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

[lugita15]

They must all have some importance, otherwise I suppose that you wouldn't bother expressing them.

Yes, it's just that step 3 is a relatively trivial and unimportant step, at least to my mind.

No, that's not in dispute. What's in dispute is the manner in which some have chosen to restrict the argument. Is it possible that the transitive property of equality expressed in terms of things that we can count at our level of macroscopic apprehension might have nothing to do with locality/nonlocality in a realm of behavior removed from our sensory apprehension and, presumably, underlying instrumental behavior -- at least wrt the way that the dilemma has so far been framed?

In step 3, I'm not "restricting the argument" or assuming anything at all about locality, nonlocality, or independence. All I'm doing is applying the transitive property of equality.

There's at least one other way of conceptualizing the reason for identical detection attributes at identical settings. Namely, that the separated polarizers are analyzing, filtering exactly the same thing wrt any given pair of entangled particles. In which case, the expected result would be in line with the QM predictions and Malus Law.

OK, but whatever you're talking about it has absolutely nothing to do with step 3.

I think so too. But you're the one who's including determinism in this.

Yes, and determinism to me means that the future can be determined with complete certainty given the present.

I'm not aware of any contention or hypothesis of instantaneous action at a distance associated, by Newton, with the relationships that his equations specify. For those who want to infer nonlocality from the equations, then that's on them. The equations express an observationally confirmed relationship. Is it possible that that relationship might be due to local interactions/transmissions? Yes, of course it is, in the sense of gravitational systems.

Again, this is irrelevant for our discussion, but if Newton's theory of gravitation were correct, we could use it to send messages instantaneously: just move around a mass here, and the gravitational field all over the universe would be immediately measured to have a change.

Yes, insofar as dBB is interpreted to explicate nonlocality, then it's nondeterministic. Just relational, just as standard QM is relational, not causal.

I think your view of determinism is not how the term is generally understood.

But that's where it takes a particular form that must affect the conclusion. Simply assuming locality, in terms of independence, is inconsequential until that assumption is put into a form that will impact the reasoning or the experimental predictions.

As I said, in step 3 I am not at all putting the assumption of locality or independence into any form. I am not invoking such notions in any way. All I am doing is starting from step 2, which says that that the particles have agreed on what angles to go through, and applying the transitive property of equality.

Yes, but step 2 doesn't put it into a form that will impact the reasoning. Step 3 does that.

No, step 3 does nothing of the sort.

Then again, I suppose you could say that step 2 in some sense implies step 3.

Yes, it certainly does.

So, maybe we should look more closely at step 2. The way it's stated is rather ... pedestrian and a bit too anthropomorphic, I must say. What are some other ways of stating the inference(s) that might be drawn from step 1?

I agree that the phrasing in step 2 is a little anthropomorphic, but we can easily change the phrasing without changing the meaning. For instance, instead of saying that the particles have AGREED in advance what angles to go through and not to go through, we can say that it is DETERMINED in advance what angles both particles will go through and what angles they will not go through.

 

[harrylin]

Neither am I. DrC is pretty familiar/fluent wrt the experiment and simulation you mentioned. I think he might agree with:
"Since the correlation is perfect at θ = 0° for the entangled pairs that are detected, then I see no reason to assume that it would be different if all entangled pairs could be detected."
But I don't know.

Hi I'm pretty sure that the experimental results to which I referred disqualified that statement in the way as I indicated. But apparently my example wasn't sufficiently clear, so I'll rephrase it.

Many pairs that were detected in Weih's experiment, were interpreted as "non-entangled"; removing those from the analysis yielded a result conform the prediction of QM, while including them yielded a different result. "Local realistic" simulations were shown to be capable of matching all those results.

Because of that kind of subtleties my comment was (and still is):
It seems to me that here (that is, in your above-mentioned comment) is a partial misunderstanding, for there is a "twist" on this: the correlation may be perfect for those pairs that are called "entangled pairs".

 

[harrylin]

[..] Let me say this right now. I feel pretty certain that ttn (Travis Norsen), DrC (David Schneider), zonde, lugita, Demystifier (Nikolic), billschnieder, Gordan Watson, unusualname, harrylin (and anybody I left out) and all the other contributors to this and other 'Bell' threads know a lot more about this stuff than I do. [..]

I thought the same of you and others! The only thing I do in this group while I'm trying to learn more by listening is to give now and then my 2cts which I happened to pick up elsewhere...

 

[DrChinese]

Many pairs that were detected in Weih's experiment, were interpreted as "non-entangled"; removing those from the analysis yielded a result conform the prediction of QM, while including them yielded a different result. "Local realistic" simulations were shown to be capable of marching all those results.

Because of that kind of subtleties my comment was (and still is):
It seems to me that here (that is, in your above-mentioned comment) is a partial misunderstanding, for there is a "twist" on this: the correlation may be perfect for those pairs that are called "entangled pairs".

Yes, that is true. And as you widen the window, you get a lower correlation rate.

But we wouldn't expect perfect correlations from pairs that are not entangled, would we! (Unentangled pairs have a match rate closer to 75%) It is pretty clear that we need some way to define what is an entangled pair. That definition is a time coincidence window. The window ultimately defines the correlation, not the other way around. Logically, pairs in which one arrives quite late might be suspect as to whether they are still polarization entangled. On the other hand, no source is perfect.

Please note that it is also possible to convert the same source into entangled pairs that are NOT polarization entangled. Using Type I PDC, simply align both crystals identically and they will produce pairs with known polarization in the Product State. You can then look at that sample and see that the time coincidence window is reasonable (since you will see the same distribution of times).

Ultimately, you only get Bell state stats with entanglement. It would not be reasonable to include pairs that are not entangled if you can avoid it.

 

[harrylin]

Yes, that is true. And as you widen the window, you get a lower correlation rate.
[..]
Ultimately, you only get Bell state stats with entanglement. It would not be reasonable to include pairs that are not entangled if you can avoid it.

Obviously; I certainly would not suggest the contrary!

It was merely to illustrate that the argument that Thomas presented can look good due to lack of knowledge of the very thing that it is about.
Another example that is less close to home: it could have looked good to state over a century ago that since Newton's mechanics work so well, we see no reason to assume that it doesn't work for MMX and the extremely unlikely possibility of length contraction is a loophole that soon will be closed.

 

[ThomasT]

@ DrC and harrylin,

Regarding the statement:

Since the correlation is perfect at θ = 0° for the entangled pairs that are detected, then I see no reason to assume that it would be different if all entangled pairs could be detected.

The situation seems to be that for entangled pairs that are detected, then the correlation is perfect at θ = 0°. Is this the case?

If so, then if detection and pairing efficiencies were perfect, then would you expect anything to be different regarding experimental BI violation and the incompatibility of the predictions of QM and LR?

 

[DrChinese]

@ DrC and harrylin,

Regarding the statement:

The situation seems to be that for entangled pairs that are detected, then the correlation is perfect at θ = 0°. Is this the case?

If so, then if detection and pairing efficiencies were perfect, then would you expect anything to be different regarding experimental BI violation and the incompatibility of the predictions of QM and LR?

I wouldn't expect anything to be different, no. As we move towards better setups, the number of standard deviations of violation should increase. It does. IIRC, Aspect was about 5, Weihs et al was 30 and we are over 100 now in some experiments.

 

[San K]

As we move towards better setups, the number of standard deviations of violation should increase.

did not get this.

are you saying that the (photons) violation of BI will increase with better instruments?

i.e. we will detect photons that violated BI with even greater degree?

however don't they have to be within the cosine curve?

 

[DrChinese]

did not get this.

are you saying that the (photons) violation of BI will increase with better instruments?

i.e. we will detect photons that violated BI with even greater degree?

Sort of. Keep in mind that there are a lot of components along the way from the source to the detector. We would like to catch and detect as many pairs as possible. Early setups had problems producing a good stream of pairs. Also problems detecting them reliably. Clearly, the issue is that we don't want there to be some kind of hidden bias in the mechanism. But in a perfect world, I would expect better detection of bigger streams to lead to higher deviations from the related BI boundary value.

 

[jtbell]

As we move towards better setups, the number of standard deviations of violation should increase. It does. IIRC, Aspect was about 5, Weihs et al was 30 and we are over 100 now in some experiments.

did not get this.

are you saying that the (photons) violation of BI will increase with better instruments?

i.e. we will detect photons that violated BI with even greater degree?

As experimental setups get better and we collect more data, the uncertainty in our measurements (measured in terms of "standard deviation") becomes smaller. Pulling numbers out of my hat, suppose the Bell inequality requires local realistic theories to predict x > 0.7 for a particular setup.

Now suppose we actually measure x = 0.5 ± 0.2 where 0.2 is the standard deviation of our measurements. We've violated the BI by one standard deviation. Most physicists would consider that suggestive, but not very conclusive.

We improve our measurements and collect more data, and we now get x = 0.51 ± 0.02. It's consistent with our previous measurement in absolute terms, but now we've violated the BI by about ten standard deviations. Most physicists would consider this to be very significant. As DrC has pointed out, some experiments have actually done much better than this.

 

[San K]

the uncertainty in our measurements (measured in terms of "standard deviation") becomes smaller.

...yes that is what i was asking, thanks jtbell. so the uncertainty becomes smaller...vastly improving credibility of the experiment/hypothesis...agreed..

however stays within the cosine curve?

 

[DrChinese]

however stays within the cosine curve?

Yup, tighter and tighter.

 

[ThomasT]

did not get this.

are you saying that the (photons) violation of BI will increase with better instruments?

i.e. we will detect photons that violated BI with even greater degree?

however don't they have to be within the cosine curve?

What DrC and jtbell said, and here's my two cents since it was my statement that was being questioned.

One can calculte proximity to some BI associated with some experimental setup assuming that everything is perfect, and those calculations show that Bell LR models satisfy BIs by a certain amount and QM violates BIs by a certain amount.

My statement in question was basically that if everything was perfect, then that wouldn't be expected to change.

But everything isn't perfect. A Bell inequality associated with a particular experiment (eg., Aspect 1982) might express something like S <= 0. The average for a number of runs was S_exp = .101 ± .020, with the experiment violating the BI by 5 standard deviations.

Regarding agreement between the QM predictions and the observed results, there's usually some slight difference. For example, wrt the above experiment the QM prediction was S_qm = .112 .

Improving efficiencies in Bell tests both decreases the standard deviation, and increases the agreement between the QM curve and the result curve.

 

[San K]

thanks DrC and ThomasT

- can we think of an experiment (that/which uses a different logic/aspect of reality) other than Bell's test that supports QE?

- for entanglement it occur, do the two particles have to always physically (locally) interact first?

- what is swapping of entanglement between two pairs of photons?

the below is not important/central to our discussion/goal (there are still some parts of BI that I have not read yet):

- why is it convenient/easier to discuss BI in terms of mismatches rather than matches?
- why do we deal with only three orientations (0, 120, 240)? --- this i guess is just for illustration purposes, we could go with more than 3
- why will the photon always pass through exactly 2 of the 3 orientations?

 

[harrylin]

@ DrC and harrylin,

Regarding the statement:

The situation seems to be that for entangled pairs that are detected, then the correlation is perfect at θ = 0°. Is this the case?

If so, then if detection and pairing efficiencies were perfect, then would you expect anything to be different regarding experimental BI violation and the incompatibility of the predictions of QM and LR?

The situation seems to be that for those pairs that are detected and labeled "entangled pairs" after following a tight data selection procedure, the correlation is nearly perfect for θ = 0°.
In one LR model to which I already referred - if I understood it correctly - then if detection and pairing efficiencies were 100% (which may be impossible according to the model) and with correct labeling according to that model, the result would be different from QM.

 

[ThomasT]

- can we think of an experiment (that/which uses a different logic/aspect of reality) other than Bell's test that supports QE?

Not sure what you mean. Are you asking whether there are quantum experiments that produce quantum entanglement that aren't specifically designed to be tests of a Bell inequality?

- for entanglement it occur, do the two particles have to always physically (locally) interact first?

Afaik, no. But there has to be some common, locally propagated, influence, or force, or torque ... whatever, introduced and interacting with certain entities (and it might be thousands, even millions, of atoms) that produces an entanglement relationship between those entities.

- what is swapping of entanglement between two pairs of photons?

Your most recent questions are beyond the scope of the subject of this thread. Start a new thread, and hopefully some more knowledgeable people will reply to your question(s). But first do a search of PF and arxiv.org articles on this.

- why is it convenient/easier to discuss BI in terms of mismatches rather than matches?

I don't know that it is. Afaik, Bell test results are usually reported in terms of coincidental matches not coincidental mismatches.

I suppose you might be referring to Herbert's proof, which, to me, makes no sense. I mean that wrt what he says his proof proves. Herbert says that his proof proves that nature is nonlocal. Which, imho, is just silly. Wrt Herbert's proof proving that an LR model of entanglement is incompatible with QM, then, yes, it is one way of demonstrating that. It's also a way of demonstrating that a particular expression of locality is incompatible with experimental results. The reason(s) why this is not a disproof of locality in nature wrt quantum entanglement, or support for nonlocality in nature wrt quantum entanglement are subtle, pertaining to the relationship between an LR-restricted formalism and an experimental design, and beyond the scope of this thread.

- why do we deal with only three orientations (0, 120, 240)? --- this i guess is just for illustration purposes, we could go with more than 3

Well, three is all you need. But it doesn't have to be those particular settings. In fact, if you want to go with Herbert's proof, then you just need two values of θ ... some θ, and then 2θ.

- why will the photon always pass through exactly 2 of the 3 orientations?

Don't know what you mean. Photons don't always pass through 2 different orientations. Sometimes the result is 1,0 or 0,1. Anyway, since there are only 2 possible orientations in a given trial, ie., the settings of the polarizers a and b, then what are you referring to by "3" orientations?

 

[ThomasT]

The situation seems to be that for those pairs that are detected and labeled "entangled pairs" after following a tight data selection procedure, the correlation is nearly perfect for θ = 0°.

Ok.

In one LR model to which I already referred - if I understood it correctly - then if detection and pairing efficiencies were 100% (which may be impossible according to the model) and with correct labeling according to that model, the result would be different from QM.

Well, yeah. The calculational result is different from QM now, wrt both practical and presumed ideal situations. It's always going to be different. What's your point?

 

[harrylin]

Ok.
Well, yeah. The calculational result is different from QM now, wrt both practical and presumed ideal situations. It's always going to be different. What's your point?

My point was to answer your question and regretfully I forgot what your point was.
Anyway, getting back to the topic: I guess that in such an alternative interpretation of "ideal situation", the relationship will be linear (both in LR theory and in Weih's experimental data).

 

[ThomasT]

My point was to answer your question and regretfully I forgot what your point was.

Well, that was wrt a statement that was a bit off topic. So, nevermind.

Anyway, getting back to the topic: I guess that in such an alternative interpretation of "ideal situation", the relationship will be linear (both in LR theory and in Weih's experimental data).

You might be right about that. I don't know.

In which case I might be wrong in saying to the OP that hidden variables don't imply a linear correlation between θ and rate of coincidental detection. But the fact of the matter, afaik, is that they don't. Simply due to the fact that there are LR models of quantum entanglement which predict a cosine, not a linear, correlation between θ and rate of coincidental detection. Even DrC will agree with this. Ask him.

 

[San K]

Not sure what you mean. Are you asking whether there are quantum experiments that produce quantum entanglement that aren't specifically designed to be tests of a Bell inequality?

yes

Don't know what you mean. Photons don't always pass through 2 different orientations. Sometimes the result is 1,0 or 0,1. Anyway, since there are only 2 possible orientations in a given trial, ie., the settings of the polarizers a and b, then what are you referring to by "3" orientations?

0, 120, 240 <--- three orientations, we'll go step at a time

http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/bell.html

http://www.upscale.utoronto.ca/PVB/Harrison/BellsTheorem/BellsTheorem.html

 

[ThomasT]

Are you asking whether there are quantum experiments that produce quantum entanglement that aren't specifically designed to be tests of a Bell inequality?

yes

Ok. I would suppose so. But I don't know. DrC, jtbell, et al. can probably answer this for you. Google it.

You asked:

- why will the photon always pass through exactly 2 of the 3 orientations?

To which I answered:

Don't know what you mean. Photons don't always pass through 2 different orientations. Sometimes the result is 1,0 or 0,1. Anyway, since there are only 2 possible orientations in a given trial, ie., the settings of the polarizers a and b, then what are you referring to by "3" orientations?

To which you replied:

0, 120, 240 <--- three orientations, we'll go step at a time

Ok, now I know what you mean by the three orientations. These are the θ (the angular difference between polarizer settings) that might be used in a Bell test. They can have any values. Wrt the θ you gave, 120° and 240° are the same θ.

But your question still isn't clear to me. You said "the photon", singular.

Are you thinking of the "three orientations" as individual polarizer offsets from the horizontal 0° setting?

 

[lugita15]

(I'm replying to a post from another thread)

Hi lugita. How do you go from step 2 to step 3?

I told you, all that's required is a trivial use of the transitive property of equality. Based on step 2, we assume it is determined in advance what angles both particles would go through and what angles they both wouldn't go through. Let us call the angles the two particles will go through "good angles", and the angles the two particles will not go through "bad angles". So for each angle, we can ask the question "Is the angle good?" or in other words "Is this one of the angles for which it is determined in advance that the particles will go through?" This is of course a yes or no question. If the answer to this question is "yes" for a particular angle, i.e. if the angle is "good", we say the instruction at that angle is "yes". If the answer is "no" for a particular angle, i.e. the angle is "bad", then we say that the instruction at that angle is "no' (The word "instruction" is just an arbitrary word I'm using. If you want to use another word for it that's fine.)

So for instance, if the two particles would go through the angle 30 degrees, then we say the instruction at 30 is "yes". If the two particles would not go through the angle 20 degees, then we say that the instruction at 20 is "no". (Remember, by step 2 the instructions, i.e. the answers to the yes or no question "would the particle go through at this angle", are determined in advance.) Are we good up to here? So far I've just been talking about what we can say based on step 2.

Now for the logic to go from step 2 to step 3. By the transitive property of equality, we get the following:
(*)If the instruction at -30 is the same as the instruction at 0, and the instruction at 0 is the same as the instruction at 30, then the instruction at -30 is the same as the instruction at 30.

(For example, if the instructions at -30 and 0 are both "yes", and the instructions at 0 and 30 are both "yes", then all three instructions are yes, and similarly if you replace all the yes's with no's.)

I hope you see that the statement (*) really is trivial, and I hope you also see that the statement (*) is completely equivalent to the statement in my step 3. If you don't see either of these, please tell me.

 

[ThomasT]

(I'm replying to a post from another thread)
I told you, all that's required is a trivial use of the transitive property of equality. Based on step 2, we assume it is determined in advance what angles both particles would go through and what angles they both wouldn't go through. Let us call the angles the two particles will go through "good angles", and the angles the two particles will not go through "bad angles". So for each angle, we can ask the question "Is the angle good?" or in other words "Is this one of the angles for which it is determined in advance that the particles will go through?" This is of course a yes or no question. If the answer to this question is "yes" for a particular angle, i.e. if the angle is "good", we say the instruction at that angle is "yes". If the answer is "no" for a particular angle, i.e. the angle is "bad", then we say that the instruction at that angle is "no' (The word "instruction" is just an arbitrary word I'm using. If you want to use another word for it that's fine.)

So for instance, if the two particles would go through the angle 30 degrees, then we say the instruction at 30 is "yes". If the two particles would not go through the angle 20 degees, then we say that the instruction at 20 is "no". (Remember, by step 2 the instructions, i.e. the answers to the yes or no question "would the particle go through at this angle", are determined in advance.) Are we good up to here? So far I've just been talking about what we can say based on step 2.

Now for the logic to go from step 2 to step 3. By the transitive property of equality, we get the following:
(*)If the instruction at -30 is the same as the instruction at 0, and the instruction at 0 is the same as the instruction at 30, then the instruction at -30 is the same as the instruction at 30.

(For example, if the instructions at -30 and 0 are both "yes", and the instructions at 0 and 30 are both "yes", then all three instructions are yes, and similarly if you replace all the yes's with no's.)

I hope you see that the statement (*) really is trivial, and I hope you also see that the statement (*) is completely equivalent to the statement in my step 3. If you don't see either of these, please tell me.

Suppose you assume, from the observation in step 1, that the angular difference between polarizer settings is measuring a relationship between entangled photons that isn't varying from entangled pair to entangled pair. How would you proceed from that assumption?

 

[lugita15]

Suppose you assume, from the observation in step 1, that the angular difference between polarizer settings is measuring a relationship between entangled photons that isn't varying from entangled pair to entangled pair. How would you proceed from that assumption?

Um, I'm not sure what this has to do with what I'm saying. Are you saying that you agree or disagree that step 2 follows from step 1? If you disagree, I can try to explain the logic.

Also, given what I said in the post you're responding to, do you now accept that step 3 is a trivial consequence of step 2?

 

[ThomasT]

Um, I'm not sure what this has to do with what I'm saying. Are you saying that you agree or disagree that step 2 follows from step 1? If you disagree, I can try to explain the logic.

Your step 1 says:
1. Entangled photons behave identically at identical polarizer settings.

There are a number of things that might be inferred from this. Your step 2 says:
2. The believer in local hidden variables says that the polarizer angles the photons will and won't go through are agreed upon in advanced by the two entangled photons.

Ok, so there's something wrt the incident optical disturbances that will determine whether or not they're transmitted by the polarizing filters.

Your step 3 says:
3. In order for the agreed-upon instructions (to go through or not go through) at -30 and 30 to be different, either the instructions at -30 and 0 are different or the instructions at 0 and 30 are different.

Where did that come from? What's the relationship between your step 3 and your step 2? Please express steps 2 and 3 in non-anthropomorphic terms. Ie., from the observation noted in step 1, then what might you infer about the properties of the optical disturbances incident on the polarizers in any given coindicence interval?

 

[harrylin]

[I wrote: [..] getting back to the topic: I guess that in such an alternative interpretation of "ideal situation", the relationship will be linear (both in LR theory and in Weih's experimental data).]

[..] You might be right about that. I don't know.

In which case I might be wrong in saying to the OP that hidden variables don't imply a linear correlation between θ and rate of coincidental detection. But the fact of the matter, afaik, is that they don't. Simply due to the fact that there are LR models of quantum entanglement which predict a cosine, not a linear, correlation between θ and rate of coincidental detection. Even DrC will agree with this. Ask him.

I hope that he will see this remark and comment on it, as it's very much on topic; I do think that hidden variables imply a linear relationship with 100% detection and without "data picking".

 

[lugita15]

Where did that come from? What's the relationship between your step 3 and your step 2? Please express steps 2 and 3 in non-anthropomorphic terms.

I think I spelled out the logic in going from step 2 to step 3 pretty well in post #142, and I did so in non-anthropomorphic terms. Tell me if you don't understand something in that post.

 

[lugita15]

I hope that he will see this remark and comment on it, as it's very much on topic; I do think that hidden variables imply a linear relationship with 100% detection and without "data picking".

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. So you CAN have some kind of nonlocal correlation, but that will only make the result differ even further from the predictions of QM. The local deterministic theory that has the closest possible agreement with QM is one in which the correlation is linear, and we know that even such a theory differs significantly from the predictions of QM.

 

[ThomasT]

I hope that he will see this remark and comment on it, as it's very much on topic; I do think that hidden variables imply a linear relationship with 100% detection and without "data picking".

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

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

 

[harrylin]

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. So you CAN have some kind of nonlocal correlation, but that will only make the result differ even further from the predictions of QM. The local deterministic theory that has the closest possible agreement with QM is one in which the correlation is linear, and we know that even such a theory differs significantly from the predictions of QM.

Ok, that makes sense!