Understanding bell's theorem: why hidden variables imply a linear relationship?

[harrylin]

Here's a hidden variable model, assuming ideal efficiencies, that produces a nonlinear relationship.

https://www.physicsforums.com/showpost.php?p=3856186&postcount=54

Thanks, that's rather complex, I'll have to study it...
In what way does the model differ from the linear one in Wikipedia?
- http://en.wikipedia.org/wiki/Bell's_theorem#Overview

 

[ThomasT]

Thanks, that's rather complex, I'll have to study it...
In what way does the model differ from the linear one in Wikipedia?
- http://en.wikipedia.org/wiki/Bell's_theorem#Overview

I do think that hidden variables imply a linear relationship with 100% detection and without "data picking".

On second thought, I think you might be right about that.
If you stick to Bell's form, and make ρ(λ) the uniform distribution (all λ equally likely, meaning no 'data picking') then I think you get this:
C(a,b) = ρ(λ) ∫ sign [cos²(a-λ)] sign [cos²(b-λ)] dλ = 1 - ((4θ)/∏) , where θ = |a-b|, the angular difference between the polarizer settings,
for the correlation coefficient (in bold), which, in the words of the OP, is a linear relationship (or linear correlation between θ and rate of coincidental detection).

Now, how does this differ from zonde's treatment at, https://www.physicsforums.com/showpost.php?p=3856186&postcount=54 ?

 

[harrylin]

On second thought, I think you might be right about that.
If you stick to Bell's form, and make ρ(λ) the uniform distribution (all λ equally likely, meaning no 'data picking') then I think you get this:
C(a,b) = ρ(λ) ∫ sign [cos²(a-λ)] sign [cos²(b-λ)] dλ = 1 - ((4θ)/∏) , where θ = |a-b|, the angular difference between the polarizer settings,
for the correlation coefficient (in bold), which, in the words of the OP, is a linear relationship (or linear correlation between θ and rate of coincidental detection).

Now, how does this differ from zonde's treatment at, https://www.physicsforums.com/showpost.php?p=3856186&postcount=54 ?

OK ... and then, was Lugita's correction in #148 perhaps not right? (and if so, why not?). I thought that I knew this, but now I'm confused. Hopefully all this confusion will help to next understand this better.

 

[ThomasT]

OK ... and then, was Lugita's correction in #148 perhaps not right? (and if so, why not?).

I think it was ok. (But I must stress: do not take my word for anything wrt this stuff. I'm trying to sort things out as I go. Eg., I'm apparently missing some, probably simple, thing wrt Herbert's and lugita's proofs, because I still don't understand how they arrive at the conclusions they do.)
You said:

I do think that hidden variables imply a linear relationship with 100% detection and without "data picking".

And lugita replied:

There's a small subtlety here that was covered earlier this thread. You should really use the term "sublinear" or "at most linear". That's because the Bell inequality in (say) Herbert's proof is of the form A is less than or equal to B+C, not of the form A=B+C, which is what's required for linearity.

Which is a more strictly correct way of saying what you said because (I'm assuming) there are other things that can be tweaked besides assumed efficiencies, and data picking which (I assume) can result in different predictions.

But it seems that if Bell's archetypal LR formulation is adhered to, and perfect efficiencies, etc. are assumed, then it predicts a linear correlation between θ and the rate of identical (and nonidentical) paired detections.

I thought that I knew this, but now I'm confused. Hopefully all this confusion will help to next understand this better.

I'm confused too, but I do feel that I'm learning certain, hopefully important, bits of the puzzle from these threads.

 

[lugita15]

But it seems that if Bell's archetypal LR formulation is adhered to, and perfect efficiencies, etc. are assumed, then it predicts a linear correlation between θ and the rate of identical (and nonidentical) paired detections.

No, even if everything is perfect and nothing is tweaked, Bell's inequality is still an INequality, not an equality, so it cannot guarantee linearity. The reason people often say that local determinism implies a linear correlation is that the linear relationship is the one that gives the closest possible agreement with quantum mechanics. The nonlinear relationships allowed by the Bell inequality lead to an even bigger disagreement, so they're uninteresting.

Anyway ThomasT, have you looked over my reasoning in post #142 in going from step 2 to step 3? Is there anything you disagree with or do not understand in that post?

 

[ThomasT]

No, even if everything is perfect and nothing is tweaked, Bell's inequality is still an INequality, not an equality, so it cannot guarantee linearity.

I wasn't referring to a Bell inequality, but rather to the calculation of the correlation coefficient (which is what I take the OP to be about), which, in the ideal, using Bell's form, is a linear function.

The reason people often say that local determinism implies a linear correlation is that the linear relationship is the one that gives the closest possible agreement with quantum mechanics.

In one sense, yes, because the linear correlation matches the QM prediction wrt 3 data points. But in another sense, no, because the nonlinear correlations are nonlinear (cosine) correlations (just as the QM correlation is a nonlinear, cosine, correlation), even though they only match the QM prediction at most, afaik, at one data point.

Anyway ThomasT, have you looked over my reasoning in post #142 in going from step 2 to step 3? Is there anything you disagree with or do not understand in that post?

Lets look at step 2. I wouldn't phrase the inference(s) that might be made given step 1 in that way. Would this make any difference wrt how one eventually frames the situation and what might thereby be inferred about nature from the experimental results?

 

[lugita15]

Lets look at step 2. I wouldn't phrase the inference(s) that might be made given step 1 in that way. Would this make any difference wrt how one eventually frames the situation and what might thereby be inferred about nature from the experimental results?

Well, you said you thought the phrasing in step 2 was a bit too anthropomorphic, and I responded that instead of saying the particles AGREE in advance what angles to both go through and what angles not to, we can instead say that it is DETERMINED in advance what angles the particles would both go through and what angles they would not go through. Given this rephrasing do you see how step 2 follows from step 1? If not, I can explan the reasoning. (The reasoning from step 1 to step 2, by the way, is the famous EPR argument, which predates Bell.)

 

[ThomasT]

Well, you said you thought the phrasing in step 2 was a bit too anthropomorphic, and I responded that instead of saying the particles AGREE in advance what angles to both go through and what angles not to, we can instead say that it is DETERMINED in advance what angles the particles would both go through and what angles they would not go through. Given this rephrasing do you see how step 2 follows from step 1? If not, I can explan the reasoning. (The reasoning from step 1 to step 2, by the way, is the famous EPR argument, which predates Bell.)

Either phrasing allows nonlocal communication between particles.

So, this is what we're on now: what can be inferred from step 1, and be phrased in a non-anthropocentric way, that corresponds to an assumption of locality.

The point, and what I'm wondering about, is exactly where the assumption of locality enters into your and Herbert's proofs.

Your step 2 doesn't qualify as a locality assumption. It just says that whether the photons will or won't be transmitted by the polarizers is determined in advance of the photons' interaction with the polarizers. Such determination might be due to nonlocal transmissions between the paired photons. Your step 2, as stated, doesn't exclude this.

 

[lugita15]

Either phrasing allows nonlocal communication between particles.

So, this is what we're on now: what can be inferred from step 1, and be phrased in a non-anthropocentric way, that corresponds to an assumption of locality.

The point, and what I'm wondering about, is exactly where the assumption of locality enters into your and Herbert's proofs.

Your step 2 doesn't qualify as a locality assumption. It just says that whether the photons will or won't be transmitted by the polarizers is determined in advance of the photons' interaction with the polarizers. Such determination might be due to nonlocal transmissions between the paired photons. Your step 2, as stated, doesn't exclude this.

You're right, there can be some nonlocal theories in which it is determined in advance which angles the photons would both go through and which ones they won't. But Step 2 doesn't say that ONLY local deterministic theories are such that the good and bad angles are determined in advance. It just says that ALL local deterministic theories have the good and bad angles determined in advance. Other kinds of theories may also satisfy this condition, in which case they too will lead to a Bell inequality.

 

[ThomasT]

You're right, there can be some nonlocal theories in which it is determined in advance which angles the photons would both go through and which ones they won't. But Step 2 doesn't say that ONLY local deterministic theories are such that the good and bad angles are determined in advance. It just says that ALL local deterministic theories have the good and bad angles determined in advance. Other kinds of theories may also satisfy this condition, in which case they too will lead to a Bell inequality.

Ok, so your step 2 doesn't provide a criterion for distinguishing local from nonlocal. So how does any step that follows from your step 2 do that?

Suppose we restate the inference (step 2) from step 1 to say that entangeled photons have a relationship due to a local common cause (such as emission from the same atom during the same atomic transition a la Aspect and atomic cascades)? Then how might step 3 be stated? (Keeping in mind that, wrt the restated step 2, we still haven't encoded a locality condition into our line of reasoning.)

Since step 4 is the conclusion, then step 3 has to be the explicitly stated, and uniquely local, locality condition.

What's your locality condition/criterion?

It can't be your current step 3, because it doesn't follow from the known behavior of light wrt crossed polarizers, or from steps 1 and 2.

What I'm, eventually, getting at is that your and Herbert's proofs are insufficient. Bell's analysis is necessary, and sufficient, to show that Bell-LR models of quantum entanglement are ruled out. What it might mean beyond that is, imo, an open question.

 

[Delta Kilo]

The reason people often say that local determinism implies a linear correlation is that the linear relationship is the one that gives the closest possible agreement with quantum mechanics.

In one sense, yes, because the linear correlation matches the QM prediction wrt 3 data points. But in another sense, no, because the nonlinear correlations are nonlinear (cosine) correlations (just as the QM correlation is a nonlinear, cosine, correlation), even though they only match the QM prediction at most, afaik, at one data point.

It's a closest match in terms of smallest difference between the two functions, either absolute or RMS. Let's say we have some arbitrary function F and pin down just two points F(0) = 1 (100% match) and F(∏/2) = 0 (100% mismatch). Then it is easy to show that Bell's theorem requires F(∏/(2n)) ≤ 1 - 1/n, n=1..∞. Points 0, ∏/4 and ∏/2 match QM prediction of cos^2(x) and all other points are strictly below cos^2(x). Behaviour of F between those points can vary in a non-obvious ways, but straight line is still the closest fit.

 

[ThomasT]

It's a closest match in terms of smallest difference between the two functions, either absolute or RMS.

Yes, and thanks for the illustration. My point was only that rate of coincidental detection would be expected, given the known behavior of light, to vary nonlinearly as θ varies, and that a linear correlation would, given the known behavior of light, be unexpected -- in a, presumably, local deterministic world.

 

[lugita15]

Ok, so your step 2 doesn't provide a criterion for distinguishing local from nonlocal. So how does any step that follows from your step 2 do that?

It's true that local deterministic theories are not the only theories that satisfy the condition described in step 2; some nonlocal theories satisfy it as well. But the important point is that ALL local deterministic theories satisfy it. If I wanted to, I could have stated my conclusion as, all local deterministic theories and some nonlocal theories contradict the experimental predictions of QM. But nonlocal theories that satisfy the condition described in step 2 aren't really interesting, so I don't usually talk about them.

Since step 4 is the conclusion, then step 3 has to be the explicitly stated, and uniquely local, locality condition.

No, step 3 is not a locality condition, just a trivial application of the transitive property of equality.

What's your locality condition/criterion?

Strictly speaking, none of my steps constitute a locality condition. Step 2 describes a condition that is satisfied by ALL local deterministic theories, but not ONLY local deterministic theories.

It can't be your current step 3, because it doesn't follow from the known behavior of light wrt crossed polarizers, or from steps 1 and 2.

Step 3 does follow from step 2. If you still disagree, look at my post #142 and tell me what you don't understand or disagree with.

What I'm, eventually, getting at is that your and Herbert's proofs are insufficient.

Why are they insufficient?

 

[lugita15]

ThomasT and I continued this discussion for a bit offline, but now I'm returning to this thread. Here is my umpteenth restatement of http://quantumtantra.com/bell2.html, which hopefully makes clearer the salient points in the reasoning:

1. Entangled photons always have identical behavior when sent through polarizers oriented at identical angles.
2. It is determined in advance what polarizer angles the photons would both go through and what angles the photons would both not go through.
3. If the results at -30° and 0° differ, then either the results at -30° and 0° differ or the results at 0° and 30° differ. (For short we say: If C, then A or B.)
4. P(C)≤P(A or B) (P denotes probability)
5. P(A or B)≤P(A)+P(B)
6. P(C)≤P(A)+P(B)

Where A denotes statement "The result at -30° differs from the result at 0°", B denotes the statement "The result at 0° differs from the result at 30°", and C denotes the statement "The result at -30° differs from the result at 30°".

Let me walk through the proof. Step 1 is an experimental prediction of quantum mechanics, and a pretty well-tested one at that (neglecting loopholes and superdeterminism).

Step 2, probably the most important step, is a consequence of local determinism: if there are no nonlocal influences, then the behavior of photon 1 when hitting a polarizer 1 oriented at angle x cannot depend on the setting y of the distant polarizer 2, so its behavior at any given angle x must be determined in advance. And by step 1, the behavior of photon 1 when sent through a polarizer oriented at angle x must be the same as the behavior of photon 2 when sent through an identically oriented polarizer. So we can meaningfully and unambiguously speak about "the result you would get if you sent either of the photons through a polarizer oriented at angle x", or "the result at x" for short, for all angles x, regardless of what angles (say y and z) we happen to actually orient our polarizers at. This is crucial for the rest of the proof, which utilizes the notion of "the result at x" in the phrasing of statements A, B, and C. In fact, another way of saying step 2 for our purposes is "A, B, and C have well-defined predetermined truth values independent of what angles the polarizers happen to measure at."

Step 3 is a trivial consequence of the transitive property of equality. If the results at -30° and 0° are the same and the results at 0° and 30° are the same, then clearly the results at -30° and 30° must be the same. But this is just another way of saying that if the results at -30° and 30° differ, then the results at -30° and 0° must differ or else the results at 0° and 30° must differ, which is precisely step 3.

The rest is smooth sailing. One of the laws of probability says that if the statement "If M then N" is true (i.e. that N is true in all the cases that M is true.), then we can conclude that the probability of M is less than or equal to the probability of N, from which step 4 follows. Another law of probability says that the probability that M or N is true is less than or equal to the probability that M is true plus the probability that N is true, from which step 5 follows. And then step 6 follows from steps 4 and 5.

But step 6, known as a Bell inequality, is in direct contradiction with the experimental predictions of quantum mechanics. According to QM, P(A)=.25 and P(B)=.25, but P(C)=.75, so we actually have P(C)>P(A)+P(B), in direct contradiction to what we have concluded from the assumption of local determinism.

I think that pretty much the only step that can reasonably be disputed is step 2, because it involves some philosophical discussion.

 

[lugita15]

Now ThomasT has proposed an alternate scenario, in which he says local determinism is in agreement with the predictions of QM, the reasoning above notwithstanding:

Consider a situation involving the production of entangled photon pairs but where polarizers are placed between the emitter and polarizer on each side. Like this:

Alice's detector <--- polarizer a <--- polarizer a_1 <--- Emitter ---> polarizer b_1 ---> polarizer b ---> Bob's detector

Polarizers a_1 and b_1 are kept aligned.

It seems clear to me that when the photons hit a1 and b1, they either both go through or both don't go through since the two polarizers are oriented at the same angle. Either way, their entanglement is broken, and they acquire a state of definite polarization either parallel or perpendicular to the orientation of a1. Assuming they end up polarized parallel, they will go through the polarizers and reach the second pair of polarizers a and b. Then the probability for each of the photons to go through its respective polarizer is the cosine squared of the difference between the polarization angle of the photon and the setting of the polarizer it now encounters. But now the photons are no longer entangled, so these probabilities are independent of each other: even if a and b have the same orientation, it need not be true that the two photons must do the same things. Instead, each photon has an independent probability cos²(x-x1) of going through, where x is the angle setting of polarizers a and b, and x1 is the angle of polarizers a1 and b1.

Thus since we no longer have identical behavior at identical angles a and b, we can no longer speak about "the result at x" without regard to which of the two particles we're talking about, and thus we can no longer say "If C then A or B" and the rest of our reasoning, so for this scenario the logic of the proof does not prevent local determinism from matching the experimental predictions of QM.

 

[San K]

But now the photons are no longer entangled, so these probabilities are independent of each other: even if a and b have the same orientation, it need not be true that the two photons must do the same things. Instead, each photon has an independent probability cos²(x-x1) of going through, where x is the angle setting of polarizers a and b, and x1 is the angle of polarizers a1 and b1.

well explained lugita.

can we re-entangle them via entanglement swapping (prior to interaction with the second set of polarizers)?

I ask because, I feel that, this modification in the experiment might result in a stronger proof of the entanglement phenomena.

btw - the above experiment, as suggested by Thomas, provides a more convincing proof as well

 

[ThomasT]

Strictly speaking, none of my steps constitute a locality condition.

Ok, so isn't it necessary to include some sort of explicit locality assumption in order to say that your line of reasoning proves that the predictions of QM contradict local determinism?

Taking step 1 as the determinism assumption (ie., determinism means that identical antecedent conditions always produce the same results), then would it be ok to state step 2 as, eg., the assumption that the events on one side of the experiment do not affect the events on the other side?

... we no longer have identical behavior at identical angles a and b ...

Ok, you're saying that in the alternate setup that identical settings wrt the analyzing polarizers (polarizers a and b aligned) would not always produce identical results (identical results means that either both detectors register detection or both do not register detection wrt any particular coincidence window).

I'm saying that in the alternate setup you'd still get identical results at identical settings (polarizers a and b aligned), but that the rate of identical detection would be reduced.

 

[lugita15]

Ok, so isn't it necessary to include some sort of explicit locality assumption in order to say that your line of reasoning proves that the predictions of QM contradict local determinism?

The only thing I meant when I said that I don't have a locality condition in the strictest sense is that, it's possible for some (dumb) nonlocal theories to also satisfy the conditions I lay out in the proof. You can have a theory in which there are such things as FTL signals, but it just happens to be the case that photon behavior is independent of distant polarizer settings, not because of relativity, but just because that's the nature of photons. Such a nonlocal theory would satisfy the conditions of the proof, and would thus also be ruled out. So the proof rules out all local deterministic theories and some nonlocal theories as well, but the nonlocal theories it rules out are so uninteresting that I don't usually mention them.

Taking step 1 as the determinism assumption (ie., determinism means that identical antecedent conditions always produce the same results), then would it be ok to state step 2 as, eg., the assumption that the events on one side of the experiment do not affect the events on the other side?

It doesn't matter to me too much how you characterize the two assumptions. The important thing is that you agree that a local determinist must believe in both of them, so that we can unambiguously talk about "the result at x" independent of which of the two photons we're talking about and which angles y and z the polarizers are oriented at. Still, if you want to know how I characterize things, I would say that step 1 is essentially an experimental prediction of QM. I think that saying the two particles have the same antecedent conditions, and thus that step 1 is a consequence of determinism, is a bit too restrictive an assumption, but that doesn't matter one way or another for our purposes. As far as step 2 goes, I think it makes sense to call it an assumption of local determinism, because the important thing is that somehow or other we get to the conclusion that "the result at x" is determined in advance for all x.

Ok, you're saying that in the alternate setup that identical settings wrt the analyzing polarizers (polarizers a and b aligned) would not always produce identical results (identical results means that either both detectors register detection or both do not register detection wrt any particular coincidence window).

That is precisely what I'm saying. Don't you agree that as soon as their polarizations are measured by a1 and b1, the entanglement is broken off and their quantum states of the two photons become definite polarization states?

I'm saying that in the alternate setup you'd still get identical results at identical settings (polarizers a and b aligned)

What is your basis for saying this? This is a simple quantum mechanics problem in which there should be no dispute. What do you disagree with in my analysis of the problem? I could replace words with math if you like.

 

[ThomasT]

That is precisely what I'm saying. Don't you agree that as soon as their polarizations are measured by a1 and b1, the entanglement is broken off and their quantum states of the two photons become definite polarization states?

Afaik, the disturbances transmitted by polarizers a1 and b1 and incident on polarizers a and b don't have to be entangled in order for you to get identical results when a and b are aligned. Just that with a1 and b1 in the setup you get a reduced rate of identical detection.

If that's not correct, then what's the predicted rate of different results in the alternate setup?

Wrt the first two steps, ok, so step 2 includes the assumptions of locality and determinism, and step 1 is necessary to get to step 3. They're interchangeable.

I think something like the following would be clearer:

1. Assumption: Determinism holds. (ie., identical antecedent conditions always produce the same results)
2. Assumption: Locality holds. (ie., events on one side of the experiment don't affect events on the other side)

But I understand how you get to 3 from your 1 and 2.

 

[lugita15]

Afaik, the disturbances transmitted by polarizers a1 and b1 and incident on polarizers a and b don't have to be entangled in order for you to get identical results when a and b are aligned.

And what's your basis for saying that?

If that's not correct, then what's the predicted rate of different results in the alternate setup?

Let me lay out exactly what the probabilities look like. First, you start out with the photons in an entangled state encounteering polarizers a1 and b1 oriented at the same angle x1, so they either both go through or they both do not go through, and it is a 50-50 chance which of these occurs. If they don't go through, our story ends there and nothing reaches or goes through polarizers a and b. If they go through, they are now each in a quantum state with definite polarization in the x1 direction. They behave like any other photons which have come out of an x1-oriented polarizer, regardless of the fact that they were entangled at one point. So when photon 1 encounters polarizer a, oriented at angle x, it operates according to Malus' law and goes through with probability cos^2(x-x1). And similarly when photon 2 encounters polarizer b, also oriented at angle x, it also goes through with probability cos^2(x-x1). These are independent probabilities, since the photons are now unconnected.

Thus the probability that neither photon makes it through to the end is .5+.5sin^4(x-x1), and the probability that both photons make it through to the end is .5cos^4(x-x1). And clearly these probabilities do not add up to 1, except when x and x1 are the same or they are 90 degrees apart.

Wrt the first two steps, ok, so step 2 includes the assumptions of locality and determinism, and step 1 is necessary to get to step 3.

Good, at least we're agreed on that.

But I understand how you get to 3 from your 1 and 2.

OK, so you agree that a local determinist cannot possibly disagree with my two assumptions, and you agree that my assumptions are sufficient to get to step 3. And you agreed earlier that from step 3 onward everything is inevitable. So then what is the source of our current disagreement? Do you just believe that the proof has to have some flaw because you think it would work equally well in your alternate scenario? If so, can you present your analysis of the scenario and show how you think it has identical behavior at identical angle settings? I can present my analysis in greater detail if you like.

 

[ThomasT]

And what's your basis for saying that?

DrC mentioned in a few threads that there are PDC setups where photons not entangled in polarization give identical results when the analyzing polarizers are aligned, but at other settings the results differ from entanglement results. I vaguely remember a paper that actually recounted such an experiment, but don't have the reference. Maybe DrC will weigh in on this. Calling DrC. (I'll send him a PM on this if he hasn't posted in this thread in the next day or so.)

I just tried to present a simpler situation where, I thought, you'd also get identical results at identical polarizer settings even though the disturbances incident on the analyzing polarizers aren't entangled in polarization.

Thus the probability that neither photon makes it through to the end is .5+.5sin^4(x-x1), and the probability that both photons make it through to the end is .5cos^4(x-x1). And clearly these probabilities do not add up to 1, except when x and x1 are the same or they are 90 degrees apart.

We're only talking about the situation where x and x1 are the same (where, equivalently, x-x1 = 0, where Theta = 0, where polarizers a and b are aligned).

So, what you wrote seems to be in agreement with the prediction that the alternate setup will produce identical results at identical settings of the analyzing polarizers.

OK, so you agree that a local determinist cannot possibly disagree with my two assumptions, and you agree that my assumptions are sufficient to get to step 3. And you agreed earlier that from step 3 onward everything is inevitable.

Yes.

So then what is the source of our current disagreement? Do you just believe that the proof has to have some flaw because you think it would work equally well in your alternate scenario?

If your step 1 applies to nonentanglement setups where the local deterministic (LD) predictions agree with the QM predictions, but where both of those disagree with your prediction, then it would seem that something is wrong in saying that your proof proves that local determinism is incompatible with QM.

If you instead state your local determinism assumptions this way:

1. Assumption: Determinism holds. (ie., identical antecedent conditions always produce the same results)
2. Assumption: Locality holds. (ie., events on one side of the experiment don't affect events on the other side)

Then, do these assumptions lead to,

If C, then A or B ?

If so, then why not just do it that way?

 

[billschnieder]

1. Entangled photons always have identical behavior when sent through polarizers oriented at identical angles.
2. It is determined in advance what polarizer angles the photons would both go through and what angles the photons would both not go through.
3. If the results at -30° and 0° differ, then either the results at -30° and 0° differ or the results at 0° and 30° differ. (For short we say: If C, then A or B.)
4. P(C)≤P(A or B) (P denotes probability)
5. P(A or B)≤P(A)+P(B)
6. P(C)≤P(A)+P(B)

Where A denotes statement "The result at -30° differs from the result at 0°", B denotes the statement "The result at 0° differs from the result at 30°", and C denotes the statement "The result at -30° differs from the result at 30°".

As usual, the error is hidden in the fuzzy description of the experiment. What is the experiment we are talking about? More specifically, what does "The result at -30° differs from the result at 0°" mean? Let us examine this more closely.

It appears we are comparing an outcome at -30° with an outcome at 0°, for example we have an entangled pair of photons one heads of to the -30 instrument and another to the 0 instrument and we compare if the outcomes match. This makes sense. But then you talk of "The result at 0° differs from the result at 30°". Where did 30° come from? We had two entangled photons which we measured at -30 and 0, all of a sudden 30 appears which would suggest one of the following possibilities:

a) You were magically able to measure two photons at 3 angles (-30°, 0°, +30°)
b) You measured two different pairs of entangled photons.
c) You are not talking about an actual performable experiment but about a hypothetical theoretical what might have been for a single pair.

So which is it? Now let us examine your claim (3).

3. If the results at -30° and 0° differ, then either the results at -30° and 0° differ or the results at 0° and 30° differ. (For short we say: If C, then A or B.)

This statement is true ONLY for the possibilities (a) and (c) above. However any experiments that could ever be performed are of the type (b). QM predictions are for type (b) experiments. It is therefore not surprising that inequalities obtained for (a) and (c) contradict an experiment performed as (b). A type (b) experiment necessarily requires the use of a different set of entangled particles to measure the probabilities of A, B, C. As a result, we have 3 different Kolmogorov probability measures which can not and should not be combined in a single probability expression as has been done in (4). In other words, there is no physical reason to expect the outcome of one set of photons to place constraints on the outcome of a different set of photons in scenario (b), however two outcomes from the same set of photons like in (a) and (c) can legitimately place constrains on a third outcome from the exact same set!

So let me summarize: (3) and (4) are valid ONLY for case (a) and (c) and NOT for case (b). QM predictions and every possible experiment that can ever be performed falls under case (b). Two outcomes from a single set of particles CAN place constraints on a third outcome from the very same set of particles. However, there is no basis in physics or logic for two outcomes from one set of particles to constrain the outcome of a different set of particles not correlated with the first one!

If this is still not clear, I would encourage that you re-read the above bolded paragraph and think about it for a moment. It might also be useful to read-up on "degrees of freedom".

It continues to amaze me that this fact escapes a great majority, and even more sad is the concerted effort to prevent others from understanding, this by those who should know better. I'm tired

 

[DrChinese]

Where did 30° come from? We had two entangled photons which we measured at -30 and 0, all of a sudden 30 appears which would suggest one of the following possibilities:

a) You were magically able to measure two photons at 3 angles (-30°, 0°, +30°)
b) You measured two different pairs of entangled photons.
c) You are not talking about an actual performable experiment but about a hypothetical theoretical what might have been for a single pair.

So which is it? Now let us examine your claim (3).

This statement is true ONLY for the possibilities (a) and (c) above. However any experiments that could ever be performed are of the type (b). QM predictions are for type (b) experiments. It is therefore not surprising that inequalities obtained for (a) and (c) contradict an experiment performed as (b). A type (b) experiment necessarily requires the use of a different set of entangled particles to measure the probabilities of A, B, C. As a result, we have 3 different Kolmogorov probability measures which can not and should not be combined in a single probability expression as has been done in (4). In other words, there is no physical reason to expect the outcome of one set of photons to place constraints on the outcome of a different set of photons in scenario (b), however two outcomes from the same set of photons like in (a) and (c) can legitimately place constrains on a third outcome from the exact same set!

...

It continues to amaze me that this fact escapes a great majority, and even more sad is the concerted effort to prevent others from understanding, this by those who should know better. I'm tired

This is called realism. That would make it half of local realism, and it is part and parcel of why local realism is rejected post Bell.

Bill, once again you are posting your personal theories which do not follow the mainstream. According to the rules here, you should not post:

"Challenges to mainstream theories (relativity, the Big Bang, etc.) that go beyond current professional discussion"

You need to be citing references for your claims that are from acceptable sources, and we both know that cannot happen as you are basically fringe. You say as much in your last paragraph. I too am tired of following behind you to clean up your misleading statements for casual readers of this and other threads.

Once again, cease or I will report you as I did last time. It is a fact that there are plenty of threads for you to hijack. Please do not make us watch your every post for fringe comments like this. You are an intelligent person who could be helpful here if you would skip the soapbox. This is not the place for fringe comments. Try an unmoderated board instead.

 

[DrChinese]

DrC mentioned in a few threads that there are PDC setups where photons not entangled in polarization give identical results when the analyzing polarizers are aligned, but at other settings the results differ from entanglement results. I vaguely remember a paper that actually recounted such an experiment, but don't have the reference. Maybe DrC will weigh in on this. Calling DrC. (I'll send him a PM on this if he hasn't posted in this thread in the next day or so.)

I think the situation you are referring to is as follows:

With a single Type I PDC crystal, you get pairs of photons coming out which have the properties of being entangled on some bases (momentum for example) but are NOT polarization entangled. The input is V> and the output is the pair H>H>. Therefore they match ONLY at the special angles 0/90/etc. They do not match at 45 degrees (for both) or any other angle, instead they follow Product state statistics.

My apologies if something I said implied otherwise.

 

[billschnieder]

I will repeat myself for those who still were not paying attention:

(3) and (4) are valid ONLY for case (a) and (c) and NOT for case (b). QM predictions and every possible experiment that can ever be performed falls under case (b). Two outcomes from a single set of particles CAN place constraints on a third outcome from the very same set of particles. However, there is no basis in physics or logic for two outcomes from one set of particles to constrain the outcome of a different set of particles not correlated with the first one!

If DrC has something meaningful to say, he will respond to what I said with reasoned arguments why he thinks the above is false.

 

[lugita15]

DrC mentioned in a few threads that there are PDC setups where photons not entangled in polarization give identical results when the analyzing polarizers are aligned, but at other settings the results differ from entanglement results.

As DrChinese said, he was talking about identical behavior when both polarizers are oriented at a particular special angle. That's very different from identical behavior at identical angles for all angles.

I just tried to present a simpler situation where, I thought, you'd also get identical results at identical polarizer settings even though the disturbances incident on the analyzing polarizers aren't entangled in polarization.

Well, your scenario doesn't seem to work, and it seems like any such scenario cannot work even in principle.

We're only talking about the situation where x and x1 are the same (where, equivalently, x-x1 = 0, where Theta = 0, where polarizers a and b are aligned).

I think you're misunderstanding what x and x1 are. I'm defining x1 to be the (common) orientation of polarizers a1 and b1, and x to be the (common) orientation of a and b.

So, what you wrote seems to be in agreement with the prediction that the alternate setup will produce identical results at identical settings of the analyzing polarizers.

No, I'm not in agreement with that, unless by "identical settings" you mean that all four polarizers are aligned, and I doubt you mean that. But I maintain that if polarizers a1 and b1 are oriented at x1, and polarizers a and b are oriented at x, where x and x1 are different, then you no longer have to get identical behavior.

If your step 1 applies to nonentanglement setups where the local deterministic (LD) predictions agree with the QM predictions, but where both of those disagree with your prediction, then it would seem that something is wrong in saying that your proof proves that local determinism is incompatible with QM.

But it seems clear to me that step 1, identical behavior at identical angles for all angles, is a special property of quantum entanglement, so the logic doesn't work for nonentanglement setups.

If you instead state your local determinism assumptions this way:

1. Assumption: Determinism holds. (ie., identical antecedent conditions always produce the same results)
2. Assumption: Locality holds. (ie., events on one side of the experiment don't affect events on the other side)

Then, do these assumptions lead to,

If C, then A or B *?

These assumptions would suffice to get to "If C, then A or B" if we also assume that the two entangled particles have identical antecedent conditions, which I think is an overly restrictive assumption. I think it's cleaner just to assume identical behavior at identical angles, since it's an experimental prediction of QM.

 

[Delta Kilo]

Strangely, I am forced to agree with Bill on that. Step 3 of the 'proof' in #164 is only applicable to case (c), that is hypothetical theoretical result for a single pair which cannot be directly measured. The transition to step 4 of the proof is not made explicit and as such is wide open to the kind of attacks we see here.

The missing ingredient is the probability distribution ρ(λ), independent of measurement angles. Having stable probability distribution which does not change from one experiment to another is what holds the whole thing together. It brings the necessary repeatability and allows one to estimate expectation values for different angles in separate experiments.

Frankly, the math of Bell's theorem is so simple, I don't see a point in trying to avoid it. It only leads to unnecessary confusion.

 

[DrChinese]

Strangely, I am forced to agree with Bill on that. Step 3 of the 'proof' in #164 is only applicable to case (c), that is hypothetical theoretical result for a single pair which cannot be directly measured. The transition to step 4 of the proof is not made explicit and as such is wide open to the kind of attacks we see here.

The missing ingredient is the probability distribution ρ(λ), independent of measurement angles. Having stable probability distribution which does not change from one experiment to another is what holds the whole thing together. It brings the necessary repeatability and allows one to estimate expectation values for different angles in separate experiments.

Frankly, the math of Bell's theorem is so simple, I don't see a point in trying to avoid it. It only leads to unnecessary confusion.

Now he has managed to derail you too. LOL. You have to remember that Bill's real point has nothing to do with lugita15's proof, he is trying to say that counterfactual reasoning does not have a physical basis and also does not have a basis in QM. Both of these go without saying, counterfactual cases are not physical by definition. But they clearly exist in the local realistic world, which is what Bell rules out. lugita15's example is perfect, and Bill is using intentionally convoluted logic to push his desired conclusion.

Obviously, lugita15's example follows Bell and Bell should not be in question here. Look at the title of the thread and you will see what a good job Bill has done with his latest hijack.

 

[ThomasT]

I think the situation you are referring to is as follows:

With a single Type I PDC crystal, you get pairs of photons coming out which have the properties of being entangled on some bases (momentum for example) but are NOT polarization entangled. The input is V> and the output is the pair H>H>. Therefore they match ONLY at the special angles 0/90/etc. They do not match at 45 degrees (for both) or any other angle, instead they follow Product state statistics.

My apologies if something I said implied otherwise.

Ok, thanks. I had misunderstood the situation. Not your fault.

A bit off topic, but I'm curious, wrt this situation how do the results vary for other orientations (the orientations that don't always produce identical results) of the aligned polarizers?

 

[ThomasT]

I think you're misunderstanding what x and x1 are. I'm defining x1 to be the (common) orientation of polarizers a1 and b1, and x to be the (common) orientation of a and b.

Ok. I misunderstood.

I maintain that if polarizers a1 and b1 are oriented at x1, and polarizers a and b are oriented at x, where x and x1 are different, then you no longer have to get identical behavior.

Ok. When x and x1 are aligned, then you'd always get identical results, and as x is offset from x1 then the percentage of identical results would vary as cos²|x-x1|. Is that correct?

Anyway, it appears that my critique of your argument using the alternate setup doesn't work.

But it seems clear to me that step 1, identical behavior at identical angles for all angles, is a special property of quantum entanglement, so the logic doesn't work for nonentanglement setups.

Yes, that seems to be the case. Thanks.

As far as I understand then, your argument does seem valid wrt showing that, wrt entanglement setups, the predictions of QM are incompatible with the assumption(s) of local determinism.

But I haven't thoroughly considered bill's or DK's objections yet.