Scholarpedia article on Bell's Theorem

[Demystifier]

Your only problem, and it's a big one, is that there's absolutely no physical evidence for assuming that nature is nonlocal. The fact is that the physical evidence points to locality.

Perhaps I could digest that claim if instead of "evidence" you said "proof". But experiments demonstrating violation of Bell inequalities definitely ARE physical evidence for nonlocality, even if they are not strictly a proof of it.

But if you still disagree, then it would be helpful if you could answer the following questions:

1. In your opinion, the experimental violation of Bell inequalities is evidence for what?

2. Suppose that experimental violation of Bell inequalities has been observed before the theory of quantum mechanics has been discovered. For such experimentalists, what would be a natural interpretation of their experimental results?

 

[Demystifier]

If you were primarily a scientist (experimentalist), then you would most likely, I think, assume locality.

If I were primarily experimentalist, I would assume neither locality nor nonlocality. Instead, I would make experiments without any theoretical prejudices. And if in a particular experiment I would found correlations between spatially separated results of measurements such as those that violate Bell inequalities, then I would conclude (not assume!) that this particular experiment suggests the existence of some nonlocal influences.

 

[Demystifier]

Your only problem, and it's a big one, is that there's absolutely no physical evidence for assuming that nature is nonlocal. The fact is that the physical evidence points to locality.

Then again there are also lots of people who are primarily philosophers, not scientists/experimentalists, who disagree with the interpretation that Bell's theorem should be interpreted as the existence of nonlocality.

Even they don't say that "there's absolutely no physical evidence for assuming that nature is nonlocal". Instead, they would rather say something like "While Bell's theorem can be interpreted as a sign of nonlocality, this is not the most natural and most convincing interpretation of it."

 

[Demystifier]

But I still retain the assumption that nature is evolving in accordance with the principle of locality. Why? ... because I suppose (assume/hypothesize) that there just might be a less exotic (more parsimonious, simpler, but nonetheless subtle) explanation for why BI's are violated than the assumption that there are nonlocal transmissions happening in the reality underlying instrumental behavior.

Let me guess: But you have absolutely no idea what that explanation might be. Am I right?

And the mere fact that you have no idea how to explain it without nonlocality should already be taken as evidence (not yet a proof) that in some cases nature might be nonlocal.

Or maybe you do not understand the difference between "evidence" and "proof"?

 

[ttn]

If your article is concerned with truth then it is not review article.

OK then, I take it back. It's not a review article. It's an encyclopedia entry. Am I allowed to be concerned with truth now?

So you skipped first part of the question: what are "the relevant predictions of quantum theory"?
So let me give my answer. There are two relevant predictions for entanglement, first is perfect correlations for matching measurement settings and second is sinusoidal relationship between relative angle of measurement settings and correlation level.
Is it right?

Well, of course the details depend on exactly what the entangled state is, but for the states standardly used for EPR-Bell type experiments, I would accept that as a rough description. But what's the point? Surely there's no controversy about what the predictions of QM are??

 

[ttn]

=
But I still retain the assumption that nature is evolving in accordance with the principle of locality. Why? Because that's the world of my experience, and I don't know of any physical evidence contradicting that assumption, and also because I suppose (assume/hypothesize) that there just might be a less exotic (more parsimonious, simpler, but nonetheless subtle) explanation for why BI's are violated than the assumption that there are nonlocal transmissions happening in the reality underlying instrumental behavior.

That's exactly what everybody should think -- until they learn about Bell's theorem. In other words, your statement here reads to me like a confession that you haven't looked at or understood Bell's theorem.

What if the λ's determining rate of individual detection and rate of coincidental detection are different underlying parameters?

Sorry, but none of this makes sense. Look at the role this lambda actually plays in the theorem. It can be *anything*. So the kind of scenario you describe (there are two different "parts" to lambda, one that affects such and such, the other affecting thus and so...) is perfectly well covered already -- i.e., it is already ruled out by the theorem.

Anyway, I'll read your article, even though I disagree with the very first sentence in it. You've taught me some things before. Maybe I'm just missing something.

Yes, you should read it. It is precisely an understanding of Bell's theorem that you are currently missing.

 

[ttn]

Of course, I understand that you, and ttn, and other dBB advocates have a certain vested interest in promoting a certain interpretation of Bell's theorem.

That is patently absurd. Tell me specifically where any of us let some kind of Bohmian bias sneak into the arguments about Bell's theorem.

You assume nonlocality based on your interpretation of Bell's theorem.

This profoundly mis-states the situation. I *infer* nonlocality based on my *understanding* of Bell's theorem. You make it sound (with all this talk of "assumptions" and "vested interests") that I and others just *arbitrarily* decide we like nonlocality, so we interpret Bell's theorem that way. That is just backwards. Read the article if you want to actually understand the issues.

 

[mattt]

I don't understand what some people are arguing here.

ttn presents (in his scholarpedia article) several mathematical theorems.

As far as I see, they are all correct. I don't see any mathematical error in his mathematical proofs.

Then you have two possibilities:

1) You find a mathematical error in one of the mathematical proofs he presents. Then show it.

2) You don't find any mathematical error in his mathematical theorems, but you think that the mathematical expression he uses as a "necessary condition of locality" is actually not a necessary condition of locality based on YOUR definition of locality. Then show your definition of locality as clearly as possible and prove that his condition is not a necessary condition for your definition of locality.

I don't see anyone here doing 1) or 2)

 

[harrylin]

Hi everybody. I dropped into Physics Forums for the first time in a while just to see what was going on in one of my old hangouts. It was nice to see about 10 threads raging about Bell's theorem! But perhaps not so nice to see many people, with whom I argued at length in the old days here, saying the same exact WRONG things still after all these years! =)

Anyway, I just thought it might be helpful to advertise the existence of a really systematic, careful review article on Bell's Theorem that Goldstein, Tausk, Zanghi, and I finished last year (after working on it for more than a year). It's free online here

http://www.scholarpedia.org/article/Bell%27s_theorem

and addresses very explicitly and clearly a number of the issues being debated on the other several current Bell's Theorem threads. It is, in my hardly unbiased opinion, far and away the best and most complete existing resource for really understanding Bell's Theorem, so anybody with a remotely serious interest in the topic should study the article. I'd be happy to try to answer any questions anybody has, but post them here and base them somehow on the scholarpedia article since I won't have time to follow (let alone get entangled in) all the parallel threads.

Travis

At first sight the article looks very nice.

On a first negative note, when checking the little stuff that I know rather well (SR, not QM) by way of test, I find it nicely informative but a bit inaccurate. SR is, just like QM, an empirical theory that is based on observations as summarized in its postulates, and with a resulting prediction of observations. Which is why Einstein could (and did) flip-flop about the ether, and why Lorentz and Langevin could (and did) promote SR . However, I read something similar as what you claim in a book on QM, so I guess that it comes straight out of one or two of such books. And that brings me to a possible weak point of Scholarpedia, it seems to have a rather narrow basis. I thus expect its articles to be a high quality reflection of a narrow range of opinions.

Anyway, I think that it (Scholarpedia incl. your article) is a useful complement to Wikipedia.

 

[ttn]

I don't understand what some people are arguing here.

Thanks mattt. I share your surprise/confusion about what some people are arguing. If we were all co-authors, trying to decide how to structure a not-yet-written article, this kind of squabbling about what level of neutrality is appropriate, etc., would be quite reasonable. But... that's not the situation here. The article is written. If you don't like its style or don't think it's "fair", OK, whatever, don't read it. But there's really no point *arguing* about that.

Anyway, hopefully people will at some point get around to actually reading the thing and then raising questions about the proofs, arguments, definitions, etc.

 

[lugita15]

ttn, what's your opinion of Herbert's version of Bell's proof?
http://quantumtantra.com/bell2.html
I think it may be the simplest known proof of Bell's theorem.

 

[DrChinese]

I don't understand what some people are arguing here.

ttn presents (in his scholarpedia article) several mathematical theorems.

As far as I see, they are all correct. I don't see any mathematical error in his mathematical proofs.

Then you have two possibilities:

1) You find a mathematical error in one of the mathematical proofs he presents. Then show it.

2) You don't find any mathematical error in his mathematical theorems, but you think that the mathematical expression he uses as a "necessary condition of locality" is actually not a necessary condition of locality based on YOUR definition of locality. Then show your definition of locality as clearly as possible and prove that his condition is not a necessary condition for your definition of locality.

I don't see anyone here doing 1) or 2)

(Apologies to ttn for answering mattt's question, which is not directly related to the article itself.)

Considering that the vast majority of the scientific community, including Einstein, believed that realism IS quite relevant to the EPR Paradox (completeness of QM), and therefore to Bell, you shouldn't be surprised that they won't feel compelled to refute this argument. There are just too many side elements to the matter. In other words: unless you are prepared to say that Bell's argument is an unnecessary step to disproving EPR, you cannot ignore realism.

You will quickly see that the base argument ttn is making has both semantic and definitional overtones. I don't expect you, ttn or anyone else to accept my reasoning, but this seems so clear to the remainder of this community it more or less goes without saying:

(1) Locality + Perfect Correlations -> Realism

(2) Since Realism is deduced, and not assumed in (1), then it is not a necessary condition to arrive at the Bell result.

I agree with (1) but disagree with (2). For the leap to occur from (1) to (2), you must assume there exist *simultaneous* Perfect Correlations. That is, the individual "elements of reality" a la EPR exist simultaneously and independently of the act of observation. So the Realism requirement is actually implicit in (1). I think a more correct rendering of (1) is:

(3) Simultaneous (*see note below) Perfect Correlations -> Realism

Notice Locality is dropped as not being a necessary condition for this conclusion. On the other hand, Locality is required so you can satisfy the EPR requirement that you are not disturbing the particle being observed. So then you end up with:

(4) Locality + Realism -> Bell result

QED. So ttn argues (1) and (2) which the rest of us see as (3) and (4). Please, there is no reason anyone to refute my argument as we will just argue over words at this point. I have answered the question about what is in play here. Ultimately, depending on your perspective, you will adopt the definitions and requirements from EPR - or you will not. And that will drive what side you come down on. Note that these things almost all agree upon, regardless of the words:

a) Nature exhibits quantum non-locality (or non-separability, or whatever you call it).
b) Nature is contextual (or non-realistic, or whatever you call it).

In other words, a) and b) are today so wound up together that it takes word games to separate the conclusions of one group from the other. Note that ttn was able to acknowledge that our views are not so dissimilar even though our labels appears to be different. The only truly distinct position is that of those who are still in the local realistic camp - and even their position is not so distinct when you drill into it enough.

* Simultaneous meaning: 3 or more. EPR had only 2 and that is why the EPR Paradox was not resolved prior to Bell. Bell added the 3rd. See after Bell's (14) where this is dropped in quietly.

 

[Demystifier]

Travis, you might be interested to see that in my blog I quoted some nontechnical highlights from your paper:
https://www.physicsforums.com/blog.php?bt=5628#comment5628

 

[ThomasT]

Perhaps I could digest that claim if instead of "evidence" you said "proof". But experiments demonstrating violation of Bell inequalities definitely ARE physical evidence for nonlocality, even if they are not strictly a proof of it.

Physical evidence would entail the observation of a ftl transmission.

But if you still disagree, then it would be helpful if you could answer the following questions:

1. In your opinion, the experimental violation of Bell inequalities is evidence for what?

It's evidence for the existence of a relationship between entangled disturbances that can only be produced via certain preparations. (Hence, BI violations are used as entanglement witnesses.) Whether that relationship is produced locally or nonlocally is an open question. However, wrt Aspect 1982 the standard model seems to indicate that the relationship is produced locally via entangled disturbances being emitted by the same atom. If that's true, and if the global measurement parameter, θ, is, in effect, measuring that relationship, then assuming nonlocality seems unwarranted.

2. Suppose that experimental violation of Bell inequalities has been observed before the theory of quantum mechanics has been discovered. For such experimentalists, what would be a natural interpretation of their experimental results?

The same two options that exist now, I suppose. That either there's some underlying nonlocal transmission (in some medium other than the em medium) between entangled disturbances, or that there's something in the formalism on which BIs are based that doesn't fit the design and execution of the experiments.

 

[ThomasT]

If I were primarily experimentalist, I would assume neither locality nor nonlocality. Instead, I would make experiments without any theoretical prejudices. And if in a particular experiment I would found correlations between spatially separated results of measurements such as those that violate Bell inequalities, then I would conclude (not assume!) that this particular experiment suggests the existence of some nonlocal influences.

The experiments assume that nature is local. BIs are based on a certain formalization of that assumption which limits the correlation between θ and rate of coincidental detection. A limitation which, imho, is not in line with what's known about the behavior of light. BI violation suggests two possibilities -- either nature is nonlocal or there's something about the BI which doesn't fit the experimental design and execution. One can assume the former or the latter as a working hypothesis.

 

[ThomasT]

Let me guess: But you have absolutely no idea what that explanation might be. Am I right?

One hypothesis wrt why BIs are violated is that the limitation they place on the correlation between θ and rate of coincidental detection is unwarranted by the experimental designs and what's known about the behavior of light in scenarios involving crossed polarizers.

And the mere fact that you have no idea how to explain it without nonlocality should already be taken as evidence (not yet a proof) that in some cases nature might be nonlocal.

What's wrong with the notion that θ is measuring a relationship between entangled entities, that that relationship is produced via local interactions/transmissions, and that LR models of entanglement setups are unduly restricted?

Nonlocality is a possibility. But not the most parsimonious working assumption.

 

[ThomasT]

That's exactly what everybody should think -- until they learn about Bell's theorem. In other words, your statement here reads to me like a confession that you haven't looked at or understood Bell's theorem.

Or that I interpret it's physical meaning differently that you do.

Sorry, but none of this makes sense. Look at the role this lambda actually plays in the theorem. It can be *anything*. So the kind of scenario you describe (there are two different "parts" to lambda, one that affects such and such, the other affecting thus and so...) is perfectly well covered already -- i.e., it is already ruled out by the theorem.

Bell concludes that separable predetermination of rate of coincidental detection is ruled out. I agree. The key term here is separable. A nonseparable relationship between λ_a and λ_b can't be separated and encoded in the function determining coincidental detection vis the functions that determine individual detection, and be expected to produce the same correlation curve that using a single nonvarying and nonseparable λ would. But this is what Bell's LR formulation does. So everything isn't perfectly well covered, and resulting BIs place an unwarranted restriction on the correlation between θ and rate of coincidental detection.

I'm just suggesting that the conceptualization of the experimental situation might be more closely scrutinized.

Or, somebody can produce a nonlocal transmission and I'll just shut up.

 

[ThomasT]

I don't understand what some people are arguing here.

ttn presents (in his scholarpedia article) several mathematical theorems.

As far as I see, they are all correct. I don't see any mathematical error in his mathematical proofs.

Then you have two possibilities:

1) You find a mathematical error in one of the mathematical proofs he presents. Then show it.

2) You don't find any mathematical error in his mathematical theorems, but you think that the mathematical expression he uses as a "necessary condition of locality" is actually not a necessary condition of locality based on YOUR definition of locality. Then show your definition of locality as clearly as possible and prove that his condition is not a necessary condition for your definition of locality.

I don't see anyone here doing 1) or 2)

There's a third possibility. That there's no way to explicitly encode any locality condition in the function determining rate of coincidental detection that both clearly represents locality and which isn't at odds with the design and execution of Bell tests. At least I can't think of one.

 

[ThomasT]

I forgot to reply to these, before I take on your paper.

That is patently absurd. Tell me specifically where any of us let some kind of Bohmian bias sneak into the arguments about Bell's theorem.

This profoundly mis-states the situation. I *infer* nonlocality based on my *understanding* of Bell's theorem. You make it sound (with all this talk of "assumptions" and "vested interests") that I and others just *arbitrarily* decide we like nonlocality, so we interpret Bell's theorem that way. That is just backwards. Read the article if you want to actually understand the issues.

Ok. I retract, with apology, my statements regarding assumptions and vested interests. But I still think your inference of nonlocality might be overlooking or mistreating something important in the relationship between LR formulation and experimental design and execution.

To the paper!

 

[ttn]

ttn, what's your opinion of Herbert's version of Bell's proof?
http://quantumtantra.com/bell2.html
I think it may be the simplest known proof of Bell's theorem.

It's nice. (I hadn't seen it before, so thanks for pointing it out!) I don't think it's any simpler, though, than the proof we give in the scholarpedia article -- see the "Bell's inequality theorem" section and in particular the proof that

1/4 + 1/4 + 1/4 > 1.

Actually, this is very very closely related to what Herbert does, so probably instead of arguing about which one is simpler, we should just call them the same proof!

Incidentally, while Herbert's article is a nice proof of (what we in the scholarpedia article call) "Bell's inequality theorem", I find it less than ideal as a proof of non-locality, i.e., a full proof of (what we in the scholarpedia article call) "Bell's theorem". The reason is that it seems to tacitly rely on an assumption that there are "local deterministic hidden variables" determining the outcomes, but without explaining clearly why this is actually not an assumption at all but instead something that follows already from (a) the assumption of locality and (b) the perfect correlations one observes when the polarizers on the two sides are perfectly aligned.

 

[ttn]

I still think your inference of nonlocality might be overlooking or mistreating something important in the relationship between LR formulation and experimental design and execution.

I will be anxious to hear your diagnosis of what, exactly, was overlooked or mistreated.

 

[ttn]

when checking the little stuff that I know rather well (SR, not QM) by way of test, I find it nicely informative but a bit inaccurate.

Could you say exactly what you thought was inaccurate? I couldn't understand, from what you wrote, what you had in mind exactly.

 

[ttn]

Considering that the vast majority of the scientific community, including Einstein, believed that realism IS quite relevant to the EPR Paradox (completeness of QM), and therefore to Bell, you shouldn't be surprised ...

OK, OK, let's go through this again. It's not that complicated. There's no reason we can't all get onto the same page here.

1. Bohr asserts that "QM is complete". It's not entirely clear exactly what this is supposed to mean, but everybody agrees it at least means that particles can never possesses "simultaneous definite values" for non-commuting observables. For example, no spin 1/2 particle can ever possess, at the same time, a definite value for s_x and s_y.

2. EPR (really this is Bohm's 1951 version, but who cares) argue as follows: you can create a pair of spin 1/2 particles such that measuring s_x of particle 1 allows you to know, with certainty, what a subsequent measurement of s_x of particle 2 will yield. And similarly for s_y. So imagine the following experiment: such a pair is created, with one particle going toward Bob and one toward Alice. Now Alice is going to flip a coin (or in some other "random" way, i.e., a way that in no way relates to the physical state of the two particles here under discussion) and measure s_x or s_y on her particle depending on the outcome of the coin flip. She will thus come to know, with certainty, the value of one of these two properties of Bob's particle. So far there is nothing controversial here; it is just a summary of certain of QM's predictions. But now let us *assume locality*. This has several implications here. First, the outcome of Alice's coin flip cannot influence the state of Bob's particle. Second, Alice's subsequent measurement of either s_x or s_y on her particle cannot influence the state of Bob's particle. Now think about what all this implies. Suppose Alice got heads and so measured s_x. Now it is uncontroversial that Bob's particle now possesses a definite s_x value. But it couldn't have acquired this value as a result of anything Alice did; so it must have had it all along. And since Alice could (for all Bob's particle knows) have flipped tails instead, Bob's particle must also have possessed an s_y value all along. (Suppose it didn't. But then, if Alice had got tails, which she might have, Bob's particle wouldn't know how to "answer" if its s_y was subsequently measured... so it might sometimes answer "wrong", i.e., contrary to the perfect correlations predicted by QM.) Conclusion: locality requires Bob's particle to possesses simultaneous definite values for s_x and s_y. (A slightly more precise way to put this would be: simultaneous definite values which then simply get revealed by measurements, i.e., what are usually called "hidden variables", are the *only local way* to account for the perfect correlations that are observed when Alice and Bob measure along the same axis.) This conclusion of course contradicts Bohr's completeness doctrine, so for EPR (who took locality for granted, as an unquestioned premise) this showed that, contra Bohr, QM was actually *incomplete*.

3. Bell shows that these "simultaneously definite values that simply get revealed by measurements" (i.e., hidden variables) imply conflicts with other predictions of QM -- predictions we now know to be empirically correct. Bell concludes that these hidden variables are not the correct explanation of QM statistics, which in turn means that locality is false (since these hidden variables were the only way to locally explain *some* of the QM statistics).

Now the reason I wanted to lay this out is that you insist on grouping 1,2, and 3 together as if they were all some inseparable whole. But they're not. There are two different things going on here. The first one is: the EPR argument, which is a response to Bohr's completeness claim. The logic is simple: EPR prove that locality --> SDV (simultaneous definite values), which in turn shows that completeness is false... so long as you assume locality! Note in particular that if that's all you're talking about -- the EPR argument -- there is no implication whatsoever that reality is non-local, or anything like that. Now the second issue is Bell's theorem. This is the conjunction of 2 and 3 above: locality --> SDV, but then SDV --> conflict with QM predictions; hence locality --> conflict with QM predictions. What I want to stress here is that this completely disentangles from the "completeness doctrine" issue, the issue of whether or not there are hidden variables. That's what I wrote the other day, about "hidden variables" functioning merely as a "middle term" in the logic here. The point is, if you just run the EPR+Bell argument, i.e., prove that locality --> conflict with QM predictions, you don't make any *assumptions* about whether QM is complete or not, and you don't get to *infer* anything about whether QM is complete or not. It just doesn't speak to that at all one way or the other.

Yet you insist on repeating over and over again that "realism [i.e., hidden variables] is quite relevant to the EPR paradox and therefore to Bell". This isn't exactly the wrongest thing ever, but it sure is misleading! You make it sound as if (indeed, I'm pretty sure you believe that) one needs to make some *assumption* about "realism" in order to run Bell's argument. But that isn't the case. And the fact that Bell's argument starts by recapitulating the EPR argument, and that the EPR argument has some implications about "realism" in another discussion, don't change that at all.

unless you are prepared to say that Bell's argument is an unnecessary step to disproving EPR, you cannot ignore realism.

Here you equivocate on "EPR". Does this mean the *argument* from locality --> SDV? Or does it mean the *conclusion*, namely, SDV?

My view is that the *argument* is entirely valid. However we now know, thanks to Bell, that the premise (namely, locality) is false. So we now know that the EPR argument doesn't tell us anything one way or the other about SDV/realism/HVs.

Do you disagree? If so, where is the flaw in the *argument* (recapitulated in 2 above)?

(1) Locality + Perfect Correlations -> Realism

(2) Since Realism is deduced, and not assumed in (1), then it is not a necessary condition to arrive at the Bell result.

I agree with (1) but disagree with (2). For the leap to occur from (1) to (2), you must assume there exist *simultaneous* Perfect Correlations.

Huh? Recall that "realism" here means (for example) that Bob's particle possesses simultaneous definite pre-existing values for both s_x and s_y, which values are simply revealed if an s_x or s_y measurement is made on the particle. Nothing more than that is needed to derive a Bell inequality. (Less, actually, is needed... but this should suffice here.)

(3) Simultaneous (*see note below) Perfect Correlations -> Realism

Huh? You'll have to explain what this SPC means, and then run the proof.

Notice Locality is dropped as not being a necessary condition for this conclusion. On the other hand, Locality is required so you can satisfy the EPR requirement that you are not disturbing the particle being observed. So then you end up with:

(4) Locality + Realism -> Bell result

QED.

I'm sorry, I can't follow this at all.

Ultimately, depending on your perspective, you will adopt the definitions and requirements from EPR - or you will not. And that will drive what side you come down on.

I'm sorry, there is no such ambiguity in the definitions/requirements. The argument is clear. You haven't understood it properly.

* Simultaneous meaning: 3 or more. EPR had only 2 and that is why the EPR Paradox was not resolved prior to Bell. Bell added the 3rd. See after Bell's (14) where this is dropped in quietly.

No, no, no. 2 is plenty. You can have a Bell inequality with only 2 settings on each side; see CHSH. But it doesn't matter anyway. The same exact argument for

locality --> 2-realism

(where "2-realism" means s_x and s_y both have simultaneous definite hidden variable realistic values) also leads immediate to

locality --> 3-realism.

There is no difference at all. You are totally barking up the wrong tree.

 

[ttn]

There's a third possibility. That there's no way to explicitly encode any locality condition in the function determining rate of coincidental detection that both clearly represents locality and which isn't at odds with the design and execution of Bell tests. At least I can't think of one.

How about Bell's locality condition? (See section 6 of the scholarpedia article.)

 

[ttn]

Bell concludes that separable predetermination of rate of coincidental detection is ruled out. I agree. The key term here is separable. A nonseparable relationship between λ_a and λ_b can't be separated and encoded in the function determining coincidental detection vis the functions that determine individual detection, and be expected to produce the same correlation curve that using a single nonvarying and nonseparable λ would.

I can't parse these words, but the issue is simple: does the kind of thing you have in mind respect, or not respect, Bell's definition of locality? If it does, it will make predictions in accord with the inequality (and hence in conflict with experiment). If it doesn't, it's nonlocal and you might as well adopt this simpler characterization of it.

That's what the theorem says. And ... paraphrasing mattt ... it's a theorem. You can't just claim that you "interpret" it differently because you don't like it. Point out the flaw in the proof, or reconcile yourself to it. Those are the options.

 

[DrChinese]

No, no, no. 2 is plenty. You can have a Bell inequality with only 2 settings on each side; see CHSH.

CHSH has 4 settings: 0, 22.5, 45, 67.5. Bell used 3 for his: a, b, c. EPR-B used 2. So you are counting the wrong things. We know entangled pairs can only be measured at 2 angles at a time. But if 2 were plenty, we wouldn't have needed Bell. That is why the EPR Paradox was a "tie" until Bell arrived.

Again, my goal was not to debate the point (as we won't agree or change our minds) but to answer the question of WHY your perspective is not generally accepted. You do not define things the way the rest of us do.

 

[mattt]

OK, OK, let's go through this again. It's not that complicated. There's no reason we can't all get onto the same page here.

1. Bohr asserts that "QM is complete". It's not entirely clear exactly what this is supposed to mean, but everybody agrees it at least means that particles can never possesses "simultaneous definite values" for non-commuting observables. For example, no spin 1/2 particle can ever possess, at the same time, a definite value for s_x and s_y.

2. EPR (really this is Bohm's 1951 version, but who cares) argue as follows: you can create a pair of spin 1/2 particles such that measuring s_x of particle 1 allows you to know, with certainty, what a subsequent measurement of s_x of particle 2 will yield. And similarly for s_y. So imagine the following experiment: such a pair is created, with one particle going toward Bob and one toward Alice. Now Alice is going to flip a coin (or in some other "random" way, i.e., a way that in no way relates to the physical state of the two particles here under discussion) and measure s_x or s_y on her particle depending on the outcome of the coin flip. She will thus come to know, with certainty, the value of one of these two properties of Bob's particle. So far there is nothing controversial here; it is just a summary of certain of QM's predictions. But now let us *assume locality*. This has several implications here. First, the outcome of Alice's coin flip cannot influence the state of Bob's particle. Second, Alice's subsequent measurement of either s_x or s_y on her particle cannot influence the state of Bob's particle. Now think about what all this implies. Suppose Alice got heads and so measured s_x. Now it is uncontroversial that Bob's particle now possesses a definite s_x value. But it couldn't have acquired this value as a result of anything Alice did; so it must have had it all along. And since Alice could (for all Bob's particle knows) have flipped tails instead, Bob's particle must also have possessed an s_y value all along. (Suppose it didn't. But then, if Alice had got tails, which she might have, Bob's particle wouldn't know how to "answer" if its s_y was subsequently measured... so it might sometimes answer "wrong", i.e., contrary to the perfect correlations predicted by QM.) Conclusion: locality requires Bob's particle to possesses simultaneous definite values for s_x and s_y. (A slightly more precise way to put this would be: simultaneous definite values which then simply get revealed by measurements, i.e., what are usually called "hidden variables", are the *only local way* to account for the perfect correlations that are observed when Alice and Bob measure along the same axis.) This conclusion of course contradicts Bohr's completeness doctrine, so for EPR (who took locality for granted, as an unquestioned premise) this showed that, contra Bohr, QM was actually *incomplete*.

3. Bell shows that these "simultaneously definite values that simply get revealed by measurements" (i.e., hidden variables) imply conflicts with other predictions of QM -- predictions we now know to be empirically correct. Bell concludes that these hidden variables are not the correct explanation of QM statistics, which in turn means that locality is false (since these hidden variables were the only way to locally explain *some* of the QM statistics).

Now the reason I wanted to lay this out is that you insist on grouping 1,2, and 3 together as if they were all some inseparable whole. But they're not. There are two different things going on here. The first one is: the EPR argument, which is a response to Bohr's completeness claim. The logic is simple: EPR prove that locality --> SDV (simultaneous definite values), which in turn shows that completeness is false... so long as you assume locality! Note in particular that if that's all you're talking about -- the EPR argument -- there is no implication whatsoever that reality is non-local, or anything like that. Now the second issue is Bell's theorem. This is the conjunction of 2 and 3 above: locality --> SDV, but then SDV --> conflict with QM predictions; hence locality --> conflict with QM predictions. What I want to stress here is that this completely disentangles from the "completeness doctrine" issue, the issue of whether or not there are hidden variables. That's what I wrote the other day, about "hidden variables" functioning merely as a "middle term" in the logic here. The point is, if you just run the EPR+Bell argument, i.e., prove that locality --> conflict with QM predictions, you don't make any *assumptions* about whether QM is complete or not, and you don't get to *infer* anything about whether QM is complete or not. It just doesn't speak to that at all one way or the other.

You are right. I see it cristal clear. Even for those who don't undestand the previous explanation with words, in his scholarpedia article he proves it mathematically (with clearly stated mathematical definitions and mathematically correct proofs, as far as I could check).

The only way out I see (for those who don't like this result) is to show that what he calls "a necessary condition of locality" (and he defines it clearly in mathematical terms) is not a necessary condition of locality for YOUR definition of locality (and you must show your own definition of locality as clearly as possible and you must prove that it doesn't imply his condition).

Another way out is to think in an incredible great cosmic conspiration.

 

[unusualname]

assume the universe is a one path version of MWI.

there is no "non-locality" is there?

 

[martinbn]

ttn, in your description of the Alice and Bob experiment you keep talking about the two particles as separate systems, which they are not. I think it needs more careful phrasing.

 

[billschnieder]

Hello Travis,

In the section titled "Bell's inequality theorem" you derive Bell's inequality supposing that the experimental outcomes were non-contextual (cf "To see this, suppose that the spin measurements for both particles do simply reveal pre-existing values."). To your credit, in the section on "Bell's theorem and non-contextual hidden variables" you discuss the fact that non-contextual hidden variables are naive and unreasonable.

You then proceed to show that you can still obtain the inequalities by assuming only locality in the section titled "The CHSH–Bell inequality: Bell's theorem without perfect correlations".

(1) You say

"While the values of A1 and A2 may vary from one run of the experiment to another even for the same choice of parameters, we assume that, for a fixed preparation procedure on the two systems, these outcomes exhibit statistical regularities. More precisely, we assume these are governed by probability distributions Pα1,α2(A1,A2) depending of course on the experiments performed, and in particular on α1 and α2."

By "statistical regularities" do you mean simply a probability distribution Pα1,α2(A1,A2) exists? Or are you talking about more than that.

(2) You say

"However, if locality is assumed, then it must be the case that any additional randomness that might affect system 1 after it separates from system 2 must be independent of any additional randomness that might affect system 2 after it separates from system 1. More precisely, locality requires that some set of data λ — made available to both systems, say, by a common source16 — must fully account for the dependence between A1 and A2 ; in other words, the randomness that generates A1 out of the parameter α1 and the data codified by λ must be independent of the randomness that generates A2 out of the parameter α2 and λ ."

What if instead you assumed that λ did not originate from the source but was instantaneoulsy (non-locally) imparted from a remote planet to produce result A2 together with α2, and result A1 together with α1. How can you explain away the suggestion that the rest of your argument, will now prove the impossibility of non-locality?

(3) You proceed to derive your expectation values Eα1,α2(A1A2|λ), defined over the probability measure, Pα1,α2(⋅|λ) and ultimately Bell's inequality based on it

C(\alpha_1,\alpha_2)=E_{\alpha_1,\alpha_2}(A_1A_2)=\int_\Lambda E_{\alpha_1,\alpha_2}(A_1A_2|\lambda)\,\mathrm dP(\lambda),
...
|C(\mathbf a,\mathbf b)-C(\mathbf a,\mathbf c)|+|C(\mathbf a',\mathbf b)+C(\mathbf a',\mathbf c)|\le2,

To make the following clear, I'm going to fully specify the implied notation in the above as follows:

|C(\mathbf a,\mathbf b|\lambda)-C(\mathbf a,\mathbf c|\lambda)|+|C(\mathbf a',\mathbf b|\lambda)+C(\mathbf a',\mathbf c|\lambda)|\le2,

Which starts revealing the problem. Unless all terms in the above inequality are defined over the exact same probability measure. The above inequality does not make sense. In other words, the only way you were able to derive such an inequality was to assume that all the terms are defined over the exact same probability measure P(λ). Do you agree? If not please, show the derivation. In fact the very next "Proof" section explicitly confirms my statement.

(4) In the section titled "Experiments", you start by saying:

Bell's theorem brings out the existence of a contradiction between the empirical predictions of quantum theory and the assumption of locality.

(a) Now since you did not show it explicity in the article, I presume when you say Bell's theorem contradicts quantum theory, you mean, you have calculated the LHS of the above inequality from quantum theory and it was greater than 2. If you will be kind as to show the calculation and in the process explain how you made sure in your calculation that all the terms you used were defined over the exact same probability measure P(λ).
(b) You also discussed how several experiments have demonstrated violation of Bell's inequality, I presume by also calculating the LHS and comparing with the RHS of the above. Are you aware of any experiments in which experimenters made sure the terms from their experiments were defined over the exact same probability measure?

(5) Since you obviously agree that non-contextual hidden variables are naive and unreasonable, let us look at the inequality from the perspective of how experiments are usually performed. For this purpose, I will rewrite the four terms obtained from a typical experiment as follows:

C(\mathbf a_1,\mathbf b_1)
C(\mathbf a_2,\mathbf c_2)
C(\mathbf a_3',\mathbf b_3)
C(\mathbf a_4',\mathbf c_4)

Where each term originates from a separate run of the experiment denoted by the subscripts. Let us assume for a moment that the same distribution of λ is in play for all the above terms. However, if we were to ascribe 4 different experimental contexts to the different runs, we will have the terms.

C(\mathbf a,\mathbf b|\lambda,1)
C(\mathbf a,\mathbf c|\lambda,2)
C(\mathbf a',\mathbf b|\lambda,3)
C(\mathbf a',\mathbf c|\lambda,4)

Where we have moved the indices into the conditions. We still find that each term is defined over a different probability measure P(λ,i), i=1,2,3,4 , where i encapsulates all the different conditions which make one run of the experiment different from another.

Therefore could you please explain why this is not a real issue when we compare experimental results with the inequality.