Why is superdeterminism not the universally accepted explanation of nonlocality?

[lugita15]

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

To me the answer is clear: both rates are calculated from the individual detection results, so the only relevant parameters are those that determine the individual detection results.

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

Yes, I also feel that much of our disagreement may be due to semantics.

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.

I mean, suppose we have sent an entangled pair of photons through the polarizers, and e.g. we may get 1 on the first detector and 0 on the second detector. Then given these individual detection results, the answer to the question "Was there a mismatch?" is completely determined. So mismatches cannot be depend on any parameter that the individual detection results don't already depend on.

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.

I think this is more semantics. To my mind, the data sets are just composed of the individual data entries, i.e. the individual detection results. So how can the data set as a whole be determined by anything other than what determines each individual entry?

The setting of polarizer a or b is not the same observational context as the angular difference between a and b.

I don't get your point here. To me, it seems so obvious that teh angular difference is nothing more and nothing less than the difference of the settings of the settings of the two polarizers, so there's nothing special about it.

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

I don't know whether I can, but I can try to explain my notation and then you can ask me what you don't get. Starting from the top, QM predicts that for an entangled pair of photons, you always get identical detection results at identical polarizer settings. From this, the local determinist concludes that both photons are using the same function P(θ) to determine whether to go through the polarizer or not. P has only two values it can have, 0 and 1. If one of the photons encounters a polarizer oriented at a given angle, it plugs the angle into the function P and gets either a 0 or 1 as the answer. If 0, then it doesn't go through the polarizer, and if 1 then it does. Are you clear up to there?

So now the following experiment is done. Polarizer 1 is turned to the angle θ1, Polarizer 2 is turned to θ2, and then we send a trillion entangled pairs of photons to the two polarizers. Each experimeter writes down a list of yes or no answers as to whether each photon goes through the polarizer or not. Then we calculate R(θ1,θ2), which is the percentage of pairs whose individual detection results had a mismatch. Another way of putting this is that R(θ1,θ2) is the observed probability that a randomly selected entangled pair will have a mismatch between individual detection results. Are you clear on that?

Now remember, the individual detection results for a given pair are determined by the function P. So if the pair has a mismatch when one polarizer is oriented at θ1 and one polarizer is oriented at θ2, what that means is that P(θ1)≠P(θ2), meaning the P function for that pair is telling you to do different things at the angle θ1 versus the angle θ2. Now remember, R(θ1,θ2) is the probability that a randomly selected pair will have a mismatch when the polarizers are set at θ1 and θ2. In other words, R(θ1,θ2) is the probability that a randomly selected pair will have a P function which gives contradictory messages at θ1 and θ2, or to put it more simply R(θ1,θ2) is the probability that a randomly selected pair has a P function such that P(θ1)≠P(θ2). Are you clear on that? If you are, then step 5 follows pretty directly. (You see me frequently writing R(θ) instead of R(θ1,θ2), because R(θ1+C,θ2+C)=R(θ1,θ2) for all C, so in particular R(θ1-θ2,0)=R(θ1,θ2), so we can write R(θ1,θ2)=R(θ,0)=R(θ), where θ=θ1-θ2; I hope that's not too confusing.)

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.

But all these data sequences are composed of the individual detection results, so the only relevant parameter are whatever determines these results. I'm sorry for repeating myself, but I feel like we're communicating on different wavelengths.

 

[lugita15]

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.

OK, that's what I was trying to get at. So are there more variants or alternatives of Bohmian mechanics which make position even less contextual?

The best known example of "unfaithful" measurements in BM are the so-called surreal trajectories.

Are these the trajectories where the particle goes one way through a double slit experiment according to Bohmian mechanics, but for some reason detectors at the slits tell a different story? How do Bohmians explain that? (I'm sorry if this is another really trivial question.)

 

[Joncon]

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

ThomasT, I'm struggling to understand your reasoning here, so let me ask a simple question.
If photon A encounters polarizer A which parameter does it use to determine whether or not it passes, individual or coincidental?

 

[ThomasT]

To me the answer is clear: both rates are calculated from the individual detection results, so the only relevant parameters are those that determine the individual detection results.

But both rates are not calculated from individual detection results.

I mean, suppose we have sent an entangled pair of photons through the polarizers, and e.g. we may get 1 on the first detector and 0 on the second detector. Then given these individual detection results, the answer to the question "Was there a mismatch?" is completely determined.

Completely determined by what?

... mismatches cannot be depend on any parameter that the individual detection results don't already depend on.

But then you're ignoring the obvious inferences from the experimental results.

To my mind, the data sets are just composed of the individual data entries, i.e. the individual detection results.

Sure. And human behavior is composed of the the behavior of individual atoms that comprise human beings. But you don't seem to realize that these are different observational contexts.

Do you think that you can explain human behavior from the atomic scale?

So how can the data set as a whole be determined by anything other than what determines each individual entry?

By "data set as a whole" I suppose that you're referring to coincidental detections.

The answer to your question is that "the data set as a whole" doesn't vary as a function of underlying polarization orientation. But individual detection, presumably, does. So, how would you explain this?

I don't get your point here. To me, it seems so obvious that teh angular difference is nothing more and nothing less than the difference of the settings of the settings of the two polarizers, so there's nothing special about it.

Yes, it's the angular difference of the settings of the two polarizers. That's what makes it a different measurement parameter than the settings of the polarizers considered separately by themselves.

I'll get back to you.

 

[lugita15]

But both rates are not calculated from individual detection results.

Yes, they are. At least in my idealized setup, everything is determined by putting the individual detection results (yes or no answers) in a spreadsheet, and then applying functions on the spreadsheet data.

Sure. And human behavior is composed of the the behavior of individual atoms that comprise human beings. But you don't seem to realize that these are different observational contexts.

Do you think that you can explain human behavior from the atomic scale?

Certainly, if human behavior is composed of the behavior of individual atom, then in principle you can definitely explain all human behavior from the atomic scale. Practically of course it might be insurmountably difficult, but we are talking about whether local determinism in principle contradicts the predictions of QM, not whether currently practical Bell tests are sufficient to definitively disprove local determinism (they're not).

 

[Demystifier]

So are there more variants or alternatives of Bohmian mechanics which make position even less contextual?

If there are, I am not aware of it.

Are these the trajectories where the particle goes one way through a double slit experiment according to Bohmian mechanics, but for some reason detectors at the slits tell a different story? How do Bohmians explain that? (I'm sorry if this is another really trivial question.)

It's not trivial at all, so I would not like to discuss it in detail. For the details, see e.g. Sec. 4.1 of
http://xxx.lanl.gov/abs/quant-ph/0412119
Let me only say that Bohmians explain it by pointing out that Bohmian trajectories do not coincide with trajectories which one would naively expect from classical physics.

 

[f95toli]

But both rates are not calculated from individual detection results.

I haven't read the post above, so maybe I am misunderstanding what you are are saying.
However, coincidence measurements are usually done -in theory AND often in practice- by postprocessing of data from two individual detectors. All you need is two "streams" of time-stamped data.

 

[ThomasT]

I haven't read the post above, so maybe I am misunderstanding what you are are saying.
However, coincidence measurements are usually done -in theory AND often in practice- by postprocessing of data from two individual detectors. All you need is two "streams" of time-stamped data.

Yes, and combining the individual data sets results in a different data set, coincidental measurements, which is correlated with a different measurement parameter, angular difference.

For your convenience, here's my reasoning (from post #330):

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.

 

[ThomasT]

ThomasT, I'm struggling to understand your reasoning here, so let me ask a simple question.
If photon A encounters polarizer A which parameter does it use to determine whether or not it passes, individual or coincidental?

Individual. See post #338 for a rehash of my reasoning.

 

[ThomasT]

... we are talking about whether local determinism in principle contradicts the predictions of QM ...

The technical requirement, local realism, has been shown by Bell to contradict the predictions of QM. However, according to my current way of thinking about it (see post #338 for the line of reasoning), the assumptions of locality and determinism don't contradict the predictions of QM.

So, unless there's a flaw in my reasoning, then the assumption of superdeterminism isn't necessary.

 

[lugita15]

ThomasT, did you get through the rest of my post #331? Now do you understand my notations concerning P and R, and do you understand my reasoning for step 5 out of 7? If so, now which of my seven steps do you disagree with, for my idealized setup? Just for everyone's reference, here they are again.

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).

 

[ThomasT]

... which of my seven steps do you disagree with ... ?

I don't necessarily disagree with any of them. I just thought that Step 5. is where the independence is introduced.

For now, I'll ask a question about Step 3. :

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.

Is your P(θ) the hidden variable, the underlying parameter, that Bell originally referred to as λ?

 

[lugita15]

I don't necessarily disagree with any of them.

Well, if you agree with all my steps then I've won, because the whole point of my argument is to show that a local determinist cannot agree with all the predictions of quantum mechanics. So if you disagree with my conclusion, you have to disagree with one of my steps.

I just thought that Step 5. is where the independence is introduced.

No, step 5 is a completely trivial step, as I think you now understand. The only locality assumption I see is in step 3.

For now, I'll ask a question about Step 3. :
Is your P(θ) the hidden variable, the underlying parameter, that Bell originally referred to as λ?

Yes, P(θ) is the local hidden variable that determines the individual detection results.

 

[ThomasT]

Well, if you agree with all my steps then I've won, because the whole point of my argument is to show that a local determinist cannot agree with all the predictions of quantum mechanics.

I don't disagree with any of Bell's steps either. His program was to construct a model of entanglement than encoded a locality assumption. He proved that any such model was incompatible with QM. But he didn't prove that nature is nonlocal. He just proved that any model of entanglement that encodes an independence feature (which is how the assumption of locality is encoded) is incompatible with QM.

So if you disagree with my conclusion, you have to disagree with one of my steps.

I don't think so. To retain the assumptions of locality and determinism, I just have to show where, in your line of reasoning, the conclusion (which contradicts the known behavior of light) that there's a linear correlation between the angular difference between the polarizers and rate of coincidental detection becomes inevitable.

The only locality assumption I see is in step 3.

The first sentence in Step 3. isn't a locality assumption. It's a common cause assumption. This isn't what differentiates LR models from QM. QM assumes a common cause also, because that's what the experiments are designed to produce.

Yes, P(θ) is the local hidden variable that determines the individual detection results.

Ok.

P(θ) or λ is usually understood as the polarization vector of the optical disturbance incident on the polarizer. An LR model of rate of individual detection as determined by the polarizer orientation and the orientation of λ is compatible with QM.

But wrt rate of coincidental detection, this doesn't work. Wrt any particular value of θ, the orientation of λ can be anything, and the rate of coincidental detection will remain the same.

This can be visualized via a circle with two lines through the center representing the polarizer settings, and a third line through the center representing λ. No matter how λ is rotated, as long as θ remains the same, then the rate of coincidental detection doesn't vary.
So, λ is not determining the rate of coincidental detection.

 

[lugita15]

I don't disagree with any of Bell's steps either. His program was to construct a model of entanglement than encoded a locality assumption. He proved that any such model was incompatible with QM. But he didn't prove that nature is nonlocal. He just proved that any model of entanglement that encodes an independence feature (which is how the assumption of locality is encoded) is incompatible with QM.

OK, but regardless of what you think Bell's purpose was, I hope it's clear to you that my purpose is explicitly to show that you cannot be a local determinist and believe that all the predictions of QM are correct. The conclusion of my argument is that R(60)≤2R(30), which is in direct contradiction with the predictions of QM. So if you believe that local determinism IS compatible with the predictions of QM, then you disagree with my last step and thus you must disagree with one of the earlier steps. So which is it?

The first sentence in Step 3. isn't a locality assumption. It's a common cause assumption. This isn't what differentiates LR models from QM. QM assumes a common cause also, because that's what the experiments are designed to produce.

I hope we can stop talking about the formal models you call LR, because my goal isn't to show that some particular formal model is incompatible with QM, but rather that ANY local deterministic theory is incompatible with the predictions of QM.

But step 3 is definitely not something a believer in (an orthodox interpretation of) QM would accept. He wouldn't believe that individual detection results are predetermined by a commonly held function P(θ). Instead, he would think that the particle makes a random decision on the spot when it's measured, and then the wavefunction of the two-particle system collapses (nonlocally of course) so that the other particle will also do the same thing when put through a detector at the same setting.

Ok.

P(θ) or λ is usually understood as the polarization vector of the optical disturbance incident on the polarizer. An LR model of rate of individual detection as determined by the polarizer orientation and the orientation of λ is compatible with QM.

But wrt rate of coincidental detection, this doesn't work. Wrt any particular value of θ, the orientation of λ can be anything, and the rate of coincidental detection will remain the same.

This can be visualized via a circle with two lines through the center representing the polarizer settings, and a third line through the center representing λ. No matter how λ is rotated, as long as θ remains the same, then the rate of coincidental detection doesn't vary.
So, λ is not determining the rate of coincidental detection.

I've already tried to tell you that the percentage of mismatches is determined entirely by the individual detection results, but let's not rehash that; we may just be having some semantic differences on that point. Just tell me which of the seven steps you disagree with. Or if you prefer, which of the seven steps is such that not all local determinists would be forced to accept it?

 

[ThomasT]

I've already tried to tell you that the percentage of mismatches is determined entirely by the individual detection results, but let's not rehash that; we may just be having some semantic differences on that point.

I don't think it's just a semantic difference. Do the visualization I suggested. It becomes quite clear that λ, the underlying polarization vector, isn't determining coincidental detection.

What you're not getting is that the relationship between entangled photons and the polarization of the pair are two different underlying parameters. It's the polarization that determines individual detection, and the relationship that determines coincidental detection.

... which of the seven steps is such that not all local determinists would be forced to accept it?

We can start with the second sentence in Step 3.

 

[lugita15]

I don't think it's just a semantic difference. Do the visualization I suggested. It becomes quite clear that λ, the underlying polarization vector, isn't determining coincidental detection.

What you're not getting is that the relationship between entangled photons and the polarization of the pair are two different underlying parameters. It's the polarization that determines individual detection, and the relationship that determines coincidental detection.

But in my idealized setup, coincidental detection comes entirely from the individual detection. I thought you acknowledged that here:

To my mind, the data sets are just composed of the individual data entries, i.e. the individual detection results.

Sure. And human behavior is composed of the the behavior of individual atoms that comprise human beings. But you don't seem to realize that these are different observational contexts.

Do you think that you can explain human behavior from the atomic scale?

And my answer was yes, if human behavior is composed of the behavior of the individual atoms then in principle you can completely explain human behavior from the atomic scale. So would you similarly acknowledge that if the coincidental detection results are composed of the individual detection results, as is the case for my idealized setup, then in principle the former can be completely explained in terms of the latter?

We can start with the second sentence in Step 3.

OK, so as a local determinist, what do you find objectionable in that sentence? "If P(θ)=1, then the photon goes through the polarizer, and if it equals zero the photon does not go through."

 

[ThomasT]

But in my idealized setup, coincidental detection comes entirely from the individual detection.

Coincidental detection comes from the relationship between entangled photons. This isn't what's being measured in the individual context.

... if human behavior is composed of the behavior of the individual atoms ...

It isn't.

... then in principle you can completely explain human behavior from the atomic scale.

You can't, not even in principle.

So would you similarly acknowledge that if the coincidental detection results are composed of the individual detection results ...

I'm not arguing that. Obviously, coincidental results are composed of individual results.

... then in principle the former can be completely explained in terms of the latter?

No, because we're dealing with two different observational contexts wrt which there are two different underlying parameters.

OK, so as a local determinist, what do you find objectionable in that sentence? "If P(θ)=1, then the photon goes through the polarizer, and if it equals zero the photon does not go through."

P(θ) is supposed to be the underlying parameter determining individual detection -- which is usually understood as the underlying polarization orientation. So P(θ) would have values in degrees or radians.

 

[lugita15]

P(θ) is supposed to be the underlying parameter determining individual detection -- which is usually understood as the underlying polarization orientation. So P(θ) would have values in degrees or radians.

But P(θ) just tells the particle whether to go through the polarizer or not. So the only instruction it gives the particle is a yes or a no, or equivalently a 1 or a 0.

 

[lugita15]

But P(θ) just tells the particle whether to go through the polarizer or not. So the only instruction it gives the particle is a yes or a no, or equivalently a 1 or a 0.

Just to add to this, the function P(θ), since it is the hidden variable, can be determined by any number of things, including a polarization vector or anything else. But the input of the function must be the polarizer setting, and the output must be a yes-or-no instruction telling the particle to go through or not.

 

[Delta Kilo]

Yes, P(θ) is the local hidden variable that determines the individual detection results.

It can't be since θ is not local-realistic.

Consider a setup where settings a and b are determined immediately before the measurement A and B by sampling polarization of photons coming from distant stars and detectors A and B are some distance apart and in relative motion. Thanks to relativity of simultaneity, it can be arranged so that in reference frame of A, measurement of A happens before the photon determining b hits the target, and vice versa, in reference frame of B, measurement B is done before a is determined. So from the point of view of either detector θ does not yet exist when the measurement is done. Therefore the results of the measurement cannot be determined by θ or any function of θ, not in local-realistic sort of way.

 

[lugita15]

It can't be since θ is not local-realistic.

Consider a setup where settings a and b are determined immediately before the measurement A and B by sampling polarization of photons coming from distant stars and detectors A and B are some distance apart and in relative motion. Thanks to relativity of simultaneity, it can be arranged so that in reference frame of A, measurement of A happens before the photon determining b hits the target, and vice versa, in reference frame of B, measurement B is done before a is determined. So from the point of view of either detector θ does not yet exist when the measurement is done. Therefore the results of the measurement cannot be determined by θ or any function of θ, not in local-realistic sort of way.

Sorry about the confusion. When I say P(θ), I don't really mean θ the relative angle between the polarizers, just the angle of a single polarizer. So P(θ1) determines the behavior of particle 1, and P(θ2) determines the behavior of particle 2, so everything is local. I could say P(x) or something instead.

 

[Delta Kilo]

When I say P(θ), I don't really mean θ the relative angle between the polarizers, just the angle of a single polarizer.

Oh, I see. I though you were referring to the same θ as introduced in #338. I suspect you and ThomasT may be talking about different P(θ) then ...

 

[lugita15]

Oh, I see. I though you were referring to the same θ as introduced in #338. I suspect you and ThomasT may be talking about different P(θ) then ...

No, I think, at least I hope, that that's not the point of confusion between us.

 

[Passionflower]

Alice and Bob are created in Venice at 10am precisely. Chris and Dale are created in New York precisely (it's just an analogy of an experiment that has actually already been performed and which I referenced earlier)). The polarization of Alice and Dale are immediately checked and they both cease to exist. They never existed in a common region of space time because they were both too far apart.

Bob and Chris are sent to our space station on Mars, where they arrive about 10:03. There, an experimenter decides to entangle them or not. After deciding to entangle, we now have the situation where Alice and Dale were entangled after they were detected, and they never existed in a common area of space time.

Now of course all of the remaining apparati/observers involved were in causal contact with each other previously, no argument about that. What I want to know is by what specific mechanism is it possible for the laser that created Alice and the laser that created Dale supposed to know how to impart a different future result for each, all the while knowing which photons will later be entangled and which ones will not.

If you understand how a laser works you will understand that there is no known distinguishing factor for one photon as compared to another. They are all 100% identical, even as to polarization.

Or maybe it isn't the laser, maybe it is the BBo crystal. But the same question then applies, how does a crystal make it do one thing versus another? By definition, the inputs are identical and the crystal has no active component which is dynamic (changes). So why one result versus another?

So the question is about the mechanism. Where is it? How does it interact with known particles? Maybe we could probe it if you told us what to look for! I think once you go through this exercise a few times, you will realize the stretch you are making. Or you can simply skip my critique and continue to hold onto your (near religious) beliefs, and prove me right as I have said.

How do you propose to entangle Bob and Chris on Mars (without using new photons)?

 

[ThomasT]

But P(θ) just tells the particle whether to go through the polarizer or not. So the only instruction it gives the particle is a yes or a no, or equivalently a 1 or a 0.

That's one way of thinking about it. But the usual way, afaik, is (wrt optical Bell tests) to think of λ, the hidden variable determining individual detection, as the underlying polarization of polarizer-incident optical disturbances. So, one might denote the disturbance incident on polarizer setting a as λ_a, and the same way for the B side ... with the possible values of λ_a (λ_b) being continuous between 0° and 180° (or between 0° and 360°, depending on how you want to frame it). And the same for the values of the polarizer settings, a and b. So that, in line with standard classical and quantum optics, the photon flux at A would be denoted by the function,

cos²(a - λ_a),

and in the same way for the B side.

This LR way of modelling individual detections is compatible with QM. But when it's extended to model the joint (entanglement) context it isn't compatible with QM.

Anyway, the point I'm making here is that confining the values of the hidden variable (that presumably determines individual detection) to the discrete values, 0 and 1, might be the point at which the conclusion that the correlation between θ and rate of coincidental detection is linear becomes logically necessary. I'm not sure.

 

[lugita15]

OK, so let's say a particular entangled photon pair is sent to the two polarizers. When the one of them encounters a particular polarizer setting, it calculates the corresponding value of λ, which is an angle. Now how does it use this angle to decide whether to go through or not? Remember, there must be a deterministic way it does this. Talking about average photon flux doesn't help here, because we're talking about a single photon meeting a single polarizer.

 

[ThomasT]

Just to add to this, the function P(θ), since it is the hidden variable, can be determined by any number of things, including a polarization vector or anything else. But the input of the function must be the polarizer setting, and the output must be a yes-or-no instruction telling the particle to go through or not.

Yes. But wrt a local realistic view, the independent variables, the input, are not just the polarizer orientation, but also the polarization of the incident optical disturbance ... which are continuous within proscribed limits. The output is either the registration of a detection (usually denoted as 1), or a nondetection (usually denoted as 0), wrt any particular coincidence interval (ie., wrt paired detection attributes).

As I mentioned, maybe it's wrt this step that your particular LR line of reasoning necessitates the conclusion that the correlation between θ and rate of coincidental detection is linear.

Though I thought it was Step 5. But I could be mistaken.

I don't see how the inference of a linear correlation follows from Step 4. ...

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.

... because all Step 4. says is that the correlation between θ and rate of coincidental detection is a function of θ. Which leaves open the question of whether or not this is a linear or a nonlinear correlation.

 

[lugita15]

Yes. But wrt a local realistic view, the independent variables, the input, are not just the polarizer orientation, but also the polarization of the incident optical disturbance ... which are continuous within proscribed limits.

I'm including things like polarization vectors in the description of the particular function P for a particular entangled pair, rather than including them as an input of the function. This is just an arbitrary choice in how I'm defining things, so it shouldn't affect anything.

 

[ThomasT]

OK, so let's say a particular entangled photon pair is sent to the two polarizers. When the one of them encounters a particular polarizer setting, it calculates the corresponding value of λ, which is an angle. Now how does it use this angle to decide whether to go through or not?

λ is meant to denote the polarization (angle) of the incident optical disturbance. a (or b) denotes the polarizer setting.

So, from standard optics, individual detection is the function, cos²(a - λ_a), or in the same way for the B side.

-------------------------------

You've asked, quite rightly I think, which of your steps would a more comprehensive local deterministic view disagree with.

It's the step (in your steps) from which a linear correlation between θ and rate of coincidental detection is necessitated.

So, which step, in your opinion, is that?