation where an assumption is made in the math. This assumption is tantamount to saying that properties exist independently of measurement. This is given the fancy name of 'counterfactual definiteness'.

Im familiar with the locality assumption: P (ab|x,y,λ) = P(a|x,λ) P(b|y,λ)
Could you or anyone show the math for a realism/CFD assumption that is referred to above ?
Im only familiar with A(aλ) = - B(aλ) when both detectors are aligned with anti correlated outcomes.
While EPR presupposes spins along all angles simultaneously well defined.

 

[zonde]

Particles that have never interacted can also be entangled. Non-local theories do not really provide a reasonable explanation for that.

How you came to that conclusion?
I will make an attempt. Let's say we have measurement events A, B and C. A and B happens in the past of C (A and B are timelike separated from C). Now say experimenter has some freedom about measurement conditions of event C and say there is some choice that makes outcomes of event C inconsistent with possible combinations of outcomes A and B. However whatever choice experimenter makes he observes outcomes C that are consistent with events A and B. We could say that experimenter has no freedom of choice i.e. reality is superdeterministic. But it does not work. Experimenter can look at outcomes of A and B and make a choice that should give inconsistent outcomes of C. What will happen then?
And in actual experiments experimenter has no control over outcome of Bell state measurement (BSM) except the freedom to not perform BSM at all. And any combination of A and B outcomes is consistent with one of the possible outcomes of BSM. So A and B can non-locally influence outcome of BSM. Alternatively if C (BSM) happens first it can non-locally influence events A and B. And similarly for chain of events A->C->B.

 

[Adel Makram]

I made a graph representing local hidden variables (HV) and Bell`s inequality.

Each plan of HV is represented by a shade of gray color. This can be interpreted as the number of particles emitted from the source following the plan protocol or by the proportion of the particles following the protocol out of all particles (probability). Each plan has a predetermined spin value of U (up) or D (down) along any chosen direction.
Alice and Bob have the freedom to measure the spin along anyone of A, B or C angle. So we have 8 plans $(2^3)$. And each plan has 9 possible combinations of match between the Alice result and Bob result which forms the similarity matrix $3x3$. The form of match is either (O) opposite result or (S) similar result. Bell`s inequality arises naturally by visual looking at the graph.
The good thing about this graph is that each plan is not represented by just a single number but by a matrix of numbers. Classically, there is no difference in the values of the matrix cells with the same indices (same location in term of row and columns) between different plans. Because all values are predetermined before the measurement. This may be the graphical interpretation of the fancy word "counterfactual definiteness (CFD)". But according to uncertainty principle (UP), cells with the same indices (same location in the matrix) between different plans are not equal. For example, look at plan-3 and plan-4, classically the pink cells must be equal and represented by a single number. This is when Alice measures her particle (U) in A-direction and Bob measures his particle (U) in B-direction. It does not matter classically what value Alice would get had she measured her particle in C-direction (blue in plan-3) or Bob in C-direction (green in plan-4). In QM, however, UP will change the pink value in plan-3 if Alice measures her particle in C-direction or the pink value in plan-4 if Bob measures his particle in C-direction. And this is the reason why BI is violated.
So in order to explain the violation of BI, one must either stick to the QM prediction but with addition of the mysterious "non-locality" or to stick to the HV-plans as above but with addition of mysterious "retrocausality". The retrocausality here means the nature would change the value of corresponding cells in the matrices between different plans back in time after knowing which direction Alice or Bob will choose to measure their particles in the future.

 

[Dadface]

One of the main things I find confusing about this subject goes back to assumptions made in the EPR paper where Einstein and his co workers suggested that QM is "not complete" and that a complete description may exist. In their words:

"We believe, however, that such a theory is possible"

I think it reasonable to believe that a (hidden variable) theory may be possible but as far as I know there is no such theory at the present time. Perhaps a theory will be developed in the future and until and if that happens we have no knowledge of what that theory will be.

Will a future theory be mega complicated? Could it be a complete diversion from QM as it's understood at the present time? Perhaps there is something very simple that's been overlooked. So could it be that all that's needed is to give QM a bit of a tweak? The answer to all questions of this type is that we don't know the answers. We can say nothing concrete about a theory that does not exist. All we can do is speculate. And isn't speculation what Bell did?

I suppose the assumptions Bell made are considered to be plausible but can those assumptions be justified? And what exactly are those assumptions. That's what I'm still stuck on. And did Bell provide evidence to question the credibility of hidden variable theories, or certain types of hidden variable theories? If so how can you question the credibility of a theory when you have no knowledge of what that theory is?

 

[jerromyjon]

And did Bell provide evidence to question the credibility of hidden variable theories, or certain types of hidden variable theories?

It's quite conclusive that local hidden variables cannot explain the probabilities. Loophole free tests confirmed the facts, local realistic variables are ruled out. For example: https://www.nature.com/articles/nature15759

 

[zonde]

I think it reasonable to believe that a (hidden variable) theory may be possible but as far as I know there is no such theory at the present time.

There are interpretations of QM. Google for Bohmian mechanics. It has hidden variable for position.

 

[DrChinese]

How you came to that conclusion?
I will make an attempt. Let's say we have measurement events A, B and C. A and B happens in the past of C (A and B are timelike separated from C). Now say experimenter has some freedom about measurement conditions of event C and say there is some choice that makes outcomes of event C inconsistent with possible combinations of outcomes A and B. However whatever choice experimenter makes he observes outcomes C that are consistent with events A and B. We could say that experimenter has no freedom of choice i.e. reality is superdeterministic. But it does not work. Experimenter can look at outcomes of A and B and make a choice that should give inconsistent outcomes of C. What will happen then?
And in actual experiments experimenter has no control over outcome of Bell state measurement (BSM) except the freedom to not perform BSM at all. And any combination of A and B outcomes is consistent with one of the possible outcomes of BSM. So A and B can non-locally influence outcome of BSM. Alternatively if C (BSM) happens first it can non-locally influence events A and B. And similarly for chain of events A->C->B.

"Particles that have never interacted can also be entangled. Non-local theories do not really provide a reasonable explanation for that."

I don't follow your question/example. Are you discussing superdeterminism? Or Bohmian concepts? Those are completely different, but you seem to be mentioning them both in a single post.

Assuming you are talking about Bohmian ideas: Yes, it is true that particles that have never interacted and are spacelike separated "might" be explainable under a non-local theory. However, the obvious problem is: "why would particles that have never interacted be entangled?" If that were so, you expect everything to be entangled with everything. And they aren't! Only particles that interact under very specific conditions - conditions that strictly respect c - can become entangled.

 

[zonde]

However, the obvious problem is: "why would particles that have never interacted be entangled?" If that were so, you expect everything to be entangled with everything. And they aren't! Only particles that interact under very specific conditions - conditions that strictly respect c - can become entangled.

Ok, I now understand your objections.
But the problem first of all is why would any particles stay entangled. It seems silly to imagine that they exchange some sort of private keys when they interact to recognize each other later over some non-local network. And I don't think Bohmian mechanics have answer for that problem.
So my working hypothesis is that this is some sort of resonance. And resonance can make particles that have never interacted become entangled. At least that's how I see it. But surely it's not the answer given by any proposed interpretation. So your objections are valid, but on the other hand you can't base any conclusion on that objection as it is too soft.

 

[Dadface]

One of the main things I find confusing about this subject goes back to assumptions made in the EPR paper where Einstein and his co workers suggested that QM is "not complete" and that a complete description may exist. In their words:

"We believe, however, that such a theory is possible"

I think it reasonable to believe that a (hidden variable) theory may be possible but as far as I know there is no such theory at the present time. Perhaps a theory will be developed in the future and until and if that happens we have no knowledge of what that theory will be.

Will a future theory be mega complicated? Could it be a complete diversion from QM as it's understood at the present time? Perhaps there is something very simple that's been overlooked. So could it be that all that's needed is to give QM a bit of a tweak? The answer to all questions of this type is that we don't know the answers. We can say nothing concrete about a theory that does not exist. All we can do is speculate. And isn't speculation what Bell did?

I suppose the assumptions Bell made are considered to be plausible but can those assumptions be justified? And what exactly are those assumptions. That's what I'm still stuck on. And did Bell provide evidence to question the credibility of hidden variable theories, or certain types of hidden variable theories? If so how can you question the credibility of a theory when you have no knowledge of what that theory is?

It's quite conclusive that local hidden variables cannot explain the probabilities. Loophole free tests confirmed the facts, local realistic variables are ruled out. For example: https://www.nature.com/articles/nature15759

Thank you jerromyjon. I am familiar with what you are saying but it does not address the point I'm trying to put across. Let me try to express it differently:

At some time in the future a successful theory and by that I mean a theory which conforms to the observations, perhaps removes the "weirdness" that Einstein et al disliked and a theory which has all the other features that a successful should have, may be developed. To my knowledge such a theory is not known of at present.

Now nobody can prove that any future theory is incorrect without knowing exactly what the theory is. I know that may seem very obvious but it can seem that that is what Bell tried to prove. He assumed that there are certain features that any attempt at formulating a successful theory should have and then went on to prove that the theory can't be successful at all because it does not conform to the observations. And that's where I'm stuck. What exactly are the assumptions that Bell assumed the theory should have?

 

[DrChinese]

What exactly are the assumptions that Bell assumed a successful theory should have?

The assumptions are call locality and realism. In his paper, they are the separability condition - Bell's (2) is associated with "locality". And the condition that there is "realism" is expressed around Bell's (14) when he says "let there be another unit vector c" which is to say that there are other elements of reality (in addition to a and b) that cannot be simultaneously be measured, but could have been predicted with certainty a la EPR.

 

[Vanadium 50]

So my working hypothesis personal theory is that this is some sort of resonance.

Fixed that for you.

 

[Dadface]

The assumptions are call locality and realism. In his paper, they are the separability condition - Bell's (2) is associated with "locality". And the condition that there is "realism" is expressed around Bell's (14) when he says "let there be another unit vector c" which is to say that there are other elements of reality (in addition to a and b) that cannot be simultaneously be measured, but could have been predicted with certainty a la EPR.

Thank you DrChinese. EPR postulated that there may be "hidden variables". I can only imagine what these hidden variables may be but different possibilities come to mind for example subtle changes in known properties or properties yet to be discovered, or currently accepted assumptions, which are incorrect, about the nature properties. It seems to me that in the realism aspect of Bell's analysis the existence of all hidden variables, whatever they may be, are implied.

I don't see how Bell's work can take into account all possible hidden variables. In fact do we know exactly what any hidden variables are?

Did bell make any assumptions about the properties of the entangled objects. I refer to them as objects for want of a better word but the use of a label for example object or photons or electron implies that the entangled objects are real with real properties. And this pins down more closely where I'm stuck. What is assumed about the properties of the quantum objects referred to in Bells analysis?

Getting tired and I don't think I'm expressing myself very clearly here. Never mind and night night.

 

[DrChinese]

Did bell make any assumptions about the properties of the entangled objects. I refer to them as objects for want of a better word but the use of a label for example object or photons or electron implies that the entangled objects are real with real properties. And this pins down more closely where I'm stuck. What is assumed about the properties of the quantum objects referred to in Bells analysis?

No, not really any particular constraints regarding the nature or number of hidden variables or what form they might take. They could conceivably even be global properties (a hidden variable accessible to all particles). The only real requirement he attempted to enforce was: the setting of a measurement device here NOT influence the outcome of a measurement there. I.e. there could be no influences faster that light.

 

[DrChinese]

I don't see how Bell's work can take into account all possible hidden variables.

You can see the problem yourself if you simply attempt to hand manipulate outcomes. I call it the DrChinese challenge. If you pick certain settings, you can't even HAND PICK results that match QM. You only need about 8 or so examples to see the impossibility. Once you hit that wall, you quickly see why Bell rules out ALL local hidden variable theories. The only way to "win" the challenge is to "cheat". That is, you hand pick knowing what you plan to measure in advance.

 

[jerromyjon]

To my knowledge such a theory is not known of at present.

Sure, there are various interpretations of QM which describe everything observable, and the wavefunction which gives insight into the nature of the quantum world, but I know what you mean from previous posts about a new "more in-depth description", but that theory would still encompass that "spooky action at a distance"! The weirdness of quantum mechanics will always be strange compared to classical mechanics. QM does not follow the macroscopic laws of nature that everyone is used to. Spacelike and timelike separated events coincide with greater probability than is classically possible!

 

[Nugatory]

Now nobody can prove that any future theory is incorrect without knowing exactly what the theory is. I know that may seem very obvious but it can seem that that is what Bell tried to prove.

That is neither what Bell proved, nor what he set out to prove, nor what he said that he proved. What he asserted and then proved is that any theory in which what happens at detector A is independent of the setting of detector B (and vice versa) must disagree with the prediction of quantum mechanics. You don't need to know exactly what that theory is to prove this result, you just have to consider the consequences of having the result at one detector be independent of the other detector and compare them with the quantum mechanical prediction.

He assumed that there are certain features that any attempt at formulating a successful theory should have and then went on to prove that the theory can't be successful at all because it does not conform to the observations.

That's not right either, because the observations in question didn't even exist when Bell came up with his inequality. Bell showed that one class of theories (those in which the results at A are independent of the setting at B) must obey the inequality while quantum mechanics would violate the inequality. Only then did experimentalists go looking for violations (and I consider the most important words in Bell's original paper to be "The example considered above has the advantage that it requires little imagination to envisage the measurements involved actually being made").

And that's where I'm stuck. What exactly are the assumptions that Bell assumed the theory should have?

He didn't assume that any theory "should have" any particular assumption. Instead, he considered the consequences of one assumption, namely that the results at A are independent of the setting at B. Here we can let Bell speak for himself, from the first paragraph of his paper: "It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past"

 

[morrobay]

From the original paper : http://www.drchinese.com/David/Bell_Compact.pdf
(2) P (a,b) = ∫ dλp(λ) A (a,λ) B (b,λ) For locality condition.
Then with (13) A(a,λ) = - B (a,λ) For aligned detectors anti correlations ( see graph, post #84)
(2) is re written in (14) :
(14) P (a,b) = - ∫ dλp(λ) A(a,λ) A (b,λ) For realism condition.
How does (14) describe the realism condition and why is B in (2) replaced by A in (14) ?

 

[jerromyjon]

(14) P (a,b) = - ∫ dλp(λ) A(a,λ) A (b,λ) For realism condition.
How does (14) describe the realism condition and why is B in (2) replaced by A in (14) ?

Because they are anticorrelated... but, yeah, that doesn't make sense to me either but it's late and I'm tired...

 

[Dadface]

Thank you for your comments everyone. I think (hope) things are starting to make a bit more sense but I need time to try to mull over the comments made.

 

[stevendaryl]

From the original paper : http://www.drchinese.com/David/Bell_Compact.pdf
(2) P (a,b) = ∫ dλp(λ) A (a,λ) B (b,λ) For locality condition.
Then with (13) A(a,λ) = - B (a,λ) For aligned detectors anti correlations ( see graph, post #84)
(2) is re written in (14) :
(14) P (a,b) = - ∫ dλp(λ) A(a,λ) A (b,λ) For realism condition.
How does (14) describe the realism condition and why is B in (2) replaced by A in (14) ?

Line 13 explains it. In the anti-correlated case (which I assume is the one that Bell is talking about here), one experimenter (who I always call Alice) measures the spin of one particle, and another experimenter (who I always call Bob) measures the spin of the other particle. Experimentally, they always get the opposite results whenever they measure their spins along the same axis. If Alice gets +1, then Bob gets -1. In the formula

$P(a,b) = \int d\lambda \rho(\lambda) A(a, \lambda) B(b, \lambda)$

$a$ is the spin-direction chosen by Alice and $b$ is the spin-direction chosen by Bob, and $P(a,b)$ is the correlation, which is the average value of the product of Alice's result and Bob's result. (I really hate the use of the letter P here, because that suggests probability, but nevermind...)

Perfect anti-correlation means that when b=a, you always get -1. That means that

$P(a,a) = \int d\lambda rho(\lambda) A(a, \lambda) B(a, \lambda) = -1$

It's just a mathematical fact that that's impossible unless $B(a,\lambda) = -A(a,\lambda)$.

 

[DrChinese]

From the original paper : http://www.drchinese.com/David/Bell_Compact.pdf
(2) P (a,b) = ∫ dλp(λ) A (a,λ) B (b,λ) For locality condition.
Then with (13) A(a,λ) = - B (a,λ) For aligned detectors anti correlations ( see graph, post #84)
(2) is re written in (14) :
(14) P (a,b) = - ∫ dλp(λ) A(a,λ) A (b,λ) For realism condition.
How does (14) describe the realism condition and why is B in (2) replaced by A in (14) ?

Well, the Realism condition is the part just after (14) where it says: "It follows that c is another unit vector [in addition to a and b already referenced]". That's when Bell sets up the relationship between 3 observables. Those 3 can't simultaneously exhibit the Quantum Mechanical expectation value.

 

[Dadface]

That is neither what Bell proved, nor what he set out to prove, nor what he said that he proved. What he asserted and then proved is that any theory in which what happens at detector A is independent of the setting of detector B (and vice versa) must disagree with the prediction of quantum mechanics. You don't need to know exactly what that theory is to prove this result, you just have to consider the consequences of having the result at one detector be independent of the other detector and compare them with the quantum mechanical prediction.

That's not right either, because the observations in question didn't even exist when Bell came up with his inequality. Bell showed that one class of theories (those in which the results at A are independent of the setting at B) must obey the inequality while quantum mechanics would violate the inequality. Only then did experimentalists go looking for violations (and I consider the most important words in Bell's original paper to be "The example considered above has the advantage that it requires little imagination to envisage the measurements involved actually being made").

He didn't assume that any theory "should have" any particular assumption. Instead, he considered the consequences of one assumption, namely that the results at A are independent of the setting at B. Here we can let Bell speak for himself, from the first paragraph of his paper: "It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past"

I'm still not getting the conclusions that seem to be drawn from EPR and Bell and I think its down to the meaning of "hidden variable theories". In my opinion a hidden variable theory would be, at the very least, just as powerful as any currently accepted quantum theory in that, for example, it would make exactly the same predictions. If it doesn't conform to the observations it shouldn't be called a theory. But I think if any hidden variable theories are developed they would have some extra feature(s), the hidden variables, which quantum theory doesn't have at the present time. Those hidden variables will account for any perceived apparent discrepancies or weirdness in quantum theories.

I think Einstein said that QM is not complete and I think that is as true today as when he first said it. I think quantum theorists are still hard at work. And there is a possibility that a hidden variable theory will be developed. But until that happens we have no knowledge of what that theory is. And that illustrates one of my sticking points because the impression is often given that Bell, and the subsequent testing of his theory ruled out the possibility of a certain type of hidden variable theory. But to rule out a theory you've got to know the full details of that theory not just one assumption you think is made in developing that theory.

 

[stevendaryl]

I think Einstein said that QM is not complete and I think that is as true today as when he first said it. I think quantum theorists are still hard at work. And there is a possibility that a hidden variable theory will be developed. But until that happens we have no knowledge of what that theory is. And that illustrates one of my sticking points because the impression is often given that Bell, and the subsequent testing of his theory ruled out the possibility of a certain type of hidden variable theory. But to rule out a theory you've got to know the full details of that theory not just one assumption you think is made in developing that theory.

Well, that's the power of mathematics. If you have a real number in mind, I can tell you that its square is not equal to -1. I don't need to know all the decimal places of your number to reach that conclusion. In Newtonian mechanics, if there is a force $F(\vec{x})$ acting on a particle that is constant in time and independent of the velocity of the particle, then I can tell you that the combination of potential energy and kinetic energy is constant. I don't need to know the details of the force.

Mathematics allows you to prove facts about a huge class of situations. You don't need to know precise details about your situation---it's enough to know that it's of a particular type.

 

[Dadface]

Well, that's the power of mathematics. If you have a real number in mind, I can tell you that its square is not equal to -1. I don't need to know all the decimal places of your number to reach that conclusion. In Newtonian mechanics, if there is a force $F(\vec{x})$ acting on a particle that is constant in time and independent of the velocity of the particle, then I can tell you that the combination of potential energy and kinetic energy is constant. I don't need to know the details of the force.

Mathematics allows you to prove facts about a huge class of situations. You don't need to know precise details about your situation---it's enough to know that it's of a particular type.

Your analogies might be OK for the situations you describe but I don't think they're necessarily relevant to the point I'm trying to make.

A hidden variables theory, should one be developed, will accept all experimental results and will be just as good, if not better, than existing quantum theories at predicting those results. The theory will encompass the findings that have been made during the experimental testing of Bells paper and will encompass all other relevant experimental findings. In fact I can't imagine Einstein et al envisioning a theory that did not conform to observations because such a theory would be a failure.

I think hidden variables is an aspect of a theory, which has not yet been realized and which accounts for any perceived apparent weirdness or discrepancies in the currently known theories. It may be something complicated or it may be something very simple, perhaps an additional sentence or two pointing out something that has been overlooked. Exactly what the hidden variable are remains to be seen. Or perhaps will never be seen. I like to keep an open mind.

 

[zonde]

Well, the Realism condition is the part just after (14) where it says: "It follows that c is another unit vector [in addition to a and b already referenced]". That's when Bell sets up the relationship between 3 observables. Those 3 can't simultaneously exhibit the Quantum Mechanical expectation value.

While EPR is talking about reality Bell's argument talks about theories. It says: "This [non-local structure] is charateristic, according to the results to be proved here, of any such theory which reproduces exactly quantum mechanical predictions."
And predictions of a theory we can recalculate as many times as we want. So as long as we can take description of initial state the same for different test parameters (description of initial state does not depend on later test parameters i.e. theory is not superdeterministic) we can redo the calculation for different test parameters and have all these results at our disposal at the same time. And this is exactly what is expected of that hypothetical theory when it says: "It follows that c is another unit vector" i.e. we get prediction for different test parameter but with the same $\lambda$.