Can grandpa understand the Bell's Theorem?

[zonde]

And just a reminder to everyone reading this thread that might have some doubts about entanglement as a non-classical state:

a) You can entangle photons that have NEVER existed in the same light cone (so there is no joint point of origin) and therefore there is no classical mechanism to build from;

This classical mechanism is called postselection.

Rude analogy with colored balls:
We have pair of boxes where in one box there is red ball but in other blue.
These would be A and B boxes.
Then we have similar pair of boxes C and D.

We open box A and box D at two remote locations and send boxes B and C to third location.
Then we mix together content of boxes B and C and the look at it. If we see two balls with different colors we keep them, if they are the same color we discard them.
Then if we look at the sets where we didn't discarded boxes B and C sure thing we see that A and D boxes contained balls with different colors.

b) You can freely choose to entangle photons AFTER they have both been detected, violating the classical cause-effect sequence;
c) And finally, you can do BOTH a) and b) in the same experiment, which should be enough to throw any classical explanation out the window.

No problem with postselection.
But what explanation do you propose?

 

[zonde]

I'm ready to believe that you are right but I don't know what Type I and Type II sources are.

In simple words Type I PDC source produces photon pairs with two having the same polarization but Type II PDC source produces photon pairs with two having the opposite polarization.

I'm really arguing here from analogy to electrons. There is the singlet state |up*dn> - |dn*up> and there is the triplet state with a plus sign instead of a minus sign. I'm guessing that the weird correlations for electrons occur only with the singlet state, and that there is something analogous with photons.

Plus or minus sign just means that you have symmetry between two analyzers if you rotate them in the same direction or in opposite direction (both clockwise vs one clockwise/other counterclockwise).

 

[DrChinese]

In the Alain Aspect’s article “BELL’S THEOREM : THE NAIVE VIEW OF AN EXPERIMENTALIST”
http://arxiv.org/ftp/quant-ph/papers/0402/0402001.pdf the Fig. 3 shows "Polarisation correlation coefficient, as a function of the relative
orientation of the polarisers..." Based on what assumptions did Aspect derive DIFFERENT Polarisation correlation coefficient for QM and for the naive model?

Good question!

The QM function happens to be cos^2(theta), which you will notice is the same as Malus. This is both a coincidence and not a coincidence. Although the math is a bit complicated, once you factor in rotational issues etc, it reduces to this for the QM expectation value. This is a consequence of the quantum formalism and there is no direct analog in a classical model.

(On the other hand, you are free to add in by hand in any LR model BUT you must put forth something so it can be applied in an LR manner. You cannot just say "agrees to QM" because the QM model is NOT realistic, deterministic, etc. This is where many people go wrong because they just say it agrees and go no further. This will not fly.)

The Local Realistic value is based on Aspect's "naive" model. You will find that most naive models that come anywhere near the QM values will exactly match this linear function when averaged over a random dataset. Any such model requires a couple of things to get it started: a) perfect correlations, i.e. a value of 1 when theta=0 degrees and b) -1 when theta=90 degrees. You will probably want rotational invariance which means that there is no preferred orientation around 360 degrees. With that in mind, you will be hard pressed to get anything but a linear result against a random sample.

And as a check, remind yourself that you must supply a result for ANY 2 angles I pick! I will then average those across 360 degrees since you are saying your data is realistic and there are values for everything, even if not measured. That becomes the data point for that theta on the chart. You only need to do this exercise for 3 angles to see the issue: 0, 120 and 240 degrees. All of these are 120 degrees apart. The average coincidence rate for any realistic dataset with simultaneous values for these 3 settings will never be less than 33.33% (i.e. 1/3). When you rotate around 360 degrees (i.e. 1/121/241, 2/122/242, etc.) the same result holds.

So you will plot 1/3 or higher for theta=120 degrees. The QM value is 1/4.

 

[DrChinese]

This classical mechanism is called postselection.

Rude analogy with colored balls:
We have pair of boxes where in one box there is red ball but in other blue.
These would be A and B boxes.
Then we have similar pair of boxes C and D.

We open box A and box D at two remote locations and send boxes B and C to third location.
Then we mix together content of boxes B and C and the look at it. If we see two balls with different colors we keep them, if they are the same color we discard them.
Then if we look at the sets where we didn't discarded boxes B and C sure thing we see that A and D boxes contained balls with different colors.

This is preposterous. A and D will NOT be entangled in YOUR example. For example, they will not show perfect correlations.

 

[miosim]

I have a simple question regarding attached picture. It is from

http://www.upscale.utoronto.ca/GeneralInterest/Harrison/SternGerlach/SternGerlach.html

I expect that the picture shows entangled pair of electrons (having opposite spins – I added blue arrows).
The picture indicates that both electrons passed their respective Stern-Gerlach filters.
In contrary I expect that only an electron on the right side will pass. Am I confused?

 

[miosim]

The QM function happens to be cos^2(theta), which you will notice is the same as Malus. This is both a coincidence and not a coincidence. Although the math is a bit complicated, once you factor in rotational issues etc, it reduces to this for the QM expectation value. This is a consequence of the quantum formalism and there is no direct analog in a classical model.

I understand this. But I have no CLEAR undersatnding what a classical model is? Can we avoid generalizations like LR, realism, determinism, etc. because this terminology (that is better suited to philosophy) doesn’t help to achieve a clarity?

At the same time, I think that I understand what is the EPR model is and if I understand it correctly, the EPR model dosn’t deny QM formalism, but provides ONLY a different interpretation to this formalism.
So how come EPR model couldn’t be “granted” with the same cos^2(theta) and with compliance with the Malus’ law.

I think that before discussing a correlated statistic I really need to understand the behavior of an INDIVIDUAL photon (not an entangled pair) interacting with an individual polarizer. What is a difference between QM and EPR model (which is also QM entity) interacting with a polarizer?

Thank you

 

[DrChinese]

I have a simple question regarding attached picture. It is from

http://www.upscale.utoronto.ca/GeneralInterest/Harrison/SternGerlach/SternGerlach.html

I expect that the picture shows entangled pair of electrons (having opposite spins – I added blue arrows).
The picture indicates that both electrons passed their respective Stern-Gerlach filters.
In contrary I expect that only an electron on the right side will pass. Am I confused?

In your example, only 1 passes from each pair. I believe the author says as much on the page.

 

[DrChinese]

I think that before discussing a correlated statistic I really need to understand the behavior of an INDIVIDUAL photon (not an entangled pair) interacting with an individual polarizer. What is a difference between QM and EPR model (which is also QM entity) interacting with a polarizer?

Thank you

There really isn't one. A classical realistic model can work for this fine.

For instance: I can give you a dataset that will reproduce the QM expectation value. Using my favorite polarizer angles: 0/120/240 where the photon is known to be oriented at 0 degrees. Similarly, I can provide a dataset if the photon orientation angle is unknown.

 

[DrChinese]

At the same time, I think that I understand what is the EPR model is and if I understand it correctly, the EPR model dosn’t deny QM formalism, but provides ONLY a different interpretation to this formalism.
So how come EPR model couldn’t be “granted” with the same cos^2(theta) and with compliance with the Malus’ law.

Be careful when you reference EPR. EPR concludes that if QM is complete, then there is NOT local realism. That is the opposite of the perspective you are advocating. On the other hand, what does complete really mean?

Bell goes further: if QM is correct in its predictions for a certain area, then there is NOT local realism. And this is quite specific. And testable!

Further: The meaning of a classical model is that you apply the model to Alice and Bob independently and obtain results. You don't just say: the QM and LR results are the same. Sometimes, they DO give similar predictions. But in the case of entanglement, they don't!

 

[JesseM]

I understand this. But I have no CLEAR undersatnding what a classical model is? Can we avoid generalizations like LR, realism, determinism, etc. because this terminology (that is better suited to philosophy) doesn’t help to achieve a clarity?

There is no single classical model, the proof is meant to deal with the broad class of all conceivable classical models that qualify as local realistic. As always, the crucial thing to understand is why any such model would require that the particles have predetermined results for each detector setting, even before the experimenters make the choice of which setting to use. This would need to be true in any classical model that had the property that the two experimenters are guaranteed to get the same (or opposite) result whenever they pick the same detector setting. Do you understand this part, or not?

At the same time, I think that I understand what is the EPR model is and if I understand it correctly, the EPR model dosn’t deny QM formalism, but provides ONLY a different interpretation to this formalism.

But Bell proved they must deny the QM formalism, they just didn't realize it at the time.

I think that before discussing a correlated statistic I really need to understand the behavior of an INDIVIDUAL photon (not an entangled pair) interacting with an individual polarizer. What is a difference between QM and EPR model (which is also QM entity) interacting with a polarizer?

The EPR model does assume each particle has a predetermined result for any possible measurement before the measurement is made, in order to explain how both members of the pair give perfectly correlated results when the same measurement is made on both.

 

[miosim]

Be careful when you reference EPR. EPR concludes that if QM is complete, then there is NOT local realism. That is the opposite of the perspective you are advocating. On the other hand, what does complete really mean?

I am taking notes from your comments. Regarding prospective I am coming from, they are different from EPR, but I share their belief that QM is fundamentally incomplete and I belief in LR, but this is just a belief I would like to validate or reject this.

Bell goes further: if QM is correct in its predictions for a certain area, then there is NOT local realism. And this is quite specific. And testable!

I don’t understand this yet, but I wonder if this either/or logic is because only two alternatives is in consideration. What if we have other alternatives? However I don’t want speculate about this; I am still lock a basic understanding and want to move slowly without jumps.

 

[miosim]

There is no single classical model, the proof is meant to deal with the broad class of all conceivable classical models that qualify as local realistic. As always, the crucial thing to understand is why any such model would require that the particles have predetermined results for each detector setting, even before the experimenters make the choice of which setting to use. This would need to be true in any classical model that had the property that the two experimenters are guaranteed to get the same (or opposite) result whenever they pick the same detector setting. Do you understand this part, or not?

I didn’t realize that in the EPR model the “particles have predetermined results for each detector setting”. I thought that the EPR model exhibits QM “determinism” that specifies polarization for both correlated particles but in terms of probabilistic wave function. Therefore, I thought, the result should not be fully deterministic.

My understanding is that the most critical characteristic of EPR model relevant to Bell’s theorem are those that are incorporated into initial condition of Bell’s theorem and becomes its starting point? So WHAT are these characteristics of EPR model that are incorporated in the Bell’s theorem? This is my ultimate question in this discussion. Bell didn’t “insert” in his theorem words like “determinism”, “LR”, “classical”, etc., but his mathematical formalism contains some very critical elements of EPR model. What these elements are? It looks to me like everybody knows, but me.

 

[JesseM]

I didn’t realize that in the EPR model the “particles have predetermined results for each detector setting”. I thought that the EPR model exhibits QM “determinism” that specifies polarization for both correlated particles but in terms of probabilistic wave function. Therefore, I thought, the result should not be fully deterministic.

But a two-particle wave function is by definition not a "local" entity. Imagine that each particle has to "make up its mind" about what to do when it encounters a detector using only the localized properties associated with that particle, or with the region of space in the immediate neighborhood of the particle and detector. Such localized properties are what is meant by "elements of reality" in the EPR paper, and they assume that the "elements of reality" in the region of one measurement can't be influenced by the what happens in the region of the other measurement, see p. 3 where they write:

On the other hand, since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system. This is, of course, merely a statement of what is meant by the absence of an interaction between the two systems.

And on p. 4 they arrive at the conclusion that the two particles must have had "simultaneous elements of reality" determining both their position and momentum, based on the idea that they always give perfectly correlated results if experimenters measure the position of both or the momentum of both.

My understanding is that the most critical characteristic of EPR model relevant to Bell’s theorem are those that are incorporated into initial condition of Bell’s theorem and becomes its starting point? So WHAT are these characteristics of EPR model that are incorporated in the Bell’s theorem?

Just the idea that each particle's behavior when it encounters a detector must be determined by "elements of reality" which are not in any way causally influenced by what happens in the region of the other detector.

This is my ultimate question in this discussion. Bell didn’t “insert” in his theorem words like “determinism”, “LR”, “classical”, etc., but his mathematical formalism contains some very critical elements of EPR model.

He talked about "causality and locality" (I think equivalent to what modern physicists mean by "local realism"), and says that if this assumption holds we should be able to infer that the results of each measurement must be "predetermined" by the properties of the particle being measured--just read the first page of his original paper.

 

[zonde]

So are you agreeing with jed that it would be impossible to construct a local hidden variables theory where photons exhibited perfectly correlated behavior when measured with polarizers at the same angles (ignoring what the statistics are when different angles are selected)? Again keep in mind that the hidden variables can work in any way that doesn't violate local realism, there's no reason they need to behave like measurable polarization vectors.

Yes I think that this is impossible in a way that is consistent with known experimental observations.
But then I do not exactly agree that "the hidden variables can work in any way that doesn't violate local realism". You don't want contrived theory where all it does is gives explanation for single experiment. And it should be falsifiable. That will considerably reduce all the ways how hidden variables can work.

 

[JesseM]

Yes I think that this is impossible in a way that is consistent with known experimental observations.

That wasn't my question. I was asking whether you think it's impossible to have a local hidden variables theory that predicts 100% correlation when the experimenters choose the same angle, not a theory that is "consistent with known experimental observations" on all counts. I say this is certainly possible, but when I said that to jed clampett, your reply in post #28 was "Obviously your hypothesis breaks down even before we start to consider polarization entangled state."

But then I do not exactly agree that "the hidden variables can work in any way that doesn't violate local realism". You don't want contrived theory where all it does is gives explanation for single experiment. And it should be falsifiable. That will considerably reduce all the ways how hidden variables can work.

Whether the theory is contrived or non-contrived, whether it is falsifiable or not, is irrelevant to Bell's theorem, which is the subject under discussion. Bell's theorem deals with all conceivable local realistic theories, and shows that it's logically impossible that any of them (even ones that are contrived or non-falsifiable due to the presence of hidden variables) could agree with QM in all its predictions.

 

[zonde]

That wasn't my question. I was asking whether you think it's impossible to have a local hidden variables theory that predicts 100% correlation when the experimenters choose the same angle, not a theory that is "consistent with known experimental observations" on all counts. I say this is certainly possible, but when I said that to jed clampett, your reply in post #28 was "Obviously your hypothesis breaks down even before we start to consider polarization entangled state."

Well you can formulate some theory using some abstract entities and abstract analyzers. So what?
The moment you will try to establish correspondence between your abstract entities/analyzers and photons/polarizers I will say it's not working. Photons do not behave at polarizers like your abstract entities at your abstract analyzers.
And we would be back where we started.

Whether the theory is contrived or non-contrived, whether it is falsifiable or not, is irrelevant to Bell's theorem, which is the subject under discussion. Bell's theorem deals with all conceivable local realistic theories, and shows that it's logically impossible that any of them (even ones that are contrived or non-falsifiable due to the presence of hidden variables) could agree with QM in all its predictions.

Yes, that Bell does. And?
jed clampett said he sees the problem in QM prediction about perfect correlations.
I agree. I would be very nice if this prediction would be tested experimentally so that we can see how perfect are these correlations. Unfortunately there are no reports about such tests.

 

[zonde]

This is preposterous. A and D will NOT be entangled in YOUR example. For example, they will not show perfect correlations.

Hmm, I was not implying that this analogy is about entanglement. Sorry if I didn't make it clear.
I just wanted to illustrate what I mean with postselection and how it can create correlations between two entities that does not have common past.

In case of entanglement it is a bit more complicated. You have to determine similarity of polarization for two photons and sign of interference term.

From the same paper you quoted http://arxiv.org/abs/0809.3991" :
"Each source in our experiment emits pairs of polarization entangled photons along spatial directions 1 & 2 and 3 & 4, respectively (see fig. 1). We chose the singlet state \psi^{-}, which is one of the four maximally entangled Bell states:
|\psi^{\pm}\rangle=\frac{1}{\sqrt{2}}|HV\rangle\pm|VH\rangle
|\phi^{\pm}\rangle=\frac{1}{\sqrt{2}}|HH\rangle\pm|VV\rangle (1)
A successful entanglement swapping procedure will result in photons 1 and 4 being entangled, although they never interacted with each other [13? ]. This is done by performing a Bell-state measurement on particles 2 and 3, i.e. by projecting them on one of the four Bell states. Consequently, photons 1 and 4 will be projected onto the Bell state corresponding to the BSM outcome."
Respectively in this experiment only photons corresponding to one of four possibilities are detected. The rest is discarded.
But if you calculate average correlations for all four possible states it results in no correlation at all.

 

[jed clampett]

Jed Clampett wrote:

I'm really arguing here from analogy to electrons. There is the singlet state |up*dn> - |dn*up> and there is the triplet state with a plus sign instead of a minus sign. I'm guessing that the weird correlations for electrons occur only with the singlet state, and that there is something analogous with photons.

To which Zonde replied:

Plus or minus sign just means that you have symmetry between two analyzers if you rotate them in the same direction or in opposite direction (both clockwise vs one clockwise/other counterclockwise).

We may be miscommunicating here, but I'm pretty sure I can do the algebra at least for electrons to show that the correlations are different depending on the plus or minus sign. I think I ought to start a new thread for this, which will have to wait until later today. (Although from the very last post I now see that it is almost becoming "on topic" for this thread.)

 

[DrChinese]

Hmm, I was not implying that this analogy is about entanglement. Sorry if I didn't make it clear.

...

But if you calculate average correlations for all four possible states it results in no correlation at all.

Quite true! And that is pretty fascinating of itself. You can look at A & D all day long and you will never notice that some items are perfectly correlated. Until you check to see which ones were projected into a Bell (B & C indicate this).

 

[DrChinese]

I didn’t realize that in the EPR model the “particles have predetermined results for each detector setting”. I thought that the EPR model exhibits QM “determinism” that specifies polarization for both correlated particles but in terms of probabilistic wave function. Therefore, I thought, the result should not be fully deterministic.

My understanding is that the most critical characteristic of EPR model relevant to Bell’s theorem are those that are incorporated into initial condition of Bell’s theorem and becomes its starting point? So WHAT are these characteristics of EPR model that are incorporated in the Bell’s theorem? This is my ultimate question in this discussion. Bell didn’t “insert” in his theorem words like “determinism”, “LR”, “classical”, etc., but his mathematical formalism contains some very critical elements of EPR model. What these elements are? It looks to me like everybody knows, but me.

You are supposed to deduce certain things from both EPR and Bell. Please realize that these papers were written for a specific audience of peers, not the general public. As such, they assume you already know certain ideas. Yes, EPR is essentially saying "the result is predetermined and that creates the element of reality".

Bell starts off saying that the spin of a,b has a value of +/-1, and then later (14) extends that to say that there are 3 such values simultaneously, a/b/c. He says this mathematically (assuming we will follow this), his audience at that time being very small. He does say (2nd paragraph) the following about the EPR argument, which should help: "Since we can predict in advance the result of measuring any chosen component ... it follows that the result of any such measurement must actually be predetermined." He then says that QM would be incomplete were that true since the QM formalism lacks such determinism.

If the result is predetermined, you should be able to construct a dataset by hand which will yield certain results. You can simply make up the values yourself and try to make them work out. You will soon see that is NOT possible. Please try it, it will go a long way towards greater understanding.

 

[JesseM]

Well you can formulate some theory using some abstract entities and abstract analyzers. So what?
The moment you will try to establish correspondence between your abstract entities/analyzers and photons/polarizers I will say it's not working. Photons do not behave at polarizers like your abstract entities at your abstract analyzers.

But the reason they don't behave like the abstract entities is because you are considering results at all possible angles, not just on trials where both experimenters chose the same angle. That's what Bell shows in his proof, that you can't have a local realistic theory that matches both the prediction of 100% correlation when the experimenters choose the same angle, and the QM statistics when they choose different angles. The first alone would be compatible with local realism!

Whether the theory is contrived or non-contrived, whether it is falsifiable or not, is irrelevant to Bell's theorem, which is the subject under discussion. Bell's theorem deals with all conceivable local realistic theories, and shows that it's logically impossible that any of them (even ones that are contrived or non-falsifiable due to the presence of hidden variables) could agree with QM in all its predictions.

Yes, that Bell does. And?
jed clampett said he sees the problem in QM prediction about perfect correlations.

But he said this in the context of a discussion of local realism, so I thought he was saying that even if our only restriction on theories is that they be local realistic, there is still some problem with perfect correlations. There isn't! If you want to add additional constraints like that the local realistic theory be "non-contrived", or that it says that photons have definite polarization vectors at each moment and the probability they pass through a filter depends on the relative angle between this vector and the polarizer according to Malus' law, then in that case these additional conditions might be able to rule out two photons always having perfect correlations whenever they're measured with polarizers at the same angle. But in the context of Bell's proof such additional constraints are irrelevant, and I thought jed was saying that even with the bare assumption of local realism there'd be a problem with perfect correlations, and that you were agreeing. If you're not saying that then I don't think we really disagree on anything here.

 

[miosim]

JesseM and DrChinese
I am taking notes of your comments and it helps.

One simple question.
I would like to make sure that I understand correctly the “...100% correlation when the experimenters choose the same angle..."

Is this (100%) applys only for the detected correlated photons? Or this is true for any pair of correlated photons (traveling along the line between polorizers) regardless we detect them or not and regardless of their polarization angle in reference to polarizes (set in parallel)?

Thanks

 

[JesseM]

Is this (100%) applys only for the detected correlated photons? Or this is true for any pair of correlated photons (traveling along the line between polorizers) regardless we detect them or not and regardless of their polarization angle in reference to polarizes (set in parallel)?

Thanks

How would we know if they weren't detected? Anyway, the result assumed in Bell's theorem is specifically that when you measure two entangled photons, then whenever the two polarizers are set to the same angle, the observed results of the measurement are always identical (or opposite depending on the experiment).

 

[jed clampett]

jed clampett said he sees the problem in QM prediction about perfect correlations.
I agree. I would be very nice if this prediction would be tested experimentally so that we can see how perfect are these correlations. Unfortunately there are no reports about such tests.

I'm really glad you raised this point because I was afraid to raise it myself. In reading about the history of Bell we find of course the famous Aspect experiments, and the earlier Clauser experiments of the 70's. But all of these used the 45/22.5 degree polarizers of Bell. I haven't found any reference to the original 100% correlation experiments as seen with polarizers aligned. Surely the experiment must have been done long before Aspect or Clauser; and surely it would have drawn considerable attention in its day, at the very least because it can't be an easy experiment to do.

Where and when was the first experimental demonstration of the 100% (or greater than 50% even!) correlation of photon detections?

 

[miosim]

Quick question:

According to the traditional QM, do entangled photons have identical polarization or their polarization may slightly differ?

Thanks

 

[DrChinese]

In reading about the history of Bell we find of course the famous Aspect experiments, and the earlier Clauser experiments of the 70's. But all of these used the 45/22.5 degree polarizers of Bell. I haven't found any reference to the original 100% correlation experiments as seen with polarizers aligned. Surely the experiment must have been done...

How do you think scientists optimize their Bell test setup? There is one easy way, set your polarizers to get perfect correlations. Once you get the highest match rate possible, you can continue to test the CHSH inequality or whatever. Because you know the source is truly entangled. It is so fundamental it is not usually mentioned as it has no bearing on the experiment at hand.

 

[DrChinese]

Quick question:

According to the traditional QM, do entangled photons have identical polarization or their polarization may slightly differ?

Thanks

This question has multiple correct answers.

In the sense you are asking it, QM says that entangled photons will have identical (or crossed according to the Type) polarizations. However, the fidelity of the source is a factor so this only reaches 100% in the ideal case. Also, it is possible to intentionally (or accidentally) make pairs for which the rule is looser than perfect match. If you allow a small bit of knowledge to creep into the equation, the entanglement can be reduced accordingly. In other words, you can have 72% entanglement, 49% entanglement, etc.

 

[jed clampett]

Jed Clampett asked:

Where and when was the first experimental demonstration of the 100% (or greater than 50% even!) correlation of photon detections?

DrChinese answered:

How do you think scientists optimize their Bell test setup? There is one easy way, set your polarizers to get perfect correlations. Once you get the highest match rate possible, you can continue to test the CHSH inequality or whatever. Because you know the source is truly entangled. It is so fundamental it is not usually mentioned as it has no bearing on the experiment at hand.

I understand from DrChinese that it was a dumb question for me to ask. But I'm still interested in the answer if anyone knows it.

 

[miosim]

... localized properties are what is meant by "elements of reality" in the EPR paper, and they assume that the "elements of reality" in the region of one measurement can't be influenced by the what happens in the region of the other measurement, see p. 3 where they write:
"On the other hand, since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system. This is, of course, merely a statement of what is meant by the absence of an interaction between the two systems."

I share their views.

And on p. 4 they arrive at the conclusion that the two particles must have had "simultaneous elements of reality" determining both their position and momentum ..."

I hold the same views.

... based on the idea that they always give perfectly correlated results if experimenters measure the position of both or the momentum of both.

However, in my phenomenological model the entangled photons aren't perfectly correlated when no longer interact.

I think that this is a time for me to move away form EPR model and start building my own.

In my phenomenological model any photon exhibits a random change in polarization within a limited range of values. It is why the absolute knowledge of one correlated photon doesn’t give us a knowledge of the correlated photon. However both photons should have their "simultaneous elements of reality" regardless can we observe it or not.

Another important element of my model is that the photons in my model aren’t a waves but corpuscles only. Actually, my model is a 100% corpuscular for any particle and banish wave entirely. Particles in my model are continuously jiggling in a wave-like trajectory sampling surrounding space. This wave-like motion of elementary particles determines their wave property that is matched with QM wave function. Therefore I expect that the interaction of my photons with polarizer cannot be distinguished from the traditional QM model and therefore Bell’s theorem ishould’t be valid for the photons of my local realistic model.

 

[Jonathan Scott]

Another important element of my model is that the photons in my model aren’t a waves but corpuscles only. Actually, my model is a 100% corpuscular for any particle and banish wave entirely. Particles in my model are continuously jiggling in a wave-like trajectory sampling surrounding space. This wave-like motion of elementary particles determines their wave property that is matched with QM wave function. Therefore I expect that the interaction of my photons with polarizer cannot be distinguished from the traditional QM model and therefore Bell’s theorem ishould’t be valid for the photons of my local realistic model.

You've missed the point. As we've said several times Bell's theorem does not relate to any specific classical model, although you can use particular classical models to illustrate it. Bell's theorem proves that the results of QM cannot be reproduced by ANY local realistic model.