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

DrChinese

A - B = -120 degrees
A - C = -240 "
B - C = -120 "

These are all equivalent for the function cos^2(theta), which is the formula for the QM prediction.

cos^2(-120) = .25
cos^2(-240) = .25
cos^2(-120) = .25

So no matter which pair you consider, the QM expectation is 1/4.

ThomasT

Ok. That makes it handy.

What you (and Bell and Herbert) are saying is that expressing coincidental detection in terms of a separable local predetermination is incompatible with the QM-predicted and observed correlation between θ and rate of coincidental detection. Which I agree with.

Wrt Bell's formulation, it's clear where the restrictions come from and how they affect the predictions of any LR model that encodes those restrictions. Wrt your and Herbert's proofs, it's not so clear to me -- so, if you could clarify that it would help.

More precisely, the proofs show that any model or line of reasoning embodying certain restrictions must be incompatible with QM and experiment. What are the restrictions, and how did they become part of the model or line of reasoning? Does employing these restrictions prove that nature is nonlocal? Imo, no.

It's been well established that the QM predictions are correct. Regarding determinism, it's an unfalsifiable assumption. So all you're dealing with is locality. So, what you're saying your proof proves is that nature is nonlocal (which is what Herbert says). But, what you've shown is that a particular way of conceptualizing coincidental detection is incompatible with QM and experiment. You can infer, from a certain conceptualization and line of reasoning that nature is nonlocal, but whether or not that inference is warranted depends on what's involved in the model or line of reasoning, and whether or not that inference is a fact of nature can only be ascertained by observing a nonlocal transmission.

Yet that seems to be what you said above, and it is what Herbert says his proof proves, and you present your steps as a simplified recounting of Herbert's proof.

I think you (and Bell and Herbert) have proved what I said above. If you don't claim that your proof proves that nature is nonlocal, then we're basically on the same page.

lugita15

According to (the standard interpretation of) quantum mechanics, you have a wave function for the two-particle system, so the polarizations of the particles are in a superposition of states, until one of the photons is detected by one of the polarizers (say the first polarizer). Then the wave function of the system collapses (nonlocally and instantaneously), putting both photons in the same definite polarization state. The collapse will either make both particles polarized in the direction of the first polarizer, or make both particles polarized perpendicular to the direction of the first polarizers. Which of these two things will happen is considered to be a 50-50 chance event, because wave function collapse is according to (the standard interpretation of) QM completely random.

So then if the collapse makes the photons polarized in the direction of the first polarizer, the first photon will go through the first polarizer. If the collapse makes the photons polarized perpendicular to the first polarizer, then the first photon doesn't go through. So to someone just looking at the first polarizer, he always sees random 50-50 results.

What about the second polarizer? Well, the second photon is now in a definite polarization state, either parallel or perpendicular to the angle of the first polarizer. So now if the second polarizer is oriented at the same angle as the first one, the second photon will do the same thing the first one did. If the second polarizer is oriented at a different angle, then the second photon will randomly either go through or not go through, with a probability of going through equal to the cosine squared of the difference between the polarization angle of the photon and the angle of the second polarizer. But if someone was just looking at the second polarizer they won't know what angle the first polarizer was turned to or whether the first photon went through or not, so he won't know what angle the second photon was polarized along before it hit, and thus to him it will seem to go through or not go through with random 50-50 chance.

Does that make sense?

San K

good answer, however my question was how cos came....:)

cos came from the de broglie waves......

now laws of probability are (linear/simple) additive for particles

are the laws of probability (linear) additive for waves? or do the have cosine in them?

San K

i was asking, in general, about non-entangled single photons.....sorry forgot to mention

what does "completely random" mean? is there cause and effect?

surely and it was good to rehearse the 50-50 logic

however I am asking something else.


take 100 photons un-entangled photons (in an indeterminate state)....why do only 50% of them pass through?

is that predetermined, cause and effect, or totally random? what does an indeterminate state mean? what do we mean by "inherent randomness" ?

second question: when we rotate the polarizer by a few degrees:

do the same 50 photons pass through or does the composition changes?

I also realize that we may not have to answer to any of the above questions, however can be important for future research


at some point/interaction ......says QM (?)......we/photons reach the edge.....the edge of cause/effect and enter into the world of "inherent" randomness.....

lugita15

My steps are concerned with local determinism in general. But what do you mean by separable? Do you mean in the sense of factorization of probability distributions, as Bell did in his original proof? Such a factorization is not used in Herbert's proof.I don't know how much help I can be, because I honestly don't know where I am "encoding restrictions" in my reasoning. I do ultimately arrive at a restriction, AKA the bell inequality, but I get that restriction as a conclusion of the argument, not an assumption of the argument. So if this conclusion is wrong then one of the steps must be wrong.

To go through the steps again, 1 is a prediction of QM, 3 uses the transitive property of equality, and 4 uses the laws of probability. Thus 2, in my view, is the only step that can possibly be disputed by someone. And yet it seems so obvious to me that a local determinist who accepts 1 must accept 2.I agree that there is ample experimental evidence for many of the predictions of QM. But a key prediction of QM used in the proof is the fact that you get perfect correlation at identical angle settings, and this has not been definitively demonstrated, because there are various experimental loopholes that in principle could be exploited to construct a local deterministic theory. But yes, the preponderance of evidence is in favor of the correctness of QM's experimental predictions.Determinism may be unfalsifiable on its own, but the combination of determinism and other assumptions may be falsifiable.I presume you mean locality in the sense of the principle of locality, which is just a synonym for local determinism. But it would be slightly improper to call a theory which violated local determinism a "nonlocal" theory, because "nonlocal" to my mind means that causality can propagate faster than the speed of light. But you can have, for instance, a nondeterministic theory in which there are no FTL influences. I mean exactly what I said: assuming that QM is completely correct in all its experimental predictions, we may reject local determinism.I'm still trying to find out how you think I'm "restricting" or "conceptualizing" local determinism.But I'm not deducing facts of nature here, I'm trying to make logical deductions. My argument is of the form "If the universe has property A, then it must have property B." But the question of whether it actually has property B is dependent on whether in fact it has property A. (Property A is local determinism+perfect correlations at identical angle settings, and property B is the Bell inequality.) And the question of whether local determinism is logically incompatible with the experimental predictions of QM is a question to be decided logically, not experimentally. So as I said, the burden of proof is on you to disagree with my steps or agree with my conclusion.It is a restatement of Herbert's proof, but I just feel he worded his conclusion a bit too strongly. But the only reason I do not make that claim is that current Bell tests have certain practical limitations and loopholes. But assuming that an ideal loophole-free Bell test, of the kind Herbert discusses, were to definitively demonstrate that the predictions of QM are completely correct, then I am certainly willing to reach the conclusion that we can reject local determinism (excluding unfalisifiable assumptions of superdeterminism, of course).

lugita15

OK, then the explanation is even simpler. The photon is initially described by a wave function that is in a superposition of polarization states. Then when the photon is hits the polarizer, the wave function collapses, and photon acquires a state of definite polarization, either polarized parallel to the polarizer or polarized perpendicular to the polarizer, each of these two outcomes has a 50% chance of happening.In the traditional Copenhagen interpretation, wave function collapse is truly random, in the sense that there is no cause and effect.
The Cophenhagen interpretation says there is no reason for this, there is just a non-deterministic element to the universe.In the Copenhagen interpretation collapse is totally random, so that absolutely nothing determines what the outcome will be.Copenhagen says that each photon goes through or doesn't go through at random, so which 50 (on average) of the 100 photons are going to go through is not determined in advance at all. Yes, at least according to the Copenhagen interpretation, which is the "standard" or "orthodox" view of QM. There are of course many other interpretations. For instance, Bohmian mechanics says that the apparent randomness of particle behavior is just due to our ignorance of (nonlocal) hidden variables associated with the particles. Many Worlds says that all possible outcomes occur in separate universes, and our mind is also there in these multiple universe, so the fact that we seem to observe outcome A rather than outcome B just means we happened to end up in a universe in which outcome A occurs. Roger Penrose's interpretation says that there is no randomness at all, and that the outcomes of all experiments can be determined in advanced just by calculating a mathematical function, but the only problem is that this function cannot be calculated by a computer (or Turing machine); but he believes that this uncomputable mathematical function CAN be calculated by humans, because he believes that the insightful abilities of the human mind surpasses that of any computer, because he believes that the human mind is based on quantum mechanics (see his books The Emperor's New Mind or Shadows of the Mind for more info).

San K

Lugita, i agree with all of what you said in your post.

it's hard for the human imagination/mind to comprehend (or think further)....what we mean by no cause and effect.....

the below is trivial (not important):

does not the Copenhagen interpretation say lets not go there (not talk about it, shut up and calculate) rather than its totally random? or does it depends upon who you ask about the interpretation of the Copenhagen interpretation? ..:)

lugita15

At a shallow level, the way quantum mechanics works is that you have to calculate the probability amplitude of a particle in state A to be measured in state B. In this case, state A is a superposition of polarization states and state B is the state of definite polarization parallel to the polarizer. Then you take the square of the absolute value of the probability amplitude to find the actual probability. When you do this straightforward calculation, the result happens to be cos^2.

At a more fundamental level, I suppose the cos^2 in this case arises from the the fact that photon polarization is related to spin angular momentum, and angular momentum is based on rotations, so the mathematics of rotations gives us sines and cosines.Yes, in some sense you could say that the sinusoidal functions come from the fact that in QM particles are described by wave functions, which is very different than how particles work classically. The laws of probability have the same form regardless of what kinds of objects you're talking about. The laws of probability are still (sub)additive, for both particles and waves, but that's not the question you should be asking. Rather the issue is, if you take two simple waves, is the square of the amplitude of their sum less than or equal to the sum of the squares of their amplitudes? That is, if wave 1 has amplitude A1, wave 2 has amplitude A2, and wave 3, which is the superposition of waves 1 and 2, has amplitude A3, is A3^2 always less than or equal to A1^2 +A2^2? The answer to that is no.

But we have a problem: the wave function in QM is supposed to represent the probability of getting of a particle getting a certain experimental outcome. So it seems like although we can say that the wave function does describe the probability the photon will be detected at a particular point on the screen of a double slit apparatus, it seems like we cannot say that it describes the probability of the photon going through one slit vs the other slit (where there are no detectors at the slits). The way it looks is that a wave goes through the slits, but then when then there is a detection event, then suddenly the wave function translates into probabilities of the particle being measured having certain attributes. So it's as if the quantum object behaves as a wave until it's measured, behaves as a particle with definite attributes after it's measured. Thus the Copenhagen interpretation was born. If you want to see more discussion along these lines, you can read a chapter from the Feynman Lectures on Physics (I can give you a scan), or watch one of Feynman's Character of Physical Law videos.

Bottom line, in some sense you can say that wave-particle duality is to blame for the strangeness of quantum entanglement.

lugita15

The doctrine of Copenhagen is total randomness of the results of wave function collapse. It's more when you ask questions like "is the wave function real" or "what constitutes a measurement for the purposes of wave function collapse" that Copenhagen people may give you the "shut up and calculate" attitude.

ThomasT

Yes, or some other means of pairing individual results. It's what DrC is talking about in his challenge.

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. So, it would probably be best if you don't ask me any more questions about this.

The Bell stuff is mainly a philosophical consideration, and wading through the language surrounding it, and eliminating the bs and irrelevant considerations is a daunting task. I hope that you and others stick with it so that maybe one day you can explain it to me and other laypersons in a way that we can understand it.

I've expressed my ideas/opinions, and now I will fade back into the peanut gallery, and hopefully learn something new.

ThomasT

Herbert says that his line of reasoning embodies the independence of spacelike separated events as the assumption of locality. If your proof is a recounting of Herbert's proof, then your proof does this also. In which step of your proof is this explicated?

The conclusion isn't wrong if it follows from your assumptions. You say, and I agree, that your conclusion follows from your assumptions. So, what we have to look at is your assumptions in order to ascertain whether we can conclude that an underlying reality, ie., nature, must be nonlocal.

It's not that one of your steps (assumptions) is necessarily wrong, it's that one of your steps might not necessarily be expressing what's actually happening in the underlying reality. The problem, the situation, is that we have no way of knowing, because of the limitations imposed by our sensory faculties.

Ok, so we, and most everybody else, agrees on that.

In which case, then what you might be falsifying would be a certain expression of those other assumptions, such as a particular expression of locality -- but not determinism.

Locality, in the usual sense, means that nothing propagates faster than the speed of light. Determinism means that events or states at t = 1 are a consequence of events or states at t = 0. Determinism isn't incompatible with ftl propagations. But it is incompatible with instantaneous action at a distance.

That's what I'm trying to find out. It's your proof. You tell me.

Then we're on the same page. You, and Herbert, and Bell aren't necessarily saying anything about nature.

San K

i won't ask you anymore.

it has been a great discussion/thread......:)

DrChinese, Lugita and others have passionately/patiently answered the posts and shared some interesting new information/knowledge. They has also helped, some of us, understand Bell's theorem faster. thanks DrC, Lugita and others.

ThomasT

But certainly keep on asking the others. They know a bit more. (And of course you can still ask me anything, and if I don't feel certain that I know the answer, then I'll try to find the time to look it up ... but of course you can do that also.)

If you're just starting out in this, then I hope you have the time to keep at it until you're satisfied that you fully understand it.

lugita15

Well, if by "independence of spacelike separated events" you mean that the behavior of particle 1 is independent of the setting of polarizer 2, and vice versa, then yes, I am assuming that that is part of the definition of local determinism. Is that what you dispute? If so, then step 2 would be the step you should disagree with me on. But there are some things that are the same regardless of the universe we live in. Step 1 is a prediction of quantum mechanics, and this prediction will be the same in every universe, although presumably it will be a correct prediction in some universes and an incorrect prediction in others. Step 3 is an application of the transitive property of equality, and step 4 is an application of the laws of probability, and I hope you agree that those apply equally well in all universes. So we come back to step 2.This is irrelevant to what we're discussing, but why do you think instantaneous action at a distance is incompatible with determinism? But I don't think I am assuming some restrictions on what a local deterministic theory can look like. Rather, I am CONCLUDING a restriction, the Bell inequality, from the assumptions of local determinism and perfect correlation at identical angles. We're not on the same page, because under the assumption that QM is correct in its experimental predictions, I most definitely AM saying something about nature.

ThomasT

But step 2 doesn't specify any particular way of expressing that. That comes with step 3. So, it would seem that step 3 is where the actual restriction is introduced.

Because determinism has to do with the temporal causal order of events. Instantaneous action at a distance between A and B precludes this. It says that A and B are happening simultaneously.

The restriction has to be introduced before the conclusion of a linear correlation between θ and rate of coincidental detection. Since step 2 doesn't express any particular literal restriction, then it has to be step 3. That is, you've chosen to express the assumption of independence (locality) in a certain way, and expressing it that way entails the conclusion that the correlation between θ and rate of coincidental detection must be linear.

The physical interpretation of QM has been an open question since the theory was first formulated. As far as I can tell it will always be an open question.

The only thing that can be concluded from any Bell-LR model of quantum entanglement is that something pertaining to the formal LR restrictions makes the model nonviable. The precise relationship between the model, any model, even viable ones, and an underlying reality remains unknown.

If the source of disagreement between model and results can be precisely identified as something in the model which clearly is incompatible wrt the experimental design and execution, then that should be taken as the effective cause of the nonviability.

Regarding your proof, is your step 3 the only way to conceptualize the experimental situation? If not, then does it establish that nature is nonlocal?

lugita15

ThomasT, you have the uncanny ability of focusing on steps I consider to be unimportant. Step 3 says "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." Let me ask you this, is the following general statement true? "If A and C are different, either A and B are different or B and C are different." For example, "If the color of your pants and shoes are different, then the color of your pants and socks are different, or the color of your socks and shoes are different." (In this case, A="the instruction at -30", B="the instruction at 0", and C="the instruction at 30".) If you cannot tell that this statement is true, consider the equivalent statement "If A and B are the same, and B and C are the same, then A and C are the same." You should recognize that as the transitive property of equality, and presumably that is not in despite.

You can, of course, dispute that the particles have agreed in advance which angles to go through and which angles not to go through, in which case you should dispute step 2. But if you have accepted step 2, and thus believe that the particles have instructions as to exactly which angles to go through and which angles not to go through, then the transitive property of equality forces you to accept step 3. This is entirely irrelevant to our discussion, but do you not consider Newton's theory of gravitation to be deterministic, since it has instantaneous action at a distance? And would consider Bohmian mechanics to be nondeterministic, since it too has instantaneous action at a distance? But I am not making any assumption of locality or independence in step 3! Step 2 is where I say the particles have determined in advance, exactly what angles to go through and what angles not to go through. So they have a list of the instructions of the form "If you see a 20 degree polarizer, go through", "If you see a 40 degree polarizer, don't go through", etc., a list they have agreed upon in advance when they were emitted from the source.

If you agree with me up to there, I don't know how you can disagree with step 3, which is completely trivial. Yes, there are some philosophical views that are indistinguishable experimentally from quantum mechanics, so they can fairly be called interpretations. Copenhagen, Many Worlds, and Bohmian mechanics are good examples of those. But not all possible viewpoints are indistinguishable experimentally from quantum mechanics. For instance, Schrodinger himself originally viewed the wave function as a literal gas that surrounded the nucleus of the atom, and that the magnitude of the wave function was indication the thickness of the gas. But this view was rejected after it was found that when measured, an electron was detected as a particle, not a wave.

Similarly, I see local determinism (excluding superdeterminism) as a view that IS distinguishable experimentally from quantum mechanics, and thus it can't be fairly called an "interpretation".As I said, step 3 is not the step where locality is invoked. Step 3 is an unimportant step where I'm just saying something of the form "If A≠C, then A≠B or B≠C" (an application of the transitive property of equality), so that I can apply the laws of probability and say "The probability that A≠B or B≠C is less than or equal to the probablity that A≠B plus the probability that B≠C" and thus conclude that "The probability that A≠C is less than or equal to the probablity that A≠B plus the probability that B≠C", which is the Bell inequality I'm after.