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

ThomasT

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.

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.

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

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).
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. 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

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.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.
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 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

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

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

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

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

San K

thanks lugita.....


the short answer is: the hidden variables imply a linear relationship because the laws of probability are additive?...and not cosine, exponential etc


you might want to check out...... http://physicsforums.com/showthread.php?t=592401

lugita15

Sure. The reason that your idea won't work is that if you just look at the results for one of the polarizers, then it will seem like the photon is going through or not going through the polarizer with 50-50 probability, regardless of the angle the polarizer is turned to. It is only when you compare the results of the two polarizers that you notice the correlation. That is why quantum entanglement does not allow you to do faster than light communication.

For a simple example of this, consider the case when you turn both polarizers to the same angle. Then, as you know, they'll either both go through or they'll both not go through. They have a 50 percent chance of both going through and a 50 percent chance of both not goon through. Of course, if you just looked at one of the polarizers it will just seem like the photon is just randomly going through or not going through. But if the experimenters recording the results of the two polarizers compare their data, they will find a remarkable phenomenon: the two photons always do the same thing!

lugita15

Yes, exactly. And the reason that the same logic does not apply to quantum mechanics itself is that in QM, when you turn the polarizers to -30 and 30, it makes no sense to talk about "the instruction at 0 degrees". So step 3 does not work for QM, only for hidden variable theories.

San K

oh yes....i missed that...ha ha....got it..thanks

this happens often with people....they come up with some way to do faster than light communication and then forgot they missed the fact that....both the "ends" need to be compared.....:)

lugita15

OK, so you might want to delete the new thread you started.

lostprophets

i like assumptions.
provided there used as an awareness of what may or may not be and not the be all .there how we built the world

ThomasT

Not sure what you're saying. Are you saying that there are recent Bell tests wrt which there are viable LR models? If so, then because any LR model of entanglement must disagree with QM, then that would mean that QM gets those Bell tests wrong.

Showing that LR theories of entanglement are ruled out isn't showing that nature is nonlocal. It just means that entanglement can't be modelled in a certain way. There are more parsimonious ways of understanding why this is so than resorting to the assumption that nature is nonlocal.

It's still not clear to me how, from those 4 steps, it can be deduced that nature is nonlocal.

Maybe if you put your argument in a form like: 'If the universe is local deterministic, then ___ must be the case ...', and so on.

ThomasT

The experiments can only rule out LR models of entanglement. They can't inform wrt whether or not nature is nonlocal. Both locality and nonlocality are assumptions. Locality is a generalization of all known experience. Nonlocality is, so far anyway, just a word for our ignorance. It has no physical referents.

lugita15

That is exactly what I'm saying, because currently there are various experimental loopholes that prevent the kind of ideal Bell test which would be able to definitively test whether this is a local deterministic universe. In particular, as zonde has pointed out, it is difficult to test the prediction that you get perfect correlation at identical polarizer settings, because you would have to "catch" literally all the photons that are emitted by the source, and that requires really efficient photon detectors. All we can say is that when the angles of the polarizers are the same, the correlation is perfect for the photon pairs we DO detect. But that leaves open the possibility, seized on by zonde and other local determinists, that the photon pairs we do detect are somehow special, because the detector is biased (in an unknown way) towards detecting photon pairs with certain (unspecified) characteristics, and that the photon pairs we do NOT detect would NOT display perfect correlation, and thus the predictions of QM would be incorrect. No, it just means that the experiments are not good enough to definitively answer which is right and which is wrong. They strongly suggest quantum mechanics is right, but due to practical loopholes they leave open a slim possibility for a local deterministic theory.Sorry, there's a miscommunication. When I say "local hidden variables", I mean the philosophical stance you call "local determinism", not the formal model you call "local realism", so keep that in mind when reading my posts. I'm trying to show that a the predictions of QM cannot be absolutely correct in a local deterministic universe. As I said, the thing to be deduced from my four steps is not the claim that nature is nonlocal. Rather it is the claim that if the predictions of quantum mechanics are completely correct, than nature is nonlocal or nondeterministic. Sure, I just need to change the wording of step 2 slightly.

1. Entangled photons behave identically at identical polarizer settings.
2. In a local deterministic universe, the polarizer angles the photons will and won't go through must be agreed upon in advance 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.