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

[ThomasT]

But Herbert is not just randomly expecting the correlation to be (sub)linear. Rather, he is showing how a certain assumption, namely that quantum entanglement can be explained by local hidden variables, leads to the conclusion that the correlation is (sub)linear.

Ok, honestly, I don't understand how he gets to the linear correlation from the assumption of local hidden variables. If you can clearly explain that, then you will have helped not just me but, I suspect, lots of other people interested in this stuff.

Surely you agree that some assumptions lead to correct conclusions about the world, and other assumptions lead to incorrect conclusions about the world.

Well, no. We're talking about the world that isn't amenable to our sensory perceptions. So, how could we ever know if any inferences about it are true or not?

 

[lugita15]

Ok, honestly, I don't understand how he gets to the linear correlation from the assumption of local hidden variables. If you can clearly explain that, then you will have helped not just me but, I suspect, lots of other people interested in this stuff.?

I've been trying to do that for a while now. Here's my latest attempt, from post #5 of this thread:

1. Entangled photons behave identically at identical polarizer settings.
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.
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.
4. The probability for the instructions at -30 and 30 to be different is less than or equal to the probability for the instruction at -30 and 0 to be different plus the probability for the instructions at 0 and 30 to be different.

Which of these steps do you disagree with, or which of these steps do not apply to all possible local hidden variable theories?

Well, no. We're talking about the world that isn't amenable to our sensory perceptions. So, how could we ever know if any inferences about it are true or not

I have no idea what you're talking about. All I said is that certain assumptions lead to correct conclusions about the world, and certain assumptions lead to incorrect conclusions about the world. Many arguments take the form of starting from an assumption and showing how it leads to a false conclusion about the world. For instance, Rayleigh-Jeans showed that the assumption that light is described by Maxwell's equation leads to the ultraviolet catastrophe, which does not occur in real life. Rayleigh-Jeans certainly wasn't ignoring the fact that there is no ultraviolet catastrophe for real-life blackbody radiation, but he was showing how a certain assumption led to that incorrect conclusion. Herbert's (and Bell's) proof works the same way. They are trying to show that the assumption of local hidden variables leads to a certain conclusion that is contrary to the experimental predictions of quantum mechanics, even though the predictions of QM are presumably correct.

 

[ThomasT]

I've been trying to do that for a while now. Here's my latest attempt, from post #5 of this thread:

Ok, seriously, I think you should abandon this exercise of attempting to reformulate Herbert's argument and look at the situation conceptually. The goal then is not to formulate a restricted LR model of entanglement, but to understand entanglement in terms of local interactions and transmissions. Which, I submit, is entirely possible, even if Bell-type LR models of quantum entanglement are definitively ruled out.

All I said is that certain assumptions lead to correct conclusions about the world, and certain assumptions lead to incorrect conclusions about the world.

And the question is: how could you possibly know if those conclusions are correct or incorrect? What's your criterion for ascertaining that?

The good thing about the scientific method, and also what makes it impossible for us to make definitive statements about deep reality, is that the ultimate arbiters wrt any questions or statements about nature are instrumental behaviors amplified to the level of our sensory apprehension. That's it. That's all we have. That's the data. One can assume, infer, deduce, etc. to one's passionate intent/content. Doesn't matter. The data are instrumental ... not deep.

They (Herbert and Bell) are trying to show that the assumption of local hidden variables leads to a certain conclusion that is contrary to the experimental predictions of quantum mechanics ...

And they have shown that a certain reasoning and formal encoding of the assumptions of locality and determinism are incompatible with the predictions of standard QM and the results of experiments. But their reasoning is neither deep nor all inclusive. They obviously ignore certain known facts about the behavior of light and the experimental designs. And on the basis of this reasoning we should assume that nature is nonlocal? That's not just bad reasoning, it's just silly ... and should, I think, be summarily rejected.

This is not to say that Bell has not definitively ruled out a broad class, maybe the general class, of LR models of quantum entanglement. I fully believe that this is a great accomplishment. And I further believe that the insight into the deep reality that it engenders is one of the great accomplishments/discoveries of modern scientific thinking.

Bell's point and contribution, imho, isn't that he showed that nature is nonlocal, but that he revealed an extremely subtle problem wrt the modelling of entanglement in a local deterministic universe.

 

[Joncon]

So, I don't think it's unnecessarily harsh to say that Herbert has ignored a characteristic behavior of light that's been known for a long time.

I don't understand this, why do you say Herbert ignores the behaviour of light? All he's saying (as far as I can tell) is that an LR model which predicts perfect correlation when the polarizers are at the same setting will produce a linear correlation for the angles between the polarizers.

 

[ThomasT]

I don't understand this, why do you say Herbert ignores the behaviour of light? All he's saying (as far as I can tell) is that an LR model which predicts perfect correlation when the polarizers are at the same setting will produce a linear correlation for the angles between the polarizers.

The known behavior of light suggests that, assuming an underlying polarization, vis λ, then the rate of individual detection will be a nonlinear function involving the interaction of λ and the polarizer setting. This is modeled after what's known from polariscopic experiments. The rate of individual detection doesn't vary with polarizer setting because, presumably, the value of λ is varying randomly from pair to pair (while, in most optical Bell tests, eg., Aspect, λ is assumed to be the same for both photons of an entangled pair), so the rate of individual detection remains constant at about half that with no polarizer in place.

The rate of coincidental detection in a setup where you have a source flanked by two polarizers and two detectors with one detection per detector per entangled pair is also modeled after polariscopic setups where the rate varies nonlinearly with θ, the angular difference between the polarizers.

Every optical Bell test is some variation on this theme. The conceptual and factual basis for modelling optical Bell tests comes from what's known about the behavior of light in various experiments involving crossed polarizers, all of which suggest that P(a,b) is a nonlinear function.

 

[Joncon]

The known behavior of light suggests that, assuming an underlying polarization, vis λ, then the rate of individual detection will be a nonlinear function involving the interaction of λ and the polarizer setting. This is modeled after what's known from polariscopic experiments. The rate of individual detection doesn't vary with polarizer setting because, presumably, the value of λ is varying randomly from pair to pair (while, in most optical Bell tests, eg., Aspect, λ is assumed to be the same for both photons of an entangled pair), so the rate of individual detection remains constant at about half that with no polarizer in place.

OK, but that still doesn't answer why you think Herbert ignored this, he mentions that the detectors will pick up a 50/50 random sequence of 1s and 0s. And anyway, how you would you go about constructing an LR model which uses Malus' Law and gets perfect correlations with the polarizers at identical settings?

 

[DrChinese]

It does if they're assumed to be polarized identically, via the source. Which is the case with the Aspect experiments.

Not so. Malus applies to single particles, not to coincidences between pairs. The cos^2 rule for entangled particles is derived in a different fashion.

Please note that the cos^2 rule does NOT in fact apply to identically polarized pairs that are not polarization entangled. For example, you can have pairs of photons coming from a single type I PDC crystal (so they are not polarization entangled but are otherwise entangled) that are known HH> and those follow a *completely* different formula.

 

[ThomasT]

OK, but that still doesn't answer why you think Herbert ignored this, he mentions that the detectors will pick up a 50/50 random sequence of 1s and 0s.

I don't know why Herbert ignored it. lugita15 has been trying to explain it to me. Here's lugita's latest offering:
1. Entangled photons behave identically at identical polarizer settings.
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.
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.
4. The probability for the instructions at -30 and 30 to be different is less than or equal to the probability for the instruction at -30 and 0 to be different plus the probability for the instructions at 0 and 30 to be different.

And anyway, how you would you go about constructing an LR model which uses Malus' Law and gets perfect correlations with the polarizers at identical settings?

This might be helpful:
http://en.wikipedia.org/wiki/Local_hidden_variable_theory

 

[lugita15]

I don't know why Herbert ignored it.

No, Joncon was asking what your reason was for thinking that Herbert ignored the fact that you always get 50-50 random results when you view the results of one polarizer in isolation, when he very clearly stated it.

 

[ThomasT]

It does if they're assumed to be polarized identically, via the source. Which is the case with the Aspect experiments.

Not so. Malus applies to single particles, not to coincidences between pairs. The cos^2 rule for entangled particles is derived in a different fashion.

Yes, I understand that.
http://plato.stanford.edu/entries/bell-theorem/

I'm reasoning from what I take to be similarities between an idealized simplified (Aspect-type) optical Bell test setup, and a polariscope.

By the way thanks for the link:
http://departments.colgate.edu/phys... research/Quantumlan07/lab5entanglement09.PDF

Equations 2 and 3 already indicate that the eventual result is going to be a nonlinear function. I'm just taking a conceptual shortcut, which allows me to retain the assumption that our universe is local deterministic, while Bell-type LR models of quantum entanglement are ruled out.

Please note that the cos^2 rule does NOT in fact apply to identically polarized pairs that are not polarization entangled.

It applies to the functions that determine individual detection, which are then combined, and what results is a modified cos²θ correlation coefficient. LR gives the same results as QM in this case.

For example, you can have pairs of photons coming from a single type I PDC crystal (so they are not polarization entangled but are otherwise entangled) that are known HH> and those follow a *completely* different formula.

Apparently they're not polarized as identically as the entangled photons in Aspect experiments, but just enough to give identical results with polarizers aligned.

 

[ThomasT]

No, Joncon was asking what your reason was for thinking that Herbert ignored the fact that you always get 50-50 random results when you view the results of one polarizer in isolation, when he very clearly stated it.

Ah, thanks. He didn't ignore the randomness of the individual data sequences. What he ignored was the application of Malus Law in modeling individual detection. Even QM uses it in that case.

 

[lugita15]

What he ignored was the application of Malus Law in modeling individual detection. Even QM uses it in that case.

First of all, as DrChinese pointed out you shouldn't call any relationship that involves cos^2 of an angle "Malus' Law". Malus' Law refers specifically to the behavior of an unentangled polarized photon when sent through a polarizer.

More importantly, Herbert most certainly does not overlook the fact that quantum mechanics uses a cos^2 relationship to model quantum entanglement. In fact, the whole point of his argument is to show that while quantum mechanics use a cos^2 relationship, no local hidden variable theory can replicate that exact relationship as long as it also agrees with the quantum mechanical prediction that entangled photons display identical behavior at identical polarizer settings. Now you may disagree with whether he successfully proves that (and if so, I want to know what you object to in my now 4-step argument), but I hope you at least see that that's what he's trying to do.

 

[ThomasT]

First of all, as DrChinese pointed out you shouldn't call any relationship that involves cos^2 of an angle "Malus' Law". Malus' Law refers specifically to the behavior of an unentangled polarized photon when sent through a polarizer.

From Abner Shimony

(18b) probΦ(1,1|a,b ) = (½)| 2<θ₁|φ₁>2 |².

Finally, the expression on the right hand side of Eq. (18b) is evaluated by using the law of Malus, which is preserved in the quantum mechanical treatment of polarization states: that the probability for a photon polarized in a direction n to pass through an ideal polarization analyzer with axis of transmission n′ equals the squared cosine of the angle between n and n′. Hence

(20a) probΦ(1,1|a,b ) = (½)cos²σ,

where σ is b−a.

In fact, the whole point of his argument is to show that while quantum mechanics use a cos^2 relationship, no local hidden variable theory can replicate that exact relationship as long as it also agrees with the quantum mechanical prediction that entangled photons display identical behavior at identical polarizer settings. Now you may disagree with whether he successfully proves that (and if so, I want to know what you object to in my now 4-step argument), but I hope you at least see that that's what he's trying to do.

Seems to me that Herbert tried to show that given the assumption of locality (that spacelike separated events are independent), then the expected correlation between θ and rate of coincidental detection should be linear. But we know that's not true ... from the known behavior of light, the design and execution of optical Bell tests, and the existence of LR models of entanglement that produce a nonlinear correlation.

 

[Joncon]

And anyway, how you would you go about constructing an LR model which uses Malus' Law and gets perfect correlations with the polarizers at identical settings?

This might be helpful:
http://en.wikipedia.org/wiki/Local_hidden_variable_theory

Thanks, but there's nothing there which shows how to get perfect correlations at identical polarizers settings, using Malus.

Seems to me that Herbert tried to show that given the assumption of locality (that spacelike separated events are independent), then the expected correlation between θ and rate of coincidental detection should be linear. But we know that's not true ...

Of course it's not true. That's the whole point - Herbert is showing that this LR model can't reproduce the predictions of QM, or the results of actual experiments.

 

[harrylin]

really? i was not aware of this/that
they are more knowledgeable, dedicated and intelligent than me...so I am about to give up on LHV/EPR and join the quantum/bell bandwagon/party... [..]

Note that that comment did not refer (or should not have referred) to models that correctly predict known experimental results, but about models that attempt to produce the exact same prediction as QM is claimed to make for all situations - even possibly non-realistic ones. There are a lot of subtleties involved.

 

[DrChinese]

1. It applies to the functions that determine individual detection, which are then combined, and what results is a modified cos²θ correlation coefficient. LR gives the same results as QM in this case.

2. Apparently they're not polarized as identically as the entangled photons in Aspect experiments, but just enough to give identical results with polarizers aligned.

1. That is like saying everything comes down to 2 pi. I will write out the derivation over the next few days to show you.

2. Your use of the word "identically" is somewhat misleading in this example. Pairs are either polarization entangled (and have no known polarization) or they are not (and can have known polarization). If they are, there are entangled state statistics. If not, there are product state statistics. One violates Bell, the other does not.

Either way, the point is that you must derive the predictions to match to the expected results, and to say this has been known for 200 years or that Herbert is ignoring something is off base. There is nothing wrong with Herbert's proof, it is yet another way (like Mermin) of getting the same result as Bell. Just a bit easier to visualize.

 

[lugita15]

Seems to me that Herbert tried to show that given the assumption of locality (that spacelike separated events are independent), then the expected correlation between θ and rate of coincidental detection should be linear.

That's mostly right, except it's not just the assumption of locality, you also need the assumption of hidden variables. And just to be precise, we really mean "sublinear" or "at most linear", because the Bell inequality is of the form A is less than or equal to B+C.

But we know that's not true ... from the known behavior of light

And that is the point. The proof shows that local hidden variable theories MUST make predictions contrary to the known behavior of light predicted by quantum mechanics.

the design and execution of optical Bell tests

The practical design and execution of Bell tests is irrelevant to the question of whether, in principle, a local deterministic universe can be compatible with all the predictions of quantum mechanics, however difficult those predictions maybe to test in practice.

the existence of LR models of entanglement that produce a nonlinear correlation.

As I said, there does not exist any local hidden variable model that matches the quantum mechanical prediction of identical behavior at identical polarizer settings, and also matches the exact nonlinear relationship predicted by quantum mechanics.

If you disagree, tell me which of my four steps you do not understand or disagree with.

 

[ThomasT]

Thanks, but there's nothing there which shows how to get perfect correlations at identical polarizers settings, using Malus.

Maybe I'm not sure what you're asking. Perfect positive correlation between individual detection attributes with polarizers aligned is predicted by QM and all LR models, afaik. I thought the Wiki article covered that. Maybe not. I didn't actually read through it.

And both QM and LR use Malus wrt formulating the functions determining individual detection. There have been a couple of other links provided in this thread that inform wrt how an LR model might be constructed. Or, just go back to Bell 1964 (DrC has the a pdf of it on his website), as Bell's form is the archetypal LR form.

Of course it's not true. That's the whole point - Herbert is showing that this LR model can't reproduce the predictions of QM, or the results of actual experiments.

If that was all he said about it, then I wouldn't have a problem. But Herbert says that he (and Bell) have proven that nature is nonlocal. Which is an interpretation that I disagree with. It's wrt Herbert's interpretation of his result that I'm saying that he's ignored the known characteristic behavior of light as well as some salient features of Bell test experimental design and execution. Considering that, my take on what he actually proved is a bit more conservative, and in line with your statement above. LR models of entanglement, because of the formal requirements/restrictions on any LR model, as laid out by Bell, are definitively (afaik) ruled out.

 

[San K]

angle: LHV / QM
0: 1 / 1 ; E(0,0) perfect correlation
15: 0.66 / 0.87 ; E(-15,0) = E(0,15)
30: 0.33 / 0.5 ; E(-30,0) = E(-15,15)= E(0,30)
45: 0: / 0 ; no correlation
60: -0.33 / -0.5 ; anticorrelation

These are expectation values of correlation E(a,b) (confusingly called P(a,b) in Bell's paper)
Coincidence probabilities: P(a=b) = E(a,b)/2+0.5

thanks Delta. are the values for E(30, 30) = E(15, 15) = E(0,0) = 1/1?

can you put down the values for the entangled electron?

can you put down the values for non-entangled photons?

 

[San K]

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.

Agreed. Trying to understand 3 & 4 below...lets do it by numbers (instead of probabilities)
Say we have a mixture (polarization) of 100 photon (entangled) pairs.

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.

did I get the below correctly?

per actual results/QM

50 pairs (will) will give the same results at both the polarizers at (-30, 30) or (0,60) etc
87 pairs (will) give the same results (at both the polarizers) at (-15, 0) or (15, 0) or (45, 30) etc
100 pairs (will) give the same results (at both the polarizers) at (0, 0) or (30, 30) or (40, 40) etc

by same results we mean = either both (of the entangled photons) pass through or both don't pass through

per Bell's reasoning/deduction of LHV hypothesis

the numbers should/would be
33.333
66.666
100

4. The probability for the instructions at -30 and 30 to be different is less than or equal to the probability for the instruction at -30 and 0 to be different plus the probability for the instructions at 0 and 30 to be different.

can you rewrite the above logic in terms of "probability of the instruction to be same"?

 

[ThomasT]

That is like saying everything comes down to 2 pi. I will write out the derivation over the next few days to show you.

No need to do that, unless you just want to.

Your use of the word "identically" is somewhat misleading in this example.

Not necessarily. It seems reasonable to me to suppose that the polarizations of photons emitted in opposite directions via, say, an atomic transition from an excited state, are more closely related than nonentangled identically polarized photons. Which, it seems, might be enough to explain why you can get more or less perfect positive correlation wrt the latter, but still not the full range of entanglement stats.

... to say ... that Herbert is ignoring something is off base. There is nothing wrong with Herbert's proof, it is yet another way (like Mermin) of getting the same result as Bell. Just a bit easier to visualize.

I agree that there's nothing wrong with Herbert's proof. What I disagree with is what Herbert says it proves. Namely, that nature is nonlocal. I think we might both agree that that's still an open question.

My language might have confused things a bit. It's only wrt his interpretation that I think Herbert is off base, and has, apparently, ignored some things which I consider important wrt interpreting his result.

 

[ThomasT]

That's mostly right, except it's not just the assumption of locality, you also need the assumption of hidden variables. And just to be precise, we really mean "sublinear" or "at most linear", because the Bell inequality is of the form A is less than or equal to B+C.

Wrt Herbert's (and your) formulation I agree. But, wrt the OP, it's been demonstrated that the assumption of hidden variables doesn't imply a linear relationship between θ and rate of coincidental detection.

And that is the point. The proof shows that local hidden variable theories MUST make predictions contrary to the known behavior of light predicted by quantum mechanics.

I agree. The only question is: what does that tell us about nature? Does it mean that nature is nonlocal? Or, is there a more parsimonious explanation for BI violations? My working hypothesis is the latter. Yours seems to be the former.

The practical design and execution of Bell tests is irrelevant to the question of whether, in principle, a local deterministic universe can be compatible with all the predictions of quantum mechanics, however difficult those predictions maybe to test in practice.

I have to disagree with this. I think that the practical design and execution of Bell tests holds some important clues regarding the nonviability of LR models of entanglement. The point being that if BIs are experimentally violated because of a necessary incompatibility between LR-constrained modelling and the practical design and execution of Bell tests, then we can't conclude from BI violations that nature is nonlocal, or nondeterministic.

What I would agree with is that the practical design and execution of Bell tests is irrelevant to the question of whether, in principle, LR models of entanglement can be compatible with all the predictions of QM. It's been definitively shown, imo, that any LR model of entanglement following Bell's formal treatment is, in principle, incompatible with QM.

If you want to show that the assumption that our universe is evolving deterministically in accordance with the principle of locality is incompatible with, say, Herbert's proof, then you'll have to do more than just reiterate or elaborate on the proofs of Bell, Herbert, et al. or refer to the experimental violation of BIs. You'll have to show exactly why experimental BI violations can't possibly be due to anything other than either instantaneous action at a distance or ftl transmissions. And to do that you're going to have to, among other things, refer to the precise relationship between LR models of entanglement and the design and execution of Bell tests.

 

[lugita15]

per actual results/QM

50 pairs (will) will give the same results at both the polarizers at (-30, 30) or (0,60) etc
87 pairs (will) give the same results (at both the polarizers) at (-15, 0) or (15, 0) or (45, 30) etc
100 pairs (will) give the same results (at both the polarizers) at (0, 0) or (30, 30) or (40, 40) etc

by same results we mean = either both (of the entangled photons) pass through or both don't pass through

Most of this is wrong, so let me tell you what is correct:
If the polarizers are 0 degrees apart 100 pairs will give the same result.
If the polarizers are 30 degrees apart, 75 pairs will give the same result.
If the polarizers are 45 degrees apart, 50 pairs will give the same result.
If the polarizers are 60 degrees apart, 25 pairs will give the same result.
If the polarizers are 90 degrees apart, 0 pairs will give the same result.
In general, if the polarizers are an angle θ apart, the number of pairs that give the same result is 100cos²(θ) and the number of pairs that give different results is 100sin²(θ).

per Bell's reasoning/deduction of LHV hypothesis

the numbers should/would be
33.333
66.666
100

I don't know what you're saying here, but what Bell's (and Herbert's) reasoning shows is that local hidden variable theories, assuming they agree with all 100 pairs giving the same result when the polarizers are 0 degrees apart, MUST have the following property: the number of pairs that give opposite results when the polarizers are an angle 2θ apart is less than or equal to twice the number of pairs that give opposite results when the polarizers are an angle θ apart. In particular, the number of pairs that give opposite results when the polarizers are 60 degrees apart must be less than or equal to twice the number of pairs that give opposite results when the polarizers are 30 degrees apart, a result which you can see from my numbers flatly contradicts quantum mechanics. That's because quantum mechanics says 25 pairs give opposite results at 30 degrees, so a local hidden variables theorist would conclude, via Bell's reasoning, that at most 50 pairs give opposite results at 60 degrees. But QM says 75 pairs give opposite results at 60 degrees.

can you rewrite the above logic in terms of "probability of the instruction to be same"?

I could, but the resulting inequality would be more confusing. But let me spell out the logic of the existing inequality.

If x=y and y=z, then x=z, agreed? Thus if x≠z, then either x≠y or y≠z, agreed? (What we really mean is x≠y or y≠z or both, but in mathematics it's customary to use the word "or" to mean "A or B or both.) But by the laws of probability, the probability that x≠y or y≠z is less than or equal to the probability that x≠y plus the probability that y≠z. Thus the probability that x≠z is less than or equal to the probability that x≠y plus the probability that y≠z. Does that make sense to you? In our case, x is "the instruction at -30", y is "the instruction at 0", and z is "the instruction at 30".

 

[lugita15]

Perfect positive correlation between individual detection attributes with polarizers aligned is predicted by QM and all LR models, afaik.

No, the local hidden variable models that successfully reproduce the results of current Bell tests do NOT agree with the quantum mechanical prediction of perfect correlation at identical polarizer settings. Rather, they claim that this particular prediction of quantum mechanics is incorrect, but that various experimental loopholes like fair sampling and detector efficiency prevent this prediction from being tested by current experimental procedures. But they hold out hope that advances in experimental capabilities will prove them right and QM wrong. You can ask zonde, who is a huge fan of such models.

If that was all he said about it, then I wouldn't have a problem. But Herbert says that he (and Bell) have proven that nature is nonlocal. Which is an interpretation that I disagree with. It's wrt Herbert's interpretation of his result that I'm saying that he's ignored the known characteristic behavior of light as well as some salient features of Bell test experimental design and execution.

I actually agree with you that Herbert's conclusion is stated a bit too boldly. He says that the proof definitively show that reality is not local, i.e. local hidden variable theories are decisively ruled out. But that's not entirely true, because experimental limitations prevent us from doing loophole-free Bell tests. But his proof does demonstrate that as long as all the predictions of quantum mechanics are completely correct, then this can't be a local deterministic universe.

Considering that, my take on what he actually proved is a bit more conservative, and in line with your statement above. LR models of entanglement, because of the formal requirements/restrictions on any LR model, as laid out by Bell, are definitively (afaik) ruled out.

But in my 4 steps, I am not discussing any formal model. I am starting with the premise that this is a local deterministic universe, and I am logically deducing the consequences of this premise.

 

[lugita15]

I agree that there's nothing wrong with Herbert's proof. What I disagree with is what Herbert says it proves. Namely, that nature is nonlocal. I think we might both agree that that's still an open question.

I agree that empirically this is still an open question, albeit open by a very slim margin. But logically, the only way local determinism would not be ruled out by an ideal loophole-free experiment would be if the predictions of QM were disproven.

 

[lugita15]

Wrt Herbert's (and your) formulation I agree. But, wrt the OP, it's been demonstrated that the assumption of hidden variables doesn't imply a linear relationship between θ and rate of coincidental detection.

No, it has not been demonstrated. I maintain that it is impossible for a local hidden variable theorist who believes in identical behavior at identical polarizer settings to not accept Herbert's Bell inequaity.

And that is the point. The proof shows that local hidden variable theories MUST make predictions contrary to the known behavior of light predicted by quantum mechanics.

I agree.

Wait a minute, you agree with me that any possible local hidden variable theory must make predictions contrary to those of QM?

The only question is: what does that tell us about nature? Does it mean that nature is nonlocal? Or, is there a more parsimonious explanation for BI violations? My working hypothesis is the latter. Yours seems to be the former.

No, my conclusion is more nuanced: it is that if all the predictions of QM are right, then any possible hidden variable explanation MUST be nonlocal.

I have to disagree with this. I think that the practical design and execution of Bell tests holds some important clues regarding the nonviability of LR models of entanglement. The point being that if BIs are experimentally violated because of a necessary incompatibility between LR-constrained modelling and the practical design and execution of Bell tests, then we can't conclude from BI violations that nature is nonlocal, or nondeterministic.

I agree with you that the BI violations produced by current Bell tests, with their practical flaws, do not definitively prove that the universe is either nonlocal or nondeterministic. However, it would be a different story if BI violations were produced by perfect, loophole-free Bell tests. If the predictions of QM are correct, then the universe must be nonlocal or nondeterministic. If you disagree, tell me which of my steps you dispute.

If you want to show that the assumption that our universe is evolving deterministically in accordance with the principle of locality is incompatible with, say, Herbert's proof, then you'll have to do more than just reiterate or elaborate on the proofs of Bell, Herbert, et al. or refer to the experimental violation of BIs. You'll have to show exactly why experimental BI violations can't possibly be due to anything other than either instantaneous action at a distance or ftl transmissions. And to do that you're going to have to, among other things, refer to the precise relationship between LR models of entanglement and the design and execution of Bell tests.

But I'm not interested in showing that currently practical experimental BI violations can't possibly be due to anything other than nonlocality or nondeterminism. I'm interested in showing that in principle, the predictions of QM are incompatible with the assumptions of locality and determinism, and I claim to have done so in my four steps.

 

[San K]

If the polarizers are 0 degrees apart 100 pairs will give the same result.
If the polarizers are 30 degrees apart, 75 pairs will give the same result.
If the polarizers are 45 degrees apart, 50 pairs will give the same result.
If the polarizers are 60 degrees apart, 25 pairs will give the same result.
If the polarizers are 90 degrees apart, 0 pairs will give the same result.

agreed, this matches with the calculations...

can you put the above numbers for un-entangled photons? i guess it would be 33 pairs would give same result...

i assume that the probabilities would be unaffected by the polarizer angles, in case of un-entangled photons, is that correct?

 

[San K]

we use the laws of probability to conclude that the probability of a mismatch between -30 and 30 is less than or equal to the probability of a mismatch between -30 and 0 plus the probability of a mismatch between 0 and 30

Per QM/actual experiment -
At (-30,30) the mismatch is 75 pairs
at (-30,0) the mismatch is 25 pairs
at (0,30) the mismatch is 25 pairs

Mr. bell is saying that per LHV the result (at the most) should be 50 not 75 using additive law of probability. The extra 25 is due to entanglement.

is the above logic correct?

 

[lugita15]

Per QM/actual experiment -
At (-30,30) the mismatch is 75 pairs
at (-30,0) the mismatch is 25 pairs
at (0,30) the mismatch is 25 pairs

That's correct.

Mr. bell is saying that per LHV the result (at the most) should be 50 not 75 using additive law of probability.

Yes, according to the local hidden variable theorist the mismatch at (-30,30) must be at most 25+25=50.

The extra 25 is due to entanglement.

Yes, the fact that QM entanglement gives a mismatch that is 25 greater than the maximum possible mismatch predicted by the local hidden variable theorist is the key point.

is the above logic correct?

Yes.

 

[San K]

Yes, the fact that QM entanglement gives a mismatch that is 25 greater than the maximum possible mismatch predicted by the local hidden variable theorist is the key point.

thanks lugita...the short answer is: the hidden variables imply a linear relationship because the laws of probability are additive?...and not cosine, exponential etcyou might want to check out... https://www.physicsforums.com/showthread.php?t=592401