Local Causality and Bell's Second Theorem

[bhobba]

Hi All

I am a bit exasperated right now. On another forum a person claimed Bell's second theorem proved QM was not local. I carefully explained what local causality was, and what the theorem states: There exist quantum phenomena for which there is no theory satisfying local causality.

It of course is true - but the person simply did not get the reason it failed - namely for local causality to be applicable the particles need to be factorizeable - and that precisely is what many interpretations reject for entangled systems.

Have I got something wrong?

Thanks
Bill

 

[Lord Jestocost]

There exist quantum phenomena for which there is no theory satisfying local causality.

No classical Theory! As Murray Gell-Mann puts it in “The Quark and the Jaguar”:

"The label “nonlocal" applied by some physicists to quantum-mechanical phenomena like the EPRB effect is thus an abuse of language. What they mean is that if interpreted classically in terms of hidden variables, the result would indicate nonlocality, but of course such a classical interpretation is wrong."

 

[Demystifier]

namely for local causality to be applicable the particles need to be factorizeable

What do you mean by "factorizeable particles"?

 

[bhobba]

What do you mean by "factorizeable particles"?

Its sometimes called Bell Locality - non technically it means you have intuitive separate particles, technically its the usual condition of probabilistic Independence ie A theory θ is factorisable, i.e. satisfies factorisabilty, iff Pθ(A, B|a, b, c, λ) = Pθ(A|a, c, λ)Pθ(B|b, c, λ).

Thanks
Bill

 

[Minnesota Joe]

Hi All

I am a bit exasperated right now. On another forum a person claimed Bell's second theorem proved QM was not local. I carefully explained what local causality was, and what the theorem states: There exist quantum phenomena for which there is no theory satisfying local causality.

It of course is true - but the person simply did not get the reason it failed - namely for local causality to be applicable the particles need to be factorizeable - and that precisely is what many interpretations reject for entangled systems.

Have I got something wrong?

I too understand Bell to have ruled out nature being entirely local, but it would be good to clear up if I'm wrong. There is a bunch of ambiguous terminology surrounding it (or at least confusing to me).

Here's Tim Maudlin on the matter:

"What Bell’s theorem, together with the experimental results, proves to be impossible (subject to a few caveats we will attend to) is not determinism or hidden variables or realism but locality, in a perfectly clear sense. What Bell proved, and what theoretical physics has not yet properly absorbed, is that the physical world itself is non-local. "
https://arxiv.org/ftp/arxiv/papers/1408/1408.1826.pdf

I think some of his caveats are that super-determinism is false, that measurements have unique outcomes, and that QM correctly predicts experimental results.

 

[bhobba]

I too understand Bell to have ruled out nature being entirely local, but it would be good to clear up if I'm wrong.

It's involved in his second theorem which you can look up. Basically it says local causality is one particle can only send information to another particle at a finite speed. It proves QM fails local causality. It can fail explicitly as stated in the wording, but can fail in a more subtle way - namely you don't have two separate particles. This is sometimes stated as, not rigorously, if you have particles, one with property A and and the other with B, then Prob(AB) = P(A)P(B), ie the normal rule of probability for independent events.

Thanks
Bill

 

[Demystifier]

namely for local causality to be applicable the particles need to be factorizeable - and that precisely is what many interpretations reject for entangled systems.

Now that you explained what do you mean by factorizeable particles, I am puzzled why, in the second statement, do you say "many"? Why not all? Are you saying that some interpretations accept factorizability for entangled systems?

Take, for example, QBism or relational interpretation. They claim that QM is local, but they do not accept factorizability. The factorizability is a claim that the joint probability $p(a,b)$ is of the form
$$p(a,b)=p_1(a)p_2(b)$$
but QBism and relational interpretation do not accept it. Instead, they deny the existence of any joint probability $p(a,b)$, factorized or not. More precisely, suppose that Alice measures the result $a$, Bob measures the result $b$, and they are spatially separated. Then no single observer can measure both $a$ and $b$. According to QBism and relational interpretation, the fact that no single observer can measure both $a$ and $b$ is interpreted as a claim that the joint probability $p(a,b)$ does not make any physical sense, and hence doesn't exist.

To conclude, it seems that all interpretations reject factorizability for entangled systems, but local interpretations reject it by saying that the quantity (the factorizability of which is at stake) does not even exist.

 

[bhobba]

Now that you explained what do you mean by factorizeable particles, I am puzzled why, in the second statement, do you say "many"? Why not all? Are you saying that some interpretations accept factorizability for entangled systems?

I was thinking DBB may - but you know more about it than I do.

Thanks
Bill

 

[DrChinese]

Hi All

I am a bit exasperated right now. On another forum a person claimed Bell's second theorem proved QM was not local. I carefully explained what local causality was, and what the theorem states: There exist quantum phenomena for which there is no theory satisfying local causality.

It of course is true - but the person simply did not get the reason it failed - namely for local causality to be applicable the particles need to be factorizeable - and that precisely is what many interpretations reject for entangled systems.

Travis Norsen and a number of others push the idea that there is a "second" Bell Theorem (he published other papers obviously) and that only nonlocal interpretations are viable. That is a controversial result, and is not generally embraced outside of Norsen's circle. (He's a Bohmian, and pushes a revisionist history of the creation story for QM.)

It's not you.

 

[Demystifier]

I was thinking DBB may - but you know more about it than I do.

dBB has two "levels" of appearance.

1. The fundamental deterministic level: At this level it does not make much sense to talk about probabilities, so the concept of factorizable or non-factorizable probabilities does not make much sense.

2. The emergent probabilistic level: At this level the probabilities are given by the standard QM Born rule, so it is certainly not factorizable for entangled states.

 

[Demystifier]

Travis Norsen and a number of others push the idea that ... only nonlocal interpretations are viable.

Bell also pushed that idea, so Norsen, Maudlin and the others are at least right that the Bell theorem as interpreted by Bell himself is a proof that only nonlocal interpretations are viable.

 

[DrChinese]

Bell also pushed that idea, so Norsen, Maudlin and the others are at least right that the Bell theorem as interpreted by Bell himself is a proof that only nonlocal interpretations are viable.

Well, Bell was often a Bohmian.

But no, I don't believe he rigorously believed that nonlocal forces/actions were "proved" by his theorem. And agreeing to a degree with your point: although some of his comments could be interpreted that way, it doesn't actually matter what he personally believed any more than what luminaries such as Einstein, Bohm or Peres believed. Selecting a preferred interpretation is a matter of choice, and Bohmian Mechanics is a respected and viable choice. But so are Many Worlds, acausal/retrocausal interpretations, etc.

Today there exists a ton of experimental and theoretical demonstrations that the original result - QM is not local realistic - stands. Virtually the entire physics community accepts this, and nearly 30 years after Bell's untimely passing nothing has occurred to select any nonlocal interpretation over any non-realistic interpretation.

 

[Minnesota Joe]

Today there exists a ton of experimental and theoretical demonstrations that the original result - QM is not local realistic - stands. Virtually the entire physics community accepts this, and nearly 30 years after Bell's untimely passing nothing has occurred to select any nonlocal interpretation over any non-realistic interpretation.

Realistic is one of the ambiguous terms of art I worried about. What does "realistic" mean in your statement?

 

[DrChinese]

Realistic is one of the ambiguous terms of art I worried about. What does "realistic" mean in your statement?

"Realistic" in this context has a specific meaning, more or less the same as an "element of reality" per EPR (1935). For entangled Alice and Bob, that means:

Realism: Alice's X observable and Bob's non-commuting Y observable - either of which can be separately predicted with certainty - are both simultaneously well-defined even if they cannot both be simultaneously predicted in advance.

In QM: Alice and Bob are part of a single system of 2 entangled particles, rather than 2 separate particles, and therefore their non-commuting observables cannot be well defined. I would say most physicists believe this to be the case regardless of their preferred interpretation. Even a Bohmian would acknowledge that the choice of what to measure is a factor in the outcome. Such viewpoint is called "subjective reality", whereas "realism" requires observer independent reality (also called "counterfactual realism").

Yes, I realize the various terms are enough to make one's head explode.

 

[vanhees71]

That explains, why "realistic" is the utmost vague notion in all these debates. It's derived from the infamous EPR paper (the first author of which didn't like it himself very much) ;-)).

 

[Minnesota Joe]

"Realistic" in this context has a specific meaning, more or less the same as an "element of reality" per EPR (1935). For entangled Alice and Bob, that means:

Realism: Alice's X observable and Bob's non-commuting Y observable - either of which can be separately predicted with certainty - are both simultaneously well-defined even if they cannot both be simultaneously predicted in advance.

In QM: Alice and Bob are part of a single system of 2 entangled particles, rather than 2 separate particles, and therefore their non-commuting observables cannot be well defined. I would say most physicists believe this to be the case regardless of their preferred interpretation. Even a Bohmian would acknowledge that the choice of what to measure is a factor in the outcome. Such viewpoint is called "subjective reality", whereas "realism" requires observer independent reality (also called "counterfactual realism").

Yes, I realize the various terms are enough to make one's head explode.

Thanks, yes, but for some reason that makes it an enjoyable puzzle? Perhaps I'm a masochist.

It was my understanding that there are two assumptions in EPR: locality and that the predictions of QM are correct. Briefly, from these assumptions it follows that incompatible properties must have had their measured values all along; they must have had an "element of reality". Since QM doesn't represent things this way, QM must be incomplete. So there must be a local and complete physical theory yet to be found.

My point is that EPR-realism is derived from EPR-locality.

Then Bell comes along and--or so I thought--and demonstrates that non-locality is a requirement of theories that reproduce the predictions of QM. So you can't have a local complete physical theory.

Where does "realism" come in? In other words, is your EPR-realism required for Bell's proof to go through? Or is EPR-locality really locality and realism? Something else?

 

[DrChinese]

Where does "realism" come in? In other words, is your EPR-realism required for Bell's proof to go through? Or is EPR-locality really locality and realism?

EPR specifically discusses realism as an assumption in the 1935 paper, and Bell included realism as a requirement in his paper as well. From EPR:

"One could object to this conclusion on the grounds that our criterion of reality is not sufficiently restrictive. Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but no both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depends upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this".

EPR felt it was unreasonable that P and Q are not simultaneously real if they could be individually (but not simultaneously) predicted. You can only predict P or Q, not both. And there is no obvious contradiction when considering 2 non-commuting observables.

But Bell saw that there was a contradiction when considering *3* non-commuting observables. So he showed there were scenarios in which 3 such were considered; and there were no value sets that would yield the quantum mechanical expectation values. So yes, Bell explicitly factors in locality (separability) and realism (counterfactual definiteness or elements of reality). When you have both of those (locality, realism), you cannot get the QM expectation.

 

[Minnesota Joe]

EPR specifically discusses realism as an assumption in the 1935 paper, and Bell included realism as a requirement in his paper as well. From EPR:

"One could object to this conclusion on the grounds that our criterion of reality is not sufficiently restrictive. Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but no both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depends upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this".

EPR felt it was unreasonable that P and Q are not simultaneously real if they could be individually (but not simultaneously) predicted. You can only predict P or Q, not both. And there is no obvious contradiction when considering 2 non-commuting observables.

But Bell saw that there was a contradiction when considering *3* non-commuting observables. So he showed there were scenarios in which 3 such were considered; and there were no value sets that would yield the quantum mechanical expectation values. So yes, Bell explicitly factors in locality (separability) and realism (counterfactual definiteness or elements of reality). When you have both of those (locality, realism), you cannot get the QM expectation.

Can you point to where in On the Einstein Podolsky Paradox there is the assumption of realism? Or is it in some later paper?

One difficulty that makes some controversy understandable is that the world "realism" or "realistic" isn't even used by Bell in that paper.

 

[DrChinese]

Can you point to where in On the Einstein Podolsky Paradox there is the assumption of realism? Or is it in some later paper?

One difficulty that makes some controversy understandable is that the world "realism" or "realistic" isn't even used by Bell in that paper.

True, he didn't use that word "realism". Don't blame me for that!

You can use any word or phrase you want to describe it, but it is included specifically in the math of the Bell paper. See after (14), where Bell says "it follows that c is another unit vector". That c is in addition to a and b, which makes 3 possible observables. Which is what I pointed out in my previous post.

Again, please note that regardless of anyone's personal opinion, the following statement is an accurate summary of Bell's Theorem as generally accepted by the scientific community:

No physical theory of local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics.

Note that Hidden Variables is often used interchangeably with Realism. Whether you prefer one wording or another, doesn't really change anything at all. The point is that local realistic theories have long been ruled out by experiment. But theories that are either nonlocal and/or non-realistic have not.

 

[atyy]

"What Bell’s theorem, together with the experimental results, proves to be impossible (subject to a few caveats we will attend to) is not determinism or hidden variables or realism but locality, in a perfectly clear sense. What Bell proved, and what theoretical physics has not yet properly absorbed, is that the physical world itself is non-local. "
https://arxiv.org/ftp/arxiv/papers/1408/1408.1826.pdf

Maudlin is assuming reality, since the physical world is real. As usual there are "outs" like MWI and retrocausation which are often not explicitly discussed, since they may be common knowledge.

However, can locality be saved by not assuming reality? In Norsen and Maudlin's language, it cannot, since realism is a precondition for locality. It also does not make sense to save locality in the sense of
no faster than light communication" since one can have this even assuming reality. So is there a definition of locality that is saved by not assuming reality? Wiseman and Cavalcanti propose one in https://arxiv.org/abs/1503.06413 (see Fig 5).

 

[Minnesota Joe]

True, he didn't use that word "realism". Don't blame me for that!

I definitely wasn't! It just helps explain some of the controversy to me personally.

You can use any word or phrase you want to describe it, but it is included specifically in the math of the Bell paper. See after (14), where Bell says "it follows that c is another unit vector". That c is in addition to a and b, which makes 3 possible observables. Which is what I pointed out in my previous post.

I apologize, but I still don't follow this. Are you saying that adding c makes the realism assumption? Or the parameters themselves?

Again, please note that regardless of anyone's personal opinion, the following statement is an accurate summary of Bell's Theorem as generally accepted by the scientific community:

No physical theory of local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics.

Note that Hidden Variables is often used interchangeably with Realism. Whether you prefer one wording or another, doesn't really change anything at all. The point is that local realistic theories have long been ruled out by experiment. But theories that are either nonlocal and/or non-realistic have not.

I'm not saying changing the wording around changes anything, except my the level of confusion!

That is at least the 3rd sense of realism you have listed:
1. EPR-realism
2. Observer independent reality
3. Hidden variables

They don't seem precisely the same thing and I think that is the type of ambiguity I was trying to resolve.

I'm also not doubting your report of the consensus. I would have said the same thing and in fact did say the same thing prior to reading in quantum foundations.

ETA: Typo

 

[Minnesota Joe]

Maudlin is assuming reality, since the physical world is real. As usual there are "outs" like MWI and retrocausation which are often not explicitly discussed, since they may be common knowledge.

However, can locality be saved by not assuming reality? In Norsen and Maudlin's language, it cannot, since realism is a precondition for locality. It also does not make sense to save locality in the sense of
no faster than light communication" since one can have this even assuming reality. So is there a definition of locality that is saved by not assuming reality? Wiseman and Cavalcanti propose one in https://arxiv.org/abs/1503.06413 (see Fig 5).

Leaving aside the strangeness of physicists arguing against physical reality, I doubt they are "not assuming reality".

I mean, if only my subjective experiences exist, then I'm assuming reality--at least the reality of subjective experiences. Everyone who doesn't find cognito ergo sum convincing is assuming some reality I'd think. So there are definitely uses of 'reality' that are way too broad to apply here.

And you are correct that I'm leaving MWI out because I'm still confused about whether and how it is local and last time I wrote that it is local, it was rightly pointed out to be controversial.

As far as the other "outs"...I think some "theories" just aren't science because they undermine it: solipsism and super-determinism come to mind.

 

[DrChinese]

I definitely wasn't! It just helps explain some of the controversy to me personally.I apologize, but I still don't follow this. Are you saying that adding c makes the realism assumption? Or the parameters themselves?I'm not saying changing the wording around changes anything, except my the level of confusion!

That is at least the 3rd sense of realism you have listed:
1. EPR-realism
2. Observer independent reality
3. Hidden variables

The don't seem precisely the same thing and I think that is the type of ambiguity I was trying to resolve.

I'm also not doubting your report of the consensus. I would have said the same thing and in fact did say the same thing prior to reading in quantum foundations.

I can't do anything to resolve the ambiguity of the different words and their precise definitions, no one can. What is not ambiguous is what the various papers (EPR, Bell, Aspect, etc) do with the math. Keep in mind that Bell ASSUMED the reader knew EPR intimately. He was writing for a very small audience at the time. So he didn't bother to write for anyone other than those who would understand the main argument when published in 1965.

Everyone concluded after EPR that there was a stalemate between 2 main factions. Einstein (on one side) believed local realism was tenable and a more complete quantum theory was possible. Bohr (leading the other side) didn't. Sadly, both died without learning of Bell. With the stalemate, EPR had proved that for an entangled system (Alice and Bob, or a and b) could have 2 non-commuting observables that could be predicted in advance. EPR said that was enough to conclude that each of the entangled particles had well defined values prior to measurement - something that Quantum Theory does NOT provide. They acknowledged that their conclusion would not be valid if you require that 2 or more of the values be predicted simultaneously - something that is not possible. Of course the other faction seized on that as a good reason not to accept the EPR conclusion. No one saw a way out.

Then Bell came along. He realized that looking at JUST 2 observables wasn't good enough. If EPR was correct, EVERY POSSIBLE observable must be predetermined. I.e. not just 2, as EPR thought they proved, but 3, 4, ... infinite.

So Bell put together an example of 3. It turns out that for almost ANY 3 observables on an entangled system, there are NO values which match the quantum expectation values. I won't repeat the Bell argument here as that is not the subject of this thread. This thread is simply to assure anyone interested that a) realism is an assumption of Bell; and b) that it is is generally accepted by the scientific community.

So what you are looking for is someplace in Bell in which there are 3 possible observables that are compared to see if they could match the quantum expectation values. Bell used spin components for his observables, after Bohm's example discussing the EPR issue. Bell labels those components a, b and c. You can see them in full view in Bell's (15). Note that you cannot measure all 3 simultaneously. Bell realized that under the EPR argument, that didn't matter! That's because their assumption was explicitly saying it didn't, and everyone had simply accepted that as reasonable. But we now know it was not.

So to recap: when you attempt to imagine a large set of pairs of entangled particles, and then measure the spin components at various angles, you quickly find that if QM averages are correct, then there cannot be objective reality (i.e. independent of the measurement decision of the observer). Again, read the full Bell paper to follow the argument. Except that Bell is probably the worst paper to read to learn this for the reasons I explained. Better is to read Mermin:

Is the moon there when nobody looks? Reality and the quantum theory

or perhaps my web page:

Bell's Theorem With Easy Math

Again, there is no serious controversy over the main point (the math). The primary controversy is arguments over the best words to use to describe the Bell result. Here are the words I think do the best overall job.

No physical theory of local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics.

Or you can say that either Bell's (2 - locality or whatever you want to call it) and/or (14 - realism or whatever you want to call it) are wrong. Same thing.

 

[Minnesota Joe]

So to recap: when you attempt to imagine a large set of pairs of entangled particles, and then measure the spin components at various angles, you quickly find that if QM averages are correct, then there cannot be objective reality (i.e. independent of the measurement decision of the observer). Again, read the full Bell paper to follow the argument. Except that Bell is probably the worst paper to read to learn this for the reasons I explained. Better is to read Mermin:

Is the moon there when nobody looks? Reality and the quantum theory

or perhaps my web page:

Bell's Theorem With Easy Math

Or read more at any rate.

Again, there is no serious controversy over the main point (the math). The primary controversy is arguments over the best words to use to describe the Bell result. Here are the words I think do the best overall job.

No physical theory of local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics.

Or you can say that either Bell's (2 - locality or whatever you want to call it) and/or (14 - realism or whatever you want to call it) are wrong. Same thing.

I don't think it is merely a matter of semantics, but thank you for trying to explain it.

 

[DrChinese]

Bell's (2) was as follows:
$$ P(a,b) = \int d\lambda \rho (\lambda) A(a, \lambda) B(b, \lambda) $$
It might have been helpful if he had added a (2b) and (2c) as follows (both equally valid individually) so that the maneuver at (14) was more clear:
$$ P(a,c) = \int d\lambda \rho (\lambda) A(a, \lambda) C(c, \lambda) $$
$$ P(b,c) = \int d\lambda \rho (\lambda) B(b, \lambda) C(c, \lambda) $$
So what I'm saying is this is the part that is not ambiguous at all. You can label it with whatever words you like. I can tell you from discussions that most scientists don't really care about whether you call it hidden variables or realism or counterfactual definiteness etc. Those terms are mostly used interchangeably, and they go back to the math when they want to be more precise.

 

[Point Conception]

It seems Bell's # 13 should also be included for the realism assumption: A (a,λ) = -B(a,λ) .
That there are always opposite results when detectors at A and B are aligned.

 

[Demystifier]

However, can locality be saved by not assuming reality? In Norsen and Maudlin's language, it cannot, since realism is a precondition for locality.

To emphasize how absurd local non-realism is, Maudlin explains the anti-realist position sarcastically as: Nothing really exists, but thank God it's local!

 

[Demystifier]

The point is that local realistic theories have long been ruled out by experiment. But theories that are either nonlocal and/or non-realistic have not.

Local realistic theories have been ruled out by experiment, provided that some additional assumptions (no superdeterminism, no backward causation, ...) are taken for granted. Non-realistic theories (local or not) can never be ruled out by experiment. Whatever one observes, one can always interpret it as "it's only in my mind" or "it's only in the apparatus". Even classical mechanics can be interpreted as a non-realistic theory.

 

[Demystifier]

QM is not local realistic - stands. Virtually the entire physics community accepts this

Maybe, but they do not agree what the words "local" and "realistic" mean.

 

[DrChinese]

To emphasize how absurd local non-realism is, Maudlin explains the anti-realist position sarcastically as: Nothing really exists, but thank God it's local!

OK, I smiled at that...

Of course, that is nowhere near what anyone means by anti-realistic. All you need to do to understand the "non-realistic" side of the street is accept that the Heisenberg Uncertainty Relations for a quantum system represents the state correctly for non-commuting observables. For example: There are not simultaneous well defined values for non-commuting P and Q. (I am ignoring interpretations here, as of course the Bohmian view is that they are both well defined but unknowable.)

If you can accept that premise for a single particle, accepting it for a system of 2 entangled particles is not much different. The same issues arise in considering the interaction between the observer and the system being observed, and locality is respected in those interactions. Of course, when Alice and Bob are present, they may be situated such that locality appears to be violated.

My objective is not to convince anyone of the "non-realistic" view, but rather to convince you it is a possibility post-Bell. I have shown where the realistic assumption was included in Bell, by considering 2 additional versions of Bell's (2) that are implied as being simultaneously valid. Bell's (2) was as follows:
$$ P(a,b) = \int d\lambda \rho (\lambda) A(a, \lambda) B(b, \lambda) $$
What Bell implied was simultaneously true (accepting the EPR assumption of the second to last paragraph of EPR):
$$ P(a,c) = \int d\lambda \rho (\lambda) A(a, \lambda) C(c, \lambda) $$
$$ P(b,c) = \int d\lambda \rho (\lambda) B(b, \lambda) C(c, \lambda) $$