Why is superdeterminism not the universally accepted explanation of nonlocality?

[ThomasT]

ThomasT, I feel like we're arguing semantics.

Partly, yes. Clarifying semantics is an important part of any discussion.

Let me just ask you this: do you agree that in the idealized setup I described, there is no experimental procedure called "coincidence detection", only individual detection events?

Yes, I agree. And that's why your idealized setup is a non sequitur wrt the considerations posed in this thread.

If you're not going to look at the actual setups, or some idealization thereof, then what I'm saying won't make any sense to you. So, we can just agree to disagree on this.

 

[ThomasT]

The difficult and interesting question with respect to Bohm's concept of quantum potential (Q) is specifying which physical object(s) cause this potential and how and why. Bohm argued for an "informational field" interpretation of Q, but this view has been criticized by other "Bohmians" as being very obscure. For instance:

Reflections on the deBroglie–Bohm Quantum Potential
http://www.tcm.phy.cam.ac.uk/~mdt26/local_papers/riggs_2008.pdf

For such reasons, some "minimalist" Bohmians (Durr, Goldstein, etc.) try to dispense with Q completely but other problems arise. For example, without Q, are particle trajectories by themselves sufficient to explain quantum phenomena ("problem of trajectories")? Other "Bohmians" attempt to employ the quantum potential concept but dispense with the information field suggesting that "primitive" forces existing on their own in addition to particles (e.g. Belousek):

Energy Content of Quantum Systems and the Alleged Collapse of the Wavefunction
http://arxiv.org/ftp/arxiv/papers/0910/0910.2834.pdf

Formalism, Ontology and Methodology in Bohmian Mechanics
https://springerlink3.metapress.com...oqulc13h34tv0ihv21kj2&sh=www.springerlink.com

Again you identify a difficult consideration and provide some helpful references wrt the consideration. Nice work. You present really hard questions/considerations.

One of the reasons that it's difficult for me to label BM as a realistic theory is because of the quantum potential, because I don't understand how it can be interpreted realistically.

Thanks for the references.

 

[ThomasT]

BM does not provide such an explanation. Instead, it POSTULATES the existence of such dependence and shows that such a postulate can explain all measurable quantum phenomena.

Your contributions, wrt your view of BM, have been most helpful.

 

[lugita15]

Yes, I agree. And that's why your idealized setup is a non sequitur wrt the considerations posed in this thread.

But the point I've been trying to make is that a local determinist cannot believe in all the experimental predictions of quantum mechanics. That includes all possible experiments, including ones like my idealized setup which may be too difficult to carry out in practice. If all you want to argue is that practical Bell tests make it hard to rule out local determinism, you and I have no quarrel. But do you believe that in the case of my idealized setup, a local determinist would be able to believe that all the predictions of QM are correct?

 

[Demystifier]

Your contributions, wrt your view of BM, have been most helpful.

Thanks!

 

[ThomasT]

... do you believe that in the case of my idealized setup, a local determinist would be able to believe that all the predictions of QM are correct?

Your idealized setup doesn't address the OP. Your idealized setup only has to do with individual detecions. Bell showed that LR models of individual detections are compatible with QM.

 

[lugita15]

Your idealized setup doesn't address the OP. Your idealized setup only has to do with individual detecions. Bell showed that LR models of individual detections are compatible with QM.

So it's your assertion that in any experiment in which the only measurements done are individual detections, such as my idealized setup, local determinism is entirely compatible with the predictions of quantum mechanics? I think that is demonstrably false, and I think I demonstrated it in a previous post in this thread. Which of the following do you disagree with, in the case of my idealized setup?

1. Pretend you are a local determinist who believes that all the experimental predictions of quantum mechanics is correct.
2. One of these experimental predictions is that entangled photons are perfectly correlated when sent through polarizers oriented at the same angle.
3. From this you conclude that both photons are consulting the same function P(θ). If P(θ)=1, then the photon goes through the polarizer, and if it equals zero the photon does not go through.
4. Another experimental prediction of quantum mechanics is that if the polarizers are set at different angles, the mismatch (i.e. the lack of correlation) between the two photons is a function R(θ) of the relative angle between the polarizers.
5. From this you conclude that the probability that P(-30)≠P(0) is R(30), the probability that P(0)≠P(30) is R(30), and the probability that P(-30)≠P(30) is R(60).
6. It is a mathematical fact that if you have two events A and B, then the probability that at least one of these events occurs (in other words the probability that A or B occurs) is less than or equal to the probability that A occurs plus the probability that B occurs.
7. From this you conclude that the probability that P(-30)≠P(30) is less than or equal to the probability that that P(-30)≠P(0) plus the probability that P(0)≠P(30), or in other words R(60)≤R(30)+R(30)=2R(30).

And just for everyone else's reference, here is my idealized setup again:

Consider an idealized experiment where one photon pair is sent out every hour by a source which is exactly at the midpoint between two polarizers, which catch every single photon with perfect accuracy. In that case all each experimenter has as far as data goes is a list of yes or no answers as to whether the photon went through the polarizer or not. There are no time stamps, distance measurements, coincidence intervals, or anything like that.

 

[bohm2]

One of the reasons that it's difficult for me to label BM as a realistic theory is because of the quantum potential, because I don't understand how it can be interpreted realistically.

Bohm and Hiley felt that the quantum potential can be "seen" but only indirectly through its manifestations in the motion of particles. If one takes a "realistic" stance (as they do), how else can one explain interference effects, superconductivity, etc? They argue, that it's unlike fields or forces though because it can only affect a one-particle or a many-particle quantum system. Because of it's representation in configuration space, they regarded it more akin to a "field of information" versus a field of force. But it isn't 'information for us'; rather, it is some form of objective information that is independent of the observer:

Firstly, it must be said that in the many-particle system, the Schrodinger wave is no longer capable of being represented in the ordinary three-dimensional space. Rather, it has now to be thought of in a multidimensional space, called configuration space, in which there are three dimensions for each particle. A single point in this multi-dimensional space corresponds to a certain configuration of the entire system of particles-hence the name, configuration space. It is not possible directly to imagine such a configuration space. However, if we recall that the essential significance of the wave in the one-particle system was that it determines a kind of information, then the interpretation can readily be extended to the many-particle system. For it is well known that information, being a highly abstract sort of thing, can be organized and understood in any number of dimensions. This is a natural development of the idea that the Schrodinger wave is not to be regarded as a field of force, but rather as a field of information.

A more careful analysis of the mathematics for this case shows that the whole set of particles is now subject to a generalized sort of quantum potential. This depends on the Schrodinger field of the entire many-body system. So we have an extension of this interpretation to the many-body system, in which each particle is self-active. However, the form of its action may now depend on a common pool of information belonging to the whole system.

Meaning and Information
http://www.implicity.org/Downloads/Bohm_meaning+information.pdf

Hiley describes this view further:

How do we think about the quantum potential? It describes a field of energy so can it be regarded as producing a force on the particle? There are some problems with this view. Firstly, as we have already remarked above, the quantum potential has no external source so that there is nothing for the particle to 'push against'. The energy is internal so clearly there is something more subtle involved. Here it is more like the role the gravitational field plays in general relativity where the gravitational energy curves space-time itself...

But how are we to understand these puzzling features physically? Because there is nothing to push against we should not regard the quantum potential as giving rise to an efficient cause, ('pushing and pulling') but it should be regarded more in the spirit of providing an example of Aristotle’s formative cause. That is the quantum potential gives new form to the evolution of the trajectories, in a way that is very reminiscent of the morphogenetic fields proposed by Waddington (1956) and Thom (1975) in biology. The form is provided from within but it is, of course, shaped by the environment. Thus the quantum potential reflects the experimental conditions. Close one slit and the quantum potential changes and the subsequent evolution of the particle is different. There seems to be a kind of 'self organisation' involved. Now self-organisation requires the notion of information to be active. In the case of a biological system, this information is clearly provided by the environment, soil conditions, lack of moisture etc. In a quantum system I want to suggest that the information is provided by the experimental conditions, its environment. But this information is not passive. It is active and causes the internal energy to be redistributed between the kinetic (pB) and potential (Q) parts. Thus the quantum process is literally 'formed from within'.

From the Heisenberg Picture to Bohm: a New Perspective on Active Information and its relation to Shannon Information.
http://www.bbk.ac.uk/tpru/BasilHiley/Vexjo2001W.pdf

 

[lugita15]

It should be noted that, as I was discussing earlier with Demystifier, Bohmians are only realist with respect to the position observable; the Kochen Specker theorem leads them to reject the reality of other observables.

 

[bohm2]

It should be noted that, as I was discussing earlier with Demystifier, Bohmians are only realist with respect to the position observable; the Kochen Specker theorem leads them to reject the reality of other observables.

Realism and contextualism are not incompatible:

What is challenging about quantum physics is not that there are no objects, but that the properties of quantum objects are remarkably different from the properties that classical physics considers. For instance, in any case of quantum entanglement, conceived as a relation among quantum objects, there are no intrinsic properties of the objects concerned on which the relation of entanglement obtains. The fact, however, that quantum objects cannot be individuated, in the classical sense, does not imply their inexistence. In other words, the non-individuality of quantum objects is not and cannot be tantamount to pronouncing their non-existence.

Realism and Objectivism in Quantum Mechanics
http://philsci-archive.pitt.edu/9042/1/Realism_and_Objectivism_in_Quantum_Mechanics.pdf

 

[lugita15]

Realism and contextualism are not incompatible

But Demystifier said the Kochen-Specker theorem places some distinction between position and (say) angular momentum, because position operators commute with one another whereas angular momentum operators do not commute with one another. What conclusion do Bohmians draw from this?

 

[bohm2]

I don't think anybody can explain why position (Q) assume pre-existence in BM while everything else is contextual: spin, energy, and other non-position “observables”. It does make it easier though to describe the macroscopic physical objects that we are normally acquainted with: chairs, people, planets, etc. without necesitating collapse. From a realist perspective, a wave function existing in configuration space, by itself, seems like not the right kind of stuff to describe everyday physical objects we are acquainted with in 3-D space, I think. Either way, Bohmian is fine with KS theorem. From the link provided by Demystifier:

Thus, in general, measurements do not measure anything in the closer meaning of the term. The only exception being of course position measurements, and, in some sense momentum-measurements. The latter do indeed measure the asymptotic (Bohmian) velocities. Hence, the only properties of a Bohmian particle are its position and its velocity. Just as ψ is no classical field, the Bohmian particles are not classical particles, i.e. they are no bearers of properties other than position. Therefore a physical object like e.g. an electron should not be confused with the Bohmian particle at position Qi. It is represented by the pair (ψ,Qi). Agreed, this is a radical departure from the classical particle concept. However, within the de Broglie-Bohm theory this move is not only natural (recall that e.g. momentum and energy are concepts which arise in 2nd order Newtonian mechanics while the guidance equation of the de Broglie-Bohm theory is 1st order) but allows for an elegant circumvention of the Kochen-Specker no-go theorem, directed against hidden variable theories (see e.g. Mermin (1990). This theorem demonstrates, that a consistent assignment of possessed values to all observables for a quantum mechanical state is not possible. However, if you allow for contextuality as the de Broglie-Bohm theory does you do not expect such an assignment to exist at all.

 

[ThomasT]

@ lugita15,

In your post #266 in this thread you wrote:

But a "coincidental detection" is ... nothing more than performing "individual detections" on each of the two particles. So definitionally, what determines the result of a coincidental detection is just what determines the results of individual detections.

In your post #271 in this thread you wrote:

Since there is no such thing as coincidence detection, and correlation is nothing but correlation of individual detections, an analysis of entanglement cannot consist, even in principle, of anything other than asking what determines the results of individual detection.

There's an experimental procedure for combining the individual data streams, and it produces a datum (called coincidental detection) that's different from individual results and that's correlated to a different measurement parameter and a different underlying parameter than individual results are.

The correlation coefficient calculated wrt any Bell test refers to the correlation between θ (the angular difference between polarizers) and the rate of coincidental detection. This is what entanglement refers to. It does not refer to a correlation between individual detection attributes at A and B.

Your idealized setup confuses things because it doesn't describe Bell tests. From the results of Bell tests, it can be inferred that the underlying parameter that determines coincidental detection can't be varying from pair to pair. So, this underlying parameter (that determines coincidental detection) must be different than the underlying parameter that's inferred to be varying from pair to pair and determining individual detection.

 

[lugita15]

ThomasT, regardless of whether you think my idealized setup is a good representation of Bell tests, just answer me this: for this setup, in which there are only individual detection results, do you or do you not believe that it is possible for a local deterministic theory to be compatible with all the predictions of QM? If your answer is yes, my followup would be: which of the seven points quoted in post #307 do you disagree with and why? (And when reading that post, please keep in mind that when I say the word correlation I mean correlation between individual detections, not correlation coefficient of the rate of coincidental detection and theta.)

 

[yoron]

Ahha, so this thread was already foretold?
As well as any answers naturally.

So all discussion must then become meaningless.

"All your resistance will be futile" as the Borg says :)

Free will?

 

[ThomasT]

ThomasT, regardless of whether you think my idealized setup is a good representation of Bell tests, just answer me this: for this setup, in which there are only individual detection results, do you or do you not believe that it is possible for a local deterministic theory to be compatible with all the predictions of QM? If your answer is yes, my followup would be: which of the seven points quoted in post #307 do you disagree with and why? (And when reading that post, please keep in mind that when I say the word correlation I mean correlation between individual detections, not correlation coefficient of the rate of coincidental detection and theta.)

Your points involve mismatches (1,0 or 0,1 -- ie., paired or coincidental detection attributes) at relative angles (ie., Theta).

The problem in constructing an LR model of entanglement is that it has to encode some sort of locality condition. This is done by assuming that events (polarizer settings and individual data sequences) at A and B are independent of each other. This is manifested in your points by calculating the expected mismatches at some Theta as being no more than twice the mismatches at 1/2 Theta. It starts with point 5., where you separate the probability at Theta = 60 degrees into the probabilities at the Theta = 30 degree offsets. This is the your locality, or, more precisely, independence assumption.

Does this mean that nature is nonlocal? I don't think so. It's just that the results of Bell tests can't be understood in terms of independent events at A and B. The measurement and (assumed) underlying parameters are irreducible.

But how can one begin to understand the experimental results in a local deterministic way? Simply put, the polarizers in the joint context are measuring an underlying parameter (unlike the underlying parameter that determines individual detection and varies randomly from pair to pair) that isn't varying from pair to pair. They're measuring a relationship between photons of a pair. So, an independence assumption doesn't fit the experimental situation (even though it's a necessary constraint on standard LR models of entanglement). But the assumption that the relationship between photons of a pair is produced locally does fit the experimental situation (eg., see the emission model associated with Aspect et al. 1982). And then of course there's the experimentally documented behavior of light in polariscopic setups.

It all, reasonably I think, points to local determinism, as far as I can tell. So, no need for superdeterminism.

 

[lugita15]

Your points involve mismatches (1,0 or 0,1 -- ie., paired or coincidental detection attributes) at relative angles (ie., Theta).

That's correct, but I hope you acknowledge that in the case the mismatches are just mismatches of individual detection results, and hence the only thing that can possibly explain the mismatches are whatever parameters or hidden variables explain the individual detection results.

The problem in constructing an LR model of entanglement is that it has to encode some sort of locality condition. This is done by assuming that events (polarizer settings and individual data sequences) at A and B are independent of each other. This is manifested in your points by calculating the expected mismatches at some Theta as being no more than twice the mismatches at 1/2 Theta. It starts with point 5., where you separate the probability at Theta = 60 degrees into the probabilities at the Theta = 30 degree offsets. This is the your locality, or, more precisely, independence assumption.

OK, let me tell you my reasoning for going from step 4 to 5, and you tell me where I am making an "independence assumption". (For the record, I think my only locality assumption was in step 3, not in step 5). For each angle pair (θ1,θ2), we send a billion entangled pairs (remember, one pair every hour... that's why it's called "in principle") and by comparing the individual results of the two experimenters, we calculate the percentage of pairs that had a mismatch, called R(θ1,θ2). Note that whatever determines the individual detection results must also determine when there is and is not a mismatch, and thus determines R(θ1,θ2). After finding the function R, we find that it has the property that R((θ1+C,θ2+C)=R(θ1,θ2) for all C, so in particular R(θ1,θ2)=R(θ1-θ2,0)=R(θ,0), so we conclude that the difference between the two angles, not the individual angles, is what is most important, so we can just write R(θ). In particular, R(30)=R(0,-30)=R(30,0), R(60)=R(60,0)=R(30,-30), and R(0)=R(0,0)=0.

Does this mean that nature is nonlocal? I don't think so. It's just that the results of Bell tests can't be understood in terms of independent events at A and B. The measurement and (assumed) underlying parameters are irreducible.

What do you mean by parameters being irreducible?

But how can one begin to understand the experimental results in a local deterministic way? Simply put, the polarizers in the joint context are measuring an underlying parameter (unlike the underlying parameter that determines individual detection and varies randomly from pair to pair) that isn't varying from pair to pair.

I really don't understand you. Don't you agree that in my idealized setup, all data and calculations done come from the recording of individual detection results, and thus the parameter that determines ALL experimental findings is whatever determines whether a given photon goes through or not? I thought you agreed with this before, which is why you thought my setup didn't capture the features crucial for Bell tests.

Do you still stand by your comment below:

Your idealized setup only has to do with individual detecions. Bell showed that LR models of individual detections are compatible with QM.

 

[ThomasT]

... the only thing that can possibly explain the mismatches are whatever parameters or hidden variables explain the individual detection results.

That can't possibly be the case, as far as I can tell. The rate of mismatches, ie., the rate of coincidental detection, varies predictably as a function of Theta. But the rate of individual detection doesn't vary, no matter how the polarizers are oriented.

So how can the same underlying parameter be determining both coincidental detection and individual detection?

... tell me where I am making an "independence assumption".

Step 5. In this step you've analyzed the probability for a certain Theta into 2(Theta/2). The problem is, light doesn't behave that way.

What do you mean by parameters being irreducible?

It means that coincidental detection is being correlated with Theta, and you can't analyze it any further and get a model that agrees with the experimental results. In the joint context, Theta is the independent variable; the relationship between entangled photons is an assumed underlying constant; and the rate of coincidental detection is the the dependent variable.

... the polarizers in the joint context are measuring an underlying parameter (unlike the underlying parameter that determines individual detection and varies randomly from pair to pair) that isn't varying from pair to pair.

I really don't understand you. Don't you agree that in my idealized setup, all data and calculations done come from the recording of individual detection results, and thus the parameter that determines ALL experimental findings is whatever determines whether a given photon goes through or not?

You're missing an important part of Bell tests. The matching of the data streams. The pairing of detections at A and B. Without this there's no entanglement.

The key to understanding why the assumption of local determinism isn't incompatible with QM is that the underlying parameter that's being measured by Theta and that's determining coincidental detection isn't varying from pair to pair. Why/how can this be inferred? Because the rate of coincidental detection varies ... with Theta.

But the rate of individual detection doesn't vary ... no matter how an individual polarizer is oriented.

The key to understanding why LR theories of quantum entanglement are ruled out is that they have to encode some sort of independence assumption. But the problem is that this contradicts the experimental situation, which evidently produces a relationship between the entangled entities via local transmissions/interactions. And because this relationship is being measured by a global instrumental variable the appropriately matched data streams at A and B aren't independent of each other.

Changing the setting of the polarizer at A (or B) instantaneously changes Theta. Recording a qualitative result at A (or B) instantaneously changes the sample space at B (or A).

Quantum nonseparability refers to the irreducibility of the relationship between entangled entities, as well as the measurement parameters, and also the data associated with that relationship.

 

[DrChinese]

Free will?

Hey, nothing is free. My bank taught me that.

 

[lugita15]

That can't possibly be the case, as far as I can tell. The rate of mismatches, ie., the rate of coincidental detection, varies predictably as a function of Theta. But the rate of individual detection doesn't vary, no matter how the polarizers are oriented.

So how can the same underlying parameter be determining both coincidental detection and individual detection?

I'm having a hard time understanding this. Do you or do you not agree in my idealized setup, the mismatches are just mismatches of individual detection results. And that if the individual detection results are the same, then the mismatches are the same? And thus that the mismatches are completely determined by the individual detection results?

If you have a bunch of data in an excel spreadsheet, then the value of any function calculated from this data is entirely determined by the data. I don't know how you can reasonably disagree with this.

Step 5. In this step you've analyzed the probability for a certain Theta into 2(Theta/2). The problem is, light doesn't behave that way.

You have the uncanny ability of focusing on steps I consider trivial. To my mind, step 5 is a completely obvious consequence of step 4. I am just applying the definition of R, which is that the probability that P(θ1)≠P(θ2) is equal to R(θ1-θ2). How can you disagree with that definition?

You're missing an important part of Bell tests. The matching of the data streams. The pairing of detections at A and B. Without this there's no entanglement.

But the result of any analysis, matching, or pairing of the data is surely determined BY the data, is it not? And thus the parameters or hidden variables that determine the data must determine anything that is derived from the data, right?

 

[lugita15]

I don't think anybody can explain why position (Q) assume pre-existence in BM while everything else is contextual: spin, energy, and other non-position “observables”.

Are you saying that position is non-contextual in Bohmian mechanics?

 

[bohm2]

Are you saying that position is non-contextual in Bohmian mechanics?

Yes, position is the only non-contextual observable. So position is the only variable that can be regarded as being possessed before measurement, in such a way that “faithful measurements” just reveal it. Having said that Bohm (at least in his metaphysics) didn't appear to believe in the "reality" of particles:

We have frequently been asked the question “Didn’t Bohm believe that there was an actual classical point-like particle following these quantum trajectories?" The answer is a definite No! For Bohm there was no solid 'particle' either, but instead, at the fundamental level, there was a basic process or activity which left a ‘track’ in, say, a bubble chamber. Thus the track could be explained by the enfolding and unfolding of an invariant form in the overall underlying process.

Zeno Paradox for Bohmian Trajectories: The Unfolding of the Metatron
http://www.freewebs.com/cvdegosson/ZenoPaper.pdf

 

[lugita15]

Yes, position is the only non-contextual observable. So position is the only variable that can be regarded as being possessed before measurement, in such a way that “faithful measurements” just reveal it.

Are there "unfaithful measurements" which do not simply reveal the pre-measurement value? If so, does that mean there is still some lingering contextuality even in position? In other words, is Bohmian mechanics not exploiting the full noncontextuality allowed by the Kochen-Specker theorem for the position observable?

 

[Demystifier]

In other words, is Bohmian mechanics not exploiting the full noncontextuality allowed by the Kochen-Specker theorem for the position observable?

There is no Kochen-Specker theorem for the position observable, nor for any SINGLE observable. The Kochen-Specker theorem is a theorem about a SET of mutually NON-COMMUTING observables. If the set contains only one observable, then all observables in this set commute with each other (because [A,A]=0), so the Kochen-Specker theorem does not refer to this set.

 

[Demystifier]

I don't think anybody can explain why position (Q) assume pre-existence in BM while everything else is contextual

I believe I can explain it. See Secs. 2.1 and 2.2 of
http://xxx.lanl.gov/pdf/1112.2034.pdf

 

[Delta Kilo]

Couple of basic questions about BM:
1. What makes position "observable" special in BM? As I understand, the actual Q_k are never observed directly, instead atoms of the apparatus interact with the system (and the rest of the universe) in mysterious ways through the guiding equation. But what makes measuring position in this way any different from measuring any other observable?

2. Can guiding equation be reformulated in momentum basis?

 

[Demystifier]

Delta Kilo,
1. See my post #325 above.
2. No.

 

[lugita15]

There is no Kochen-Specker theorem for the position observable, nor for any SINGLE observable. The Kochen-Specker theorem is a theorem about a SET of mutually NON-COMMUTING observables. If the set contains only one observable, then all observables in this set commute with each other (because [A,A]=0), so the Kochen-Specker theorem does not refer to this set.

That was my point. Since the Kochen-Specker theorem does not apply to position, position can be completely non-contextual. But bohm2 seemed to imply that only "faithful" measurements reveal the pre-measurement position. So in Bohmian mechanics is position contextual for "unfaithful" measurements?

 

[Demystifier]

That was my point. Since the Kochen-Specker theorem does not apply to position, position can be completely non-contextual. But bohm2 seemed to imply that only "faithful" measurements reveal the pre-measurement position. So in Bohmian mechanics is position contextual for "unfaithful" measurements?

Yes, in BM there are also "unfaithful" measurements of positions, so BM can be said to be more contextual than necessary due to the KC theorem. The best known example of "unfaithful" measurements in BM are the so-called surreal trajectories.

 

[ThomasT]

I'm having a hard time understanding this.

I'm just wondering how the rate of individual detection and the rate of coincidental detection can be attributed to the same underlying parameter.

Do you or do you not agree in my idealized setup, the mismatches are just mismatches of individual detection results.

Yes, I agree. The language surrounding all this can get confusing. But I know what you're saying.

And that if the individual detection results are the same, then the mismatches are the same?

I'm not sure what you mean by this.

If you have a bunch of data in an excel spreadsheet, then the value of any function calculated from this data is entirely determined by the data. I don't know how you can reasonably disagree with this.

I don't disagree with it. But the individual detection sequences, considered separately, are different data than the sequences, appropriately combined, considered together. The two different data sets are correlated with different measurement parameters. The setting of polarizer a or b is not the same observational context as the angular difference between a and b.

You have the uncanny ability of focusing on steps I consider trivial. To my mind, step 5 is a completely obvious consequence of step 4. I am just applying the definition of R, which is that the probability that P(θ1)≠P(θ2) is equal to R(θ1-θ2). How can you disagree with that definition?

Your notation is a bit confusing for me. Say in words what you mean by the above notations.

But the result of any analysis, matching, or pairing of the data is surely determined BY the data, is it not?

Ultimately, yes. But the organization of the data, how it's parsed or matched, and what it's correlated with is determined by the experimental design. Individual data sequences composed of 0's and 1's aren't the same as combined data sequences composed of (1,1)'s, (0,0)'s, (1,0)'s, and (0,1)'s.

And thus the parameters or hidden variables that determine the data must determine anything that is derived from the data, right?

The individual data sequences, considered separately, are correlated with the settings of the individual polarizers, considered separately.

The combined data sequences are correlated with the angular difference between the individual polarizer settings.

In most (or at least many) LR accounts, the underlying parameter determining individual detection is assumed to be the polarization vector of polarizer-incident photons.

From the assumption of common cause, and the results when polarizers are aligned, it's assumed that this polarization vector is the same for both polarizer-incident photons of an entangled pair.

But here's the problem, the rate of coincidental detection varies only as θ, the angular difference between a and b, varies. (That is, wrt any particular θ, the common underlying polarization vector can be anything, and the rate of coincidental detection will remain the same. But if θ is changed, then the rate of coincidental detection changes as cos²θ, which, afaik, and not unimportantly, is how light would be expected to behave.)

So, it seems to me, θ must be measuring something other than the polarization vector of the polarizer-incident photons.

And it has to be something that, unlike the underlying polarization vector, isn't varying randomly from entangled pair to entangled pair.

So, I reason, θ is measuring a relationship between photons of an entangled pair -- a relationship which, wrt any particular Bell test, doesn't vary from pair to pair, and which Bell tests are designed to produce ... locally.