Assumptions of the Bell theorem

[gentzen]

So, the theorem does not prove that QM is nonlocal, given that there is at least one interpretation that the theorem does not apply to.

Even stochastic interpretations of QM are nonlocal. Or do you know one which is not?

 

[Demystifier]

So, the theorem does not prove that QM is nonlocal, given that there is at least one interpretation that the theorem does not apply to.

But as I said, I don't think that statistical interpretation denies individual events. And Ballentine explicitly says that QM is nonlocal. So it's not clear why do you think so.

 

[Fra]

I am confused again, but as we are talking abot EPR here, I assume we by local means "bell-local" right? or which version of locality are we talking about?

There was at some paper ealier in the thred this definition

"A physical theory is EPR-‐local iff according to the theory procedures carried out in one region do not immediately disturb the physical state of systems in sufficiently distant regions in any significant way"

This looks like a reasonable definition to me.

But I really suspect there is a blurring of the concept of locality and various presumptions about causality. I see its' often said that the "bell locality condition" is the factorizability or partitioning of the sum or the hidden variable paths.

Ie it seems sometimes the partition
$$ P(A |O_ {A}) = \sum_{\lambda} P(A|\lambda|O_ {A}) P(\lambda|O_ {A}) $$
is what some seem to label "bell locality condition", is this what some talk about? If so, I object to that notion as this contains also implicit assumptions of causation.

So what kind of locality definition do you use, when you say physics is non-local? One can hardly claim that we proved that Alice immediately disturbs Bobs lab, can we?

Can we step back and just sort out the basic concept?

/Fredrik

 

[PeterDonis]

My question is: is one of the assumption that the theory needs to apply to the individual objects, not the ensembles?

The theorem is about correlations between individual measurement results, not ensembles, so I would say yes. However:

the theorem does not prove that QM is nonlocal, given that there is at least one interpretation that the theorem does not apply to.

The theorem is not about QM interpretations, or indeed about QM directly at all. It is a mathematical theorem that applies to the predictions of any theory that satisfies the assumptions. The point about QM in connection with Bell's Theorem is that QM's predictions violate the Bell inequalities, so QM, as a theory that makes predictions (and QM's predictions are independent of any interpretation), must violate at least one of the assumptions of Bell's theorem. And when you look at the math of how QM makes predictions, it is obvious that QM violates the factorizability assumption, the one that says, roughly, that the joint probability distribution for results of two measurements as a function of the settings for each measurement, must factor into two distributions, each of which is only a function of the settings for one of the two measurements. Since the factorizability assumption is usually called the "locality" assumption, the fact that QM violates it means that QM is "nonlocal" in the sense that it violates that assumption.

It is, of course, possible to have QM interpretations that imply violations of other assumptions of the theorem, such as the assumption that measurements have single outcomes, which is violated by the MWI. But none of those interpretations can change the fact that QM, as a theory (independent of any interpretation), violates the factorizability assumption. So none of those interpretations can change the fact that QM is "nonlocal" in the sense of violating that assumption. Refusing to use the term "nonlocal" to describe violating the factorizability assumption does not change the fact that QM does violate it.

 

[Demystifier]

But none of those interpretations can change the fact that QM, as a theory (independent of any interpretation), violates the factorizability assumption.

What about factorizability assumption in classical physics? For instance, do Bertlmann socks violate the factorizability assumption? If they do, then Bell nonlocality is more than violation of factorizability assumption.

 

[martinbn]

Even stochastic interpretations of QM are nonlocal. Or do you know one which is not?

I am not saying that QM isn't nonlocal. I am saying that I don't follow the reasoning in the theorem. So I don't see how the theorem implies it.

But as I said, I don't think that statistical interpretation denies individual events. And Ballentine explicitly says that QM is nonlocal. So it's not clear why do you think so.

Nobody denies individual events. But EPR implies that preexisting values of observable. So, let me ask you this. Alice receives particle, one at a time, and measures the spin along the z-axis. Half of the results are "up", half "down". What is the value of spin-z? You say it is "up" or "down" for each individual particle. But that seems like amusing non-statistical description.

It is, of course, possible to have QM interpretations that imply violations of other assumptions of the theorem, such as the assumption that measurements have single outcomes, which is violated by the MWI. But none of those interpretations can change the fact that QM, as a theory (independent of any interpretation), violates the factorizability assumption. So none of those interpretations can change the fact that QM is "nonlocal" in the sense of violating that assumption. Refusing to use the term "nonlocal" to describe violating the factorizability assumption does not change the fact that QM does violate it.

The theorem in question is a bit more than the inequities and their violations. It is the combinations of the inequalities and EPR + perfect correlations implies preexisting values. All that implies that the locality assumption is wrong, hence nonlocality. Just to point out that this nonlocality is not just the factorazability in Bell's inequalities. It is the Einstein's locality.

 

[Demystifier]

Nobody denies individual events. But EPR implies that preexisting values of observable. So, let me ask you this. Alice receives particle, one at a time, and measures the spin along the z-axis. Half of the results are "up", half "down". What is the value of spin-z? You say it is "up" or "down" for each individual particle. But that seems like amusing non-statistical description.

Before answering your question, let me tell that EPR implies preexisting values if one assumes locality.

Now your question. Yes, the value of spin is either up or down for each individual particle. But it's not any less statistical than the fact that each individual classical coin is either heads or tails.

 

[martinbn]

Before answering your question, let me tell that EPR implies preexisting values if one assumes locality.

Yes, I understand what this says. And here locality is locality and not Bell's locality, right?

Now your question. Yes, the value of spin is either up or down for each individual particle. But it's not any less statistical than the fact that each individual classical coin is either heads or tails.

So, the theorem with statement "every theory that reproduces QM predictions, is nonlocal" has the additional assumption, that it is a theory which describes the dynamics of individual objects and not ensembles, right? So you can add this to your list of assumption.

 

[Demystifier]

Yes, I understand what this says. And here locality is locality and not Bell's locality, right?

Well, I'm not sure what do you mean by "locality" here. The locality used in the EPR argument is not signal locality, nor operators-commuting-at-spatial-separations locality, nor local-Hamiltonian locality. Technically it's also not Bell locality (because EPR did it much before Bell entered the scene), but it's vary akin to Bell locality.

So, the theorem with statement "every theory that reproduces QM predictions, is nonlocal" has the additional assumption, that it is a theory which describes the dynamics of individual objects and not ensembles, right? So you can add this to your list of assumption.

I would add this to the assumptions if there was at least one interpretation of QM which claims to avoid nonlocality by not dealing with individual objects. But I am not aware of any such interpretation.

 

[vanhees71]

Before answering your question, let me tell that EPR implies preexisting values if one assumes locality.

Now your question. Yes, the value of spin is either up or down for each individual particle. But it's not any less statistical than the fact that each individual classical coin is either heads or tails.

In standard QM the spin component in a given direction of a spin-1/2 particle is either determined or undetermined. If it is determined the particle is prepared in an eigenstate of this spin component's representing self-adjoint operator and the spin component then takes the corresponding eigenvalue. If the particle is not prepared in such an eigenstate the spin-component's value is indetermined, and Born's rule gives you the probabilities to get each of the possible eigenvalues when measuring this spin component.

I don't know, what EPR are thinking. The more often I try to understand this paper the less I succeed ;-).

 

[martinbn]

Well, I'm not sure what do you mean by "locality" here. The locality used in the EPR argument is not signal locality, nor operators-commuting-at-spatial-separations locality, nor local-Hamiltonian locality. Technically it's also not Bell locality (because EPR did it much before Bell entered the scene), but it's vary akin to Bell locality.

Yes, I meant that it is not Bell's factorizability. It is Bell's locality in the sense that going ons here do not affect going ons there.

I would add this to the assumptions if there was at least one interpretation of QM which claims to avoid nonlocality by not dealing with individual objects. But I am not aware of any such interpretation.

The statistical interpretation!

 

[vanhees71]

Again, we are at a point that we have to define "locality", because this word has so many meanings that it's hard to discuss without giving a clear definition whenever it's used.

For me "locality" means that there is no faster-than-light signalling in any relativistic theory, and that's fulfilled by local (sic!) relativistic QFT, which implements it in its foundations by assuming the microcausality condition, i.e., that any local observables commute at space-like separated arguments. This particularly holds for the commutator between any local observable and the Hamilton density.

 

[Demystifier]

For me "locality" means that there is no faster-than-light signalling in any relativistic theory, and that's fulfilled by local (sic!) relativistic QFT, which implements it in its foundations by assuming the microcausality condition, i.e., that any local observables commute at space-like separated arguments. This particularly holds for the commutator between any local observable and the Hamilton density.

By this definition, even Bohmian mechanics is local.

 

[Demystifier]

The statistical interpretation!

Please provide a reference claiming that statistical interpretation avoids nonlocality by not dealing with individual objects. As far as I am aware, there is none.

 

[martinbn]

Please provide a reference claiming that statistical interpretation avoids nonlocality by not dealing with individual objects. As far as I am aware, there is none.

I am not claiming that. All I said was that I don't see how the argument goes. You stubbornly refuse to explain.

 

[Demystifier]

I am not claiming that. All I said was that I don't see how the argument goes. You stubbornly refuse to explain.

If I don't explain, that's because I don't see what kind of explanation would satisfy you. (It's not that I haven't try to explain it, but obviously you were not satisfied.) That's certainly not a reason to add the assumption to the list.

 

[Demystifier]

It is Bell's locality in the sense that going ons here do not affect going ons there.

Yes, that's correct.

 

[martinbn]

If I don't explain, that's because I don't see what kind of explanation would satisfy you. (It's not that I haven't try to explain it, but obviously you were not satisfied.) That's certainly not a reason to add the assumption to the list.

An explanaition that is phrased in the language of the statistical interpretation. You keep talking about the spin of the particles.

Let's start here. Answer my question from post #146. Alice receives particles, one at a time, and measures the spin along the z-axis. Half of the results are "up", half "down". What is the value of spin-z?

 

[Demystifier]

In standard QM the spin component in a given direction of a spin-1/2 particle is either determined or undetermined. If it is determined the particle is prepared in an eigenstate of this spin component's representing self-adjoint operator and the spin component then takes the corresponding eigenvalue. If the particle is not prepared in such an eigenstate the spin-component's value is indetermined, and Born's rule gives you the probabilities to get each of the possible eigenvalues when measuring this spin component.

I don't know, what EPR are thinking. The more often I try to understand this paper the less I succeed ;-).

You overlearned quantum theory, so you are no longer able to frame your thinking in terms of concepts more general than those of standard quantum theory. That's why EPR and various quantum interpretations don't make sense to you. If someone explained it to you when you were in your early 20's, I'm sure it would be very different.

 

[Demystifier]

Let's start here. Answer my question from post #146. Alice receives particles, one at a time, and measures the spin along the z-axis. Half of the results are "up", half "down". What is the value of spin-z?

Let's start at something even more elementary. Alice receives coins, one at a time, and watches their upper side. Half of the results are "heads", half "tails". What is the upper side?

 

[stevendaryl]

For me "locality" means that there is no faster-than-light signalling in any relativistic theory...

"Local" doesn't mean "no FTL". If someone asks you what's a good local restaurant, they're not asking about any restaurant that can be reached before dinner time by traveling at slower than the speed of light.

Locality is the idea of splitting spacetime into small regions, such that what happens in one region is only affected by conditions neighboring regions. Of course, this raises the question of what it means to "affect" something...

But suppose that there were a pair of coins such that, no matter how far separated, if you flip both coins, they always produce the same result: Either both heads or both tails. (More specifically: the sequence of heads and tails produced by one coin matches the sequence produced by the other coin.) Each coin taken separately is completely random--there is no pattern to the sequence of heads and tails. I would consider that a nonlocal effect. It's an effect that is not bound by distance. A theory in which that effect is a "law of physics" is a nonlocal theory. You can't signal using it, but it's not expressible in terms of local evolution. Now, it could be that there is a "deeper" theory that explains the nonlocal behavior using local law. Maybe each coin contains a hidden mechanism that produces a deterministic sequence of heads and tails, and the mechanisms in the two coins are identical. That would explain the nonlocal correlations in terms of a local mechanism.

But the correlations themselves are nonlocal.

There is a distinction between "nonlocal effects" and "nonlocal influences". A nonlocal effect can be "implemented" or "explained" in terms of local interactions by proposing a mechanism for establishing a correlation. The connection with FTL is not at all that FTL means the same thing as locality. Rather, the implication is this:

If you can show that the nonlocal effect can be used for FTL signaling,​

then you know that there can be no local explanation for the effect.​

So it's a theory-independent conclusion, in the sense that no matter underlying "deeper theory" one proposes to explain the effect, if there is FTL signalling involved, then it can't have a local explanation.

Bell's proof is another, more general, to derive a theory-independent conclusion.

I think it's much more productive to separate the idea of a "nonlocal effect" from a "nonlocal influence".

 

[stevendaryl]

If you can show that the nonlocal effect can be used for FTL signaling,​

then you know that there can be no local explanation for the effect.​

Once people start talking about "superdeterminism", even this conclusion becomes suspect. I assume that superdeterminism could be used to "implement" FTL signaling.

 

[WernerQH]

I would add this to the assumptions if there was at least one interpretation of QM which claims to avoid nonlocality by not dealing with individual objects. But I am not aware of any such interpretation.

There's the rub: talk of "objects". There's no need for "objects" traveling from A to B, and quantum theory is absolutely silent on what happens "in between". It just gives you correlations betwen "state preparation" and "measurement" events. Objects like "electrons" and "photons" are just classical ideas that we have foisted on the microworld.

I'm aware that it is a really compelling idea to explain the correlations with "particles" carrying information with them. But I'd forgo such explanations in favour of a mere description of events, if it offers a chance of restoring common sense.

 

[Demystifier]

There's the rub: talk of "objects". There's no need for "objects" traveling from A to B, and quantum theory is absolutely silent on what happens "in between". It just gives you correlations betwen "state preparation" and "measurement" events. Objects like "electrons" and "photons" are just classical ideas that we have foisted on the microworld.

I'm aware that it is a really compelling idea to explain the correlations with "particles" carrying information with them. But I'd forgo such explanations in favour of a mere description of events, if it offers a chance of restoring common sense.

I'm not convinced that that's enough to restore common sense. Furthermore, even if we accept that there are only events and not objects existing between two events, there is still the question why are those events correlated.

 

[Fra]

After all the last pages of posts, I would say locality is not the issue or concept we should be talking about here. The "correlation" kind of non-locality, is not problematic in itself. And all seem to also agree that we do not have instant causation (FTL). So this is mostly due to communication issues.

The core issue seems to still be the lack of an qualified explanatory causation chain or physical interactions, that explains the "non-local correlations", but WITHOUT beeing stopped by Bells theorem.

We know any theory with Bells theorems ansatz of partitioning of causation chain is wrong.

We know QM makes the right predictions, but it lacks the qualified explanatory value, so in this SENSE, it is obvious that QM if not "incomplete", at least unsatisfactory to use a less confusing word.

But then QM really does not say much more about the nature of physical interactions than does classical physics, as hamiltoninans or action formulations are usually pulled from a hat. The lesson we can draw from this is that physical interactions most certainly does not work like mechanical chains or physical influence. If there is anything we can guess from QM, its that physical interaction likely involved "self-reflection", meaning that INFORMATION is an integral part of understanding actions - unlike a completelty "mechanical" paradigm: Which means, that when A and B interact, itäs not best understood in terms of them physically poking each each, but that the action A chooses towards B, depends not on B, but in A's "image" of B. Such a parardigm seem to have the potential to explain correlations, but without beeing stopped by Bells theorem, as the partitioning assumption can not be justified.

/Fredrik

 

[Demystifier]

And all seem to also agree that we do not have instant causation (FTL).

No! In Bohmian mechanics, for instance, there is instant FTL causation. But all interpretations agree that there is no instant FTL signaling. Do I have to explain the difference?

 

[WernerQH]

even if we accept that there are only events and not objects existing between two events, there is still the question why are those events correlated.

Some people may well find a QFT calculation sufficient as an explanation. I certainly do, and I don't abhor propagators reaching backwards in time.

 

[Fra]

No! In Bohmian mechanics, for instance, there is instant FTL causation. But all interpretations agree that there is no instant FTL signaling. Do I have to explain the difference?

Unless you by FTL-causation, means FTL-correlation, do you mean that causation is a one-way communication, and signaling is two-way communication?

With causation, I assume you mean that something here changes the physical situation at a remote location - not that the local INFORMATION about the remove location is changed? IS that correct?

Would your FTL causation allow us to set up an "instant" trigger for a one shot communication, where A can send a one-way "trigger signal" instantly to B? (And perhaps A and B, can pre-agree on the meaning of this trigger)

/Fredrik

 

[stevendaryl]

Unless you by FTL-causation, means FTL-correlation, do you mean that causation is a one-way communication, and signaling is two-way communication?

I'm not sure exactly what @Demystifier is meaning by the distinction, but to me, FTL causation is in terms of a proposed law of physics. If the state of one object at a future time depends on the state of distant objects in the recent past (too recently for light to propagate), then that implies FTL causation.

FTL signaling is a special case in which the conditions can be manipulated.

Here's a made-up law of physics that might illustrate the difference. Suppose that there is some weird object, a will-o-the-wisp, which just randomly appears at various locations, and then disappears, only to re-appear at some random spot. Suppose that there is a force ##F_{wow}## which acts instantaneously on electrons everywhere in the universe, and is constant in magnitude, and is directed toward the will-o-the-wisp.

This would imply FTL causation: the will-o-the-wisp affects electrons instantaneously. But it couldn't be used for FTL signaling, since there is no way to control where the will-o-the-wisp appears.

 

[vanhees71]

"Local" doesn't mean "no FTL". If someone asks you what's a good local restaurant, they're not asking about any restaurant that can be reached before dinner time by traveling at slower than the speed of light.

Locality is the idea of splitting spacetime into small regions, such that what happens in one region is only affected by conditions neighboring regions. Of course, this raises the question of what it means to "affect" something...

But suppose that there were a pair of coins such that, no matter how far separated, if you flip both coins, they always produce the same result: Either both heads or both tails. (More specifically: the sequence of heads and tails produced by one coin matches the sequence produced by the other coin.) Each coin taken separately is completely random--there is no pattern to the sequence of heads and tails. I would consider that a nonlocal effect. It's an effect that is not bound by distance. A theory in which that effect is a "law of physics" is a nonlocal theory. You can't signal using it, but it's not expressible in terms of local evolution. Now, it could be that there is a "deeper" theory that explains the nonlocal behavior using local law. Maybe each coin contains a hidden mechanism that produces a deterministic sequence of heads and tails, and the mechanisms in the two coins are identical. That would explain the nonlocal correlations in terms of a local mechanism.

But the correlations themselves are nonlocal.

There is a distinction between "nonlocal effects" and "nonlocal influences". A nonlocal effect can be "implemented" or "explained" in terms of local interactions by proposing a mechanism for establishing a correlation. The connection with FTL is not at all that FTL means the same thing as locality. Rather, the implication is this:

If you can show that the nonlocal effect can be used for FTL signaling,​

then you know that there can be no local explanation for the effect.​

So it's a theory-independent conclusion, in the sense that no matter underlying "deeper theory" one proposes to explain the effect, if there is FTL signalling involved, then it can't have a local explanation.

Bell's proof is another, more general, to derive a theory-independent conclusion.

I think it's much more productive to separate the idea of a "nonlocal effect" from a "nonlocal influence".

Well, you also didn't precisely tell us, what you mean by "locality". If we want to have chance to know what we are talking about, we'd need a precise mathematical definition. I was referring to the microcausality condition of relativistic QFTs, and that's what I think is what's "locality" with a really important meaning in the discussion about EPR.

I don't know, what you precisely mean by "splitting spacetime into small regions, such that what happens in one region is only affected by conditions neighboring regions."

That doesn't hold for any physical theory, and it doesn't hold by observation, because there are of course signals which can traveling a very far distance and are very extended. E.g., we receive light from galaxies being, the the best of our knowledge, billions of light years away, i.e., electromagnetic waves propagate from their "pretty local" sources over large distances and spread practically "all over space" given enough time.

I also agree with you that one has to distinguish clearly between "long-ranged correlations" (I try to avoid to say "non-local" here) and "long-ranged causal effects".

For me the point about the long-ranged correlations between observables of far-distant parts, fully and accurately describable by local, i.e., "microcausal", relativistic QFT, and their confirmation by various Bell tests, show that there is no contradiction between microcausal QFT, which simply by construction does not allow for faster-than-light propagating causal effects but at the same time describe the said observed long-ranged correlations. There is no spooky interaction at a distance but rather "local interactions" in the sense of microcausality. I'm not sure, whether at the time of the EPR paper microcausality was already so clearly pronounced as it is today. I guess not, because that was before the seminal work of Streater and Wightman et al, where the "axiomatic foundations" of QFT were more clearly worked out.

As I said, I have a hard time to understand what EPR really wanted to say. It's also clear that Einstein himself was not very happy with this paper. There's another paper of 1948, where Einstein himself wrote (in German, which is perhaps important, because Einstein was for sure way more fluent in German than in English) that the real point about quantum theory and entangledment that was bothering him is the "non-separability" rather than the faster-than-light spooky actions at a distance. I think "inseparability" is a much better phrase than calling it "non-local", because it precisely describes the far-distant correlations of parts of a quantum system which are stronger than any such correlations as described by what Bell called "local realistic theories". As we know today, thanks to this work by Bell, Nature rather behaves as predicted by quantum theory than as is describable by "local realistic" theories.

 

[gentzen]

Again, we are at a point that we have to define "locality", because this word has so many meanings that it's hard to discuss without giving a clear definition whenever it's used.

I understand that you have a real problem with the word "locality" independent of this current discussion on Bell's theorem. I was so happy that stevendaryl gave a clear explanation of nonlocal randomness, but somehow I already suspected that it would help you only a tiny little bit. And your response confirms that expectation. I am fully aware that I will be even less able to help you, but I will try nevertheless.

For me, the fact that QM is nonlocal is not identical to what is proved by Bell's theorem. Even worse, there is no longer a single Bell's theorem, but a variety of related theorems that all go under the heading of Bell's theorem. For each single one of those theorems, you could (and should) exactly write down the assumptions and what the specific theorem proves based on those assumptions. So you have to make precise in each case what exactly you mean by "locality". In most cases this boils down to the separability condition mentioned multiple times in this thread.

However, the word "locality" is not really unclear. It may mean slightly different things in different context, but it is still clear in the "you know it when you see it" sense. So for Bohmian mechanics, the way the trajectories influence each other is nonlocal. For Copenhagen, the way the Born rule updates the wavefunction is nonlocal. Not in the sense of a judgment, but in the sense that this is what we mean when we say that the interpretation is nonlocal. And in the sense of "conservation of difficulty", the expectation is that you will find similar elements of nonlocality in any complete and valid interpretation of quantum mechanics. As an example, consistent histories in its basic form (as a logical framework) does not contain elements of nonlocality. But if you complete it to a full interpretation, then nonlocal elements appear.

 

[Demystifier]

Unless you by FTL-causation, means FTL-correlation, do you mean that causation is a one-way communication, and signaling is two-way communication?

With causation, I assume you mean that something here changes the physical situation at a remote location - not that the local INFORMATION about the remove location is changed? IS that correct?

Would your FTL causation allow us to set up an "instant" trigger for a one shot communication, where A can send a one-way "trigger signal" instantly to B? (And perhaps A and B, can pre-agree on the meaning of this trigger)

No. Causation is influence of one physical object (or phenomenon) on another. Signaling is deliberate influence of a subject (human or intelligent animal) on something else (another subject or a physical object or phenomenon). Signaling is a very anthropomorphic concept, causation is not so much. In deterministic interpretations of QM such as Bohmian mechanics, there is FTL causation (the position of one particle influences the velocity of another particle), but there is no FTL signalling (a human does not have a control over Bohmian particle positions, i.e. she cannot deliberately put the particle here rather than there).

 

[PeterDonis]

do Bertlmann socks violate the factorizability assumption?

No.

 

[PeterDonis]

I meant that it is not Bell's factorizability. It is Bell's locality

This makes no sense. Bell's locality is Bell's factorizability.

 

[PeterDonis]

The theorem in question is a bit more than the inequities and their violations.

If you mean Bell's theorem, no, the theorm is the inequalities. Proving that any theory that satisfies the assumptions must make predictions that satisfy the inequalities is the whole point of the theorem.