Violation of Bell inequalities for classical fields?

[jarekduda]

There is a recent article (Optics July 2015) claiming violation of Bell inequalities for classical fields:
"Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields"
https://www.osapublishing.org/optica/abstract.cfm?URI=optica-2-7-611
arxiv.org/pdf/1506.01305

It says that light's electric field in orthogonal directions behaves as superposition/entanglement.
Kind of dynamics of crystal lattice: we can see it as classical oscillations, or equivalently as superpositions of normal modes: phonons, which are described by quantum mechanics - can violate Bell inequalities.

What do you think about it?

 

[jarekduda]

My email says that this there was (?) this response by Mentz114:
"I think I missed the point of this paper. How is this different from the well know polarization filtering violation ?
I always thought light was on the quantum side of the CM/QM divide."
The standard EPR violation is made by single entangled photons ... here they have a beam of light and measure correlations as intensities - as we can read in the article:
"In the classical context that we are examining, the optical field is macroscopic and correlation detection is essentially calorimetric (i.e., using a power meter, not requiring or employing individual photon recognition)."

 

[A. Neumaier]

violation of Bell inequalities for classical fields

Yes, the Bell inequalities can only be derived under a classical particle assumptions and are violated by classical fields.

See my 2008 lecture Classical and quantum field aspects of light.

 

[Mentz114]

My email says that this there was (?) this response by Mentz114:
"I think I missed the point of this paper. How is this different from the well know polarization filtering violation ?
I always thought light was on the quantum side of the CM/QM divide."
The standard EPR violation is made by single entangled photons ... here they have a beam of light and measure correlations as intensities - as we can read in the article:
"In the classical context that we are examining, the optical field is macroscopic and correlation detection is essentially calorimetric (i.e., using a power meter, not requiring or employing individual photon recognition)."

I deleted my useless reply but since you resurrected it I can add that this is not QM ( so a superposition is a real-space superposition). I quote

In this notation, the field actually looks like what it is, a two-party
superposition of products in independent vector spaces, i.e., an
entangled two-party state (actually a Bell state). Here the two
parties are the independent polarization and amplitude DOFs.

The point being that the inequalities can be violated outside the QM context because of this entanglement.

That's about all I understood, actually.

 

[jarekduda]

Yes, the Bell inequalities can only be derived under a classical particle assumptions and are violated by classical fields.
See my 2008 lecture Classical and quantum field aspects of light.

Dear Prof. Neumaier,
Thank you - I see from your slides that we agree in many points - I will have to study it deeper.
Personally, I require this violation to try to understand field configuration of particles - mainly electron ( https://www.physicsforums.com/threa...-it-a-perfect-point-what-does-it-mean.843297/ ) - get rid of Bell inequality counterargument from soliton particle models, where I base on model of prof. Manfried Faber from TU Wien you probably know (his lecture from EmQM13 in Vienna: http://www.emqm13.org/abstracts/presentation-videos/video-manfried-faber/ ).

I deleted my useless reply but since you resurrected it I can add that this is not QM ( so a superposition is a real-space superposition)

Think about normal modes of coupled pendula ... or of regular lattice of a crystal: they are called phonos, they are described by quantum formalism - superposition of normal modes (like Fourier) of classical field is very similar to quantum entanglement - and so, as the discussed article shows, it can lead to violation of Bell inequalities.

 

[stevendaryl]

Yes, the Bell inequalities can only be derived under a classical particle assumptions and are violated by classical fields.

See my 2008 lecture Classical and quantum field aspects of light.

I'm a little puzzled by your paper, because it seems to contradict what I understood about Bell's theorem. You say that it is the assumption of particles that causes problems, but it seems to me that in Bell's derivation, there is no specific assumption about particles. Instead, he's assuming that if you have two measurements A and B, each of which gives result \pm 1, that take place at a spacelike separation (so there is no possibility of one influencing the other), then the probability that A=B=1 is given by an expression:

\sum_\lambda P(\lambda) P(A=1|\alpha, \lambda) P(B=1|\beta, \lambda)

where \alpha represents whatever state variables are local to A, \beta represents the state variables local to B, and \lambda represents state variables that are in the common backward lightcone of A and B. I don't see that particles specifically come into it.

 

[Truecrimson]

There was already a thread about this paper: https://www.physicsforums.com/threa...ed-with-classical-light-in-experiment.828233/ IMO, this paper doesn't shift the quantum-classical boundary in any meaningful way.

 

[rubi]

I'm a little puzzled by your paper, because it seems to contradict what I understood about Bell's theorem. You say that it is the assumption of particles that causes problems, but it seems to me that in Bell's derivation, there is no specific assumption about particles. Instead, he's assuming that if you have two measurements A and B, each of which gives result \pm 1, that take place at a spacelike separation (so there is no possibility of one influencing the other), then the probability that A=B=1 is given by an expression:

\sum_\lambda P(\lambda) P(A=1|\alpha, \lambda) P(B=1|\beta, \lambda)

where \alpha represents whatever state variables are local to A, \beta represents the state variables local to B, and \lambda represents state variables that are in the common backward lightcone of A and B. I don't see that particles specifically come into it.

Particles come into it in the following way:
In order to prove that a violation of Bell's inequality precludes a classical relativistic particle description, one proves the following theorems:

Theorem 1: Every classical relativistic particle theory implies that probabilities are given by the expression \sum_\lambda P(\lambda) P(A=1|\alpha, \lambda) P(B=1|\beta, \lambda).
Theorem 2: Whenever probabilities are given by the expression \sum_\lambda P(\lambda) P(A=1|\alpha, \lambda) P(B=1|\beta, \lambda), Bell's inequality holds.

An experimental violation of Bell's inequality shows that probabilities can't possibly be given by that expression and by the first theorem, this precludes a classical relativistic particle description.

Now if you want to prove that a violation of Bell's inequality precludes a description by classical relativistic fields, you would have to prove the following theorem:

(tentative) Theorem 1': Every classical relativistic field theory implies that probabilities are given by the expression \sum_\lambda P(\lambda) P(A=1|\alpha, \lambda) P(B=1|\beta, \lambda).

Apparently, the paper from the OP provides a counterexample to this (tentative) theorem, so a violation of Bell's inequality doesn't preclude a description by classical relativistic fields.

It is important to note that theorem 2 (Bell's theorem) on its own does not preclude such descriptions and it is always required to have a theorems like 1 and 1' to establish the preclusion of the respective descriptions. Unfortunately, quantum theory books usually just prove theorem 2 and assume that theorem 1 is trivial. However, a real mathematical proof requires the discussion of the causal structure of spacetime and the tools for its study are usually part of an advanced treatment of GR, rather than a course on QM. Nevertheless, theorem 1 can be proven rigorously. However, theorem 1' seems to be invalid, as the paper from the OP shows. I haven't studied it thoroughly, but the math seems to be alright to me. (The answer in the thread Truecrimson linked misses the point.)

 

[stevendaryl]

Now if you want to prove that a violation of Bell's inequality precludes a description by classical relativistic fields, you would have to prove the following theorem:

(tentative) Theorem 1': Every classical relativistic field theory implies that probabilities are given by the expression \sum_\lambda P(\lambda) P(A=1|\alpha, \lambda) P(B=1|\beta, \lambda).

Apparently, the paper from the OP provides a counterexample to this (tentative) theorem, so a violation of Bell's inequality doesn't preclude a description by classical relativistic fields.

Well, Maxwell's equations are deterministic, so there is no probabilities involved at all. I don't see how Maxwell's equations could violate Bell's inequality. But where probabilities come from (and maybe this is where the point about particles comes in--but I would say it's about quantization, rather than specifically particles) is that if you combine Maxwell's equations with some additional assumptions:

1. For extremely low-intensity light, the energy associated with light is found in discrete packets of size \hbar \omega where \omega is the frequency of the light. A measurement of the energy from the electromagnetic field always results in an integer multiple of this quantum of energy.
2. The intensity of the electromagnetic field gives us the number density of these packets of energy.

So classical E&M together with the assumption that measured energy is quantized produces much (if not all) of quantum weirdness. But that combination is not classical field theory.

 

[rubi]

Well, Maxwell's equations are deterministic, so there is no probabilities involved at all.

Classical relativistic particle theories are also deterministic, yet you can derive the assumptions of Bell's inequality (and therefore Bell's inequality) from them. You just assume a classical probability distribution. In the same way, fields can be distributed according to some classical probability distribution.

I don't see how Maxwell's equations could violate Bell's inequality.

Just because you can't see it, it doesn't mean they can't. What matters is mathematical proof. If you can direct me to a mathematical proof of the (tentative) theorem 1', I will give up immediately. To me it seems that the paper from the OP really provides a counterexample. (But again, I haven't studied it thoroughly, so I might be wrong.)

But where probabilities come from (and maybe this is where the point about particles comes in--but I would say it's about quantization, rather than specifically particles) is that if you combine Maxwell's equations with some additional assumptions:

1. For extremely low-intensity light, the energy associated with light is found in discrete packets of size \hbar \omega where \omega is the frequency of the light. A measurement of the energy from the electromagnetic field always results in an integer multiple of this quantum of energy.
2. The intensity of the electromagnetic field gives us the number density of these packets of energy.

So classical E&M together with the assumption that measured energy is quantized produces much (if not all) of quantum weirdness. But that combination is not classical field theory.

You don't need quantum mechanics to make a photodetector click. This behaviour can in principle also be modeled by fields. If you point the detector towards the sun, you will also hear (lots of) clicks and not a continuous beep. That's just the way a photodetector is built. It cannot not click (unless it is broken). The fact that a photodetector clicks doesn't imply that light needs to come in discrete chunks. (Of course, there are other hints for the quantization of light, but clicking detectors are not one of them.)

 

[zonde]

Just because you can't see it, it doesn't mean they can't. What matters is mathematical proof. If you can direct me to a mathematical proof of the (tentative) theorem 1', I will give up immediately.

There is this informal proof of Bell inequality for particular angles and particular predictions of QM https://www.physicsforums.com/showthread.php?p=2817138#post2817138
It is not using probabilities. However clicks in detectors should be produced in pairs (very much particle like behavior). But experiments support that assumption so it should not be a problem.

 

[rubi]

There is this informal proof of Bell inequality for particular angles and particular predictions of QM https://www.physicsforums.com/showthread.php?p=2817138#post2817138
It is not using probabilities. However clicks in detectors should be produced in pairs (very much particle like behavior). But experiments support that assumption so it should not be a problem.

This seems to be an informal proof of theorem 2, but not a proof of the (tentative) theorem 1'. What I'm looking for is a proof like this:
1. Maxwell's equations hold.
2. ...
3. ...
4. ...
5. Therefore, probabilities are given by stevendaryl's expression.
(Also, I'm a mathematical physicist and not really interested in informal arguments. If theorem 1' holds, it shouldn't be too hard to prove it rigorously and someone certainly would have done it already. Also, I'm not excluding this possibility. I'm just not aware of such a proof and I would like to learn about it if it existed.)

 

[jarekduda]

I see there are some confusions - let's take a closer look at the experiment from the article - the measurement of correlations is here:

https://dl.dropboxusercontent.com/u/12405967/classbell.png

So beam of light comes from the top - assuming some "hidden variables" describing it, they should fulfill Bell inequalities (CHSH) ... but classical EM and experiment say that CHSH is violated (above 2).

We can see Mach-Zehnder interferometer - its path with 'a' polarizer corresponds to encoding 'a' for P(a,b) correlation by choice of the angle of the polarizer.
The 's-a' path is chosen such that the interference will remove mixed term (c_12 in (9)).
Finally the remaining 'c_11' term corresponding to P(a,b) correlation can be retrieved from all three intensities - formula (10) ... and these correlations violate CHSH.

CHSH allows for maximal value B = 2, quantum mechanics allows for at most B = 2.828.
They have reached only B = 2.54, but they explain that it is caused by imperfection of polarization:
"To be careful, we note that in our experiments the field was almost but not quite completely unpolarized; thus, not quite the same field was sketched in the Background Theory section."

 

[A. Neumaier]

where α\alpha represents whatever state variables are local to AA, β\beta represents the state variables local to BB, and λ\lambda represents state variables that are in the common backward lightcone of AA and BB. I don't see that particles specifically come into it.

The locality assumption is extremely restrictive in the context of a field theory. One can always form fields that are nonlocal in Bell's sense. I particular, beams of classical light always have this property (as they satisfy the Maxwell equations).

 

[A. Neumaier]

Well, Maxwell's equations are deterministic, so there is no probabilities involved at all. I don't see how Maxwell's equations could violate Bell's inequality.

The detection process in terms of the photo effect is fully justified for classical light and produces the probabilities.

 

[jarekduda]

Indeed - while Bell inequalities are for classical mechanics, classical field theories are much more complex - for example a single electric charge brings 1/r^2 contribution to electric field of the entire universe - making charge/soliton a highly nonlocal entity.

For EPR-like situation, Noether theorem says that the entire field guards angular momentum conservation (like Gauss law guarding charge) - this information is highly delocalized: it travels from the point of pair creation with speed of light, it is in encoded in the field - we can imagine that particle is accompanied with wave carrying information of both particles.

 

[gill1109]

Bell's argument assumes that at two widely separated locations, one can freely choose a binary setting, and then observe a binary outcome. The time interval between initiation and conclusion of the measurements must be shorter than the time it would take for the distant measurement setting to propagate to the other arm of the experiment. The Qian et al experiments do not have this format at all.

The original poster asks "what do you think of this?". Here's my answer: I think "so what?"

Of course you can find areas of classical physics where the same mathematical structures turn up as those which lead to the EPR-B correlations, so if you follow the analogy, you will be able to find violation of the CHSH inequality and achievement of the Tsirelson bound. It won't have anything at all to do with *locality*; it won't have anything to do with the raison d'etre of Bell's inequality. It quite simply won't be very interesting.

 

[jarekduda]

Sure, it doesn't have anything to do with nonlocality.
It only tests if we can assume Bell-like hidden variables to a classical field - and, like for QM, the answer is: no.

It shows that in contrast to classical mechanics, both QM and field theories are much more complex, they can be seen/decomposed as a superposition of waves.
Like seeing a crystal lattice through classical positions of balls-and-springs ... or through normal modes - phonons, described by quantum mechanics.

 

[zonde]

This seems to be an informal proof of theorem 2

No, this is not proof of theorem 2. Probably you skimmed over the proof as it is not to your taste. This proof does not use probabilities. It is using sort of "what if" type of reasoning as way to dispose of probabilities. But it needs some assumptions:
First it assumes that we speak about perfectly paired discrete detections.
Second, that observed frequencies of individual measurements of pairs reproduce predicted frequencies exactly. Say if prediction is that with particular settings 1/4 of paired detections give matching outcome then we observe 1/4 matching pairs (out of 4n individual pair measurements).
And third, there is no superdeterminism so that we can meaningfully speak about (hypothetically) having phenomena under investigation with exactly the same physical configuration while we are free to apply different measurement settings. This allows us to replace probabilities with "what if" type of reasoning.

But as the topic is classical fields I would like to point out that we can use such approach only if we have discrete "clicks" that we correlate pairwise (first assumption of the proof). But if we get measurements and correlations differently we would need different approach.

 

[zonde]

The locality assumption is extremely restrictive in the context of a field theory. One can always form fields that are nonlocal in Bell's sense. I particular, beams of classical light always have this property (as they satisfy the Maxwell equations).

What if we define locality as "any observable certainty of the state at particular spacetime point is determined solely by it's past lightcone"?
Would you still speak about nonlocality of the fields? It would contradict SR then, as I see it.

 

[stevendaryl]

The locality assumption is extremely restrictive in the context of a field theory. One can always form fields that are nonlocal in Bell's sense. I particular, beams of classical light always have this property (as they satisfy the Maxwell equations).

Sorry for being dense, but I don't understand the sense in which beams of classical light are nonlocal. To me, the sense of "local theory" assumed by Bell in his derivation of his inequality is roughly this: The complete state of the universe at any given time can be described arbitrarily closely by partitioning the universe into little regions, and for each region, give the values (and maybe a finite number of time derivatives) for each type of field, and give the number, momentum, angular momentum, etc. for each type of particle within that region. (This presupposes a choice of splitting spacetime into spacelike slices plus a time coordinate)

I don't see how an electromagnetic field fails to be local in this sense.

 

[A. Neumaier]

I don't see how an electromagnetic field fails to be local

A single beam of polarized monochromatic light may be considered local in your sense, but not the pair of beams that comes out when the beam is subjected to a beam spitter. The outcome oscillates synchronously - it is a single solution of the Maxwell equations localized on two beams rather than one, and cannot be described by giving an independent characterization of each beam. (This has nothing to do with relativity, but with coherence.) But this independence is essential for Bell-type arguments.

 

[rubi]

No, this is not proof of theorem 2. Probably you skimmed over the proof as it is not to your taste. This proof does not use probabilities. It is using sort of "what if" type of reasoning as way to dispose of probabilities. But it needs some assumptions:
First it assumes that we speak about perfectly paired discrete detections.
Second, that observed frequencies of individual measurements of pairs reproduce predicted frequencies exactly. Say if prediction is that with particular settings 1/4 of paired detections give matching outcome then we observe 1/4 matching pairs (out of 4n individual pair measurements).
And third, there is no superdeterminism so that we can meaningfully speak about (hypothetically) having phenomena under investigation with exactly the same physical configuration while we are free to apply different measurement settings. This allows us to replace probabilities with "what if" type of reasoning.

But as the topic is classical fields I would like to point out that we can use such approach only if we have discrete "clicks" that we correlate pairwise (first assumption of the proof). But if we get measurements and correlations differently we would need different approach.

Sorry, but I am unable to extract the validity of theorem 1' from this argument, not even informally. If you didn't intend it to be a proof of theorem 1', then I probably misunderstood you.

What if we define locality as "any observable certainty of the state at particular spacetime point is determined solely by it's past lightcone"?
Would you still speak about nonlocality of the fields? It would contradict SR then, as I see it.

This is exactly what I'm asking for. You claim that this would contradict SR, but a strict mathematical proof seems to be missing. Is it really impossible to construct an observable from a local field, however contrived it may be, which reproduces the predictions of QM? Bell's theorem alone certainly isn't enough to establish the truth of this statement. One definitely needs a theorem like theorem 1' in addition to it. I don't doubt that it exists, but I'm curious about the proof.

 

[zonde]

Sorry, but I am unable to extract the validity of theorem 1' from this argument, not even informally. If you didn't intend it to be a proof of theorem 1', then I probably misunderstood you.

My intention was to avoid the need to prove theorem 1'. The idea was that Herbert's proof avoids probabilities so there is no need to for theorem 1'. And the third assumption applies to particles and fields alike so there is no need for separate theorems if other two assumptions hold as well.

This is exactly what I'm asking for. You claim that this would contradict SR, but a strict mathematical proof seems to be missing. Is it really impossible to construct an observable from a local field, however contrived it may be, which reproduces the predictions of QM? Bell's theorem alone certainly isn't enough to establish the truth of this statement. One definitely needs a theorem like theorem 1'. I don't doubt that it exists, but I'm curious about the proof.

I do not follow you. If do not we restrict any source of changes to past light cone then the changes propagate FTL (or retrocausally). This violates SR.
And I suppose there are observables that can be constructed from fields in past light cone. So the theorem 1' in general sense could not be proved. So we should consider only entangled state observables (conditional observations).

 

[Heinera]

Sorry for being dense, but I don't understand the sense in which beams of classical light are nonlocal. To me, the sense of "local theory" assumed by Bell in his derivation of his inequality is roughly this: The complete state of the universe at any given time can be described arbitrarily closely by partitioning the universe into little regions, and for each region, give the values (and maybe a finite number of time derivatives) for each type of field, and give the number, momentum, angular momentum, etc. for each type of particle within that region. (This presupposes a choice of splitting spacetime into spacelike slices plus a time coordinate)

I don't see how an electromagnetic field fails to be local in this sense.

I think A. Neumaier's point is that since QED implies Maxwell's equations, it is possible to show a predicted violation of Bell's inequality using Maxwell's equations directly, without taking the detour through QM or QED. If I understand him correctly, that is.

 

[rubi]

My intention was to avoid the need to prove theorem 1'. The idea was that Herbert's proof avoids probabilities so there is no need to for theorem 1'. And the third assumption applies to particles and fields alike so there is no need for separate theorems if other two assumptions hold as well.

I see. Nevertheless, I find it desirable to have a pure mathematical proof with clearly stated assumptions and without the need for physical input. After all, it's a purely mathematical question, whether there can exist a classical relativistic field theory that can reproduce quantum mechanics or not. There must be a purely mathematical answer to it. I think we shouldn't be satisfied below this level of rigor.

I do not follow you. If do not we restrict any source of changes to past light cone then the changes propagate FTL (or retrocausally). This violates SR.
And I suppose there are observables that can be constructed from fields in past light cone. So the theorem 1' in general sense could not be proved. So we should consider only entangled state observables (conditional observations).

Yes, one must of course restrict to local observables. I find it non-trivial that no observables can possibly be constructed from relativistic fields that show non-local Bell violating correlations.

 

[A. Neumaier]

since QED implies Maxwell's equations, it is possible to show

The logic is slightly different. The possibility to show Bell violation from QED or QM does not logically imply that it can be shown from the Maxwell equations alone; it only offers the hope that one might be able to do so. But it can be shown by an independent argument, and therefore implies the Bell violation of QED.

This independent argument is relevant since it is based on classical reasoning only. It shows that Bell-type arguments are no obstacle for a possible classical field theory with hidden variables underlying quantum mechanics.

 

[jarekduda]

since QED implies Maxwell's equation

The argument is much simpler - just look at phonons in a crystal: https://en.wikipedia.org/wiki/Phonon
We can describe classical evolution of positions of atoms in a crystal lattice.
Alternatively, we can look at its normal modes as phonos - and describe them using quantum formalism - in a linear theory, sum of normal modes acts as superposition/entanglement.

Classical (lattice/field) and quantum pictures are just two equivalent descriptions of the same system.

 

[Heinera]

The logic is slightly different. The possibility to show Bell violation from QED or QM does not logically imply that it can be shown from the Maxwell equations alone; it only offers the hope that one might be able to do so. But it can be shown by an independent argument, and therefore implies the Bell violation of QED.

This independent argument is relevant since it is based on classical reasoning only. It shows that Bell-type arguments are no obstacle for a possible classical field theory with hidden variables underlying quantum mechanics.

I looked through your presentation you linked to earlier in the thread. My impression is that in your experiment you have two paths, but only one detector (the two paths are combined again before detection), and that the violation of Bell's inequality is essentially due to constructive inteference of these two beams. Is that correct?

 

[A. Neumaier]

in your experiment you have two paths, but only one detector, and that the violation of Bell's inequality is essentially due to constructive interference of these two beams. Is that correct?

Essentially yes. Though I don't get the traditional Bell inequality, I get (as in Bell's traditional setting) a different statistics from the assumption of hidden variables and for quantum mechanics, even in the case of strong laser light. Moreover, the prediction from classical field theory (which applies to the case of strong laser light) is identical with that from quantum mechanics, while it differs from that of hidden variables.

The point is that classical fields can constructively or destructively interfere, while classical particles with hidden variables cannot.

 

[Heinera]

Essentially yes. Though I don't get the traditional Bell inequality, I get (as in Bell's traditional setting) a different statistics from the assumption of hidden variables and for quantum mechanics, even in the case of strong laser light. Moreover, the prediction from classical field theory (which applies to the case of strong laser light) is identical with that from quantum mechanics, while it differs from that of hidden variables.

The point is that classical fields can constructively or destructively interfere, while classical particles with hidden variables cannot.

I certainly agree with your last sentence.

But "proper" Bell experiments use two separate detectors (the beams are never merged), with detection events being space-like separated. So interference can not play a role here. In that setup, I can't see how the classical EM-field theory can explain a violation of the inequality.

 

[A. Neumaier]

But "proper" Bell experiments use two separate detectors (the beams are never merged), with detection events being space-like separated. So interference can not play a role here. In that setup, I can't see how the classical EM-field theory can explain a violation of the inequality.

There is more to quantum electrodynamics than classical electromagnetic fields. The latter do not encode multi-photon entanglement and the associated quantum effects.

But I didn't claim more than I said in my slides. This is enough to demonstrate that arguments based on classical particles with hidden variables are not only logically vacuous for theories based on classical fields but also practically irrelevant.

If one wants to emulate quantum mechanics with classical fields in a way reproducing multiparticle entanglement one must of course equip the classical fields also with hidden variables that can carry the additional information visible in the experiments. However, I don't think there is a single published no-go theorem for field theories with hidden variables. So this is a widely open field for attempting to find a classical picture underlying quantum mechanics.

Since I find the particle-based Bohmian mechanics faulty and inflexible (free fields and free particles have incompatible ontologies!), but have strong reasons to believe that a fundamental theory of the universe cannot be inherently probabilistic, I expect that the true fundamental theory to be found one day will be a kind of classical hidden variable field theory.

 

[Simon Phoenix]

Forgive me for being dense but isn't locality simply encapsulating the notion that some change 'here' cannot affect something 'there' in a time duration smaller than it would take light to get from here to there?

I'm failing to see how Maxwell's equations could be described as non-local in the above sense in any conceivable configuration. Let's suppose we have solved for the EM field given all the boundary conditions etc. We have two remote locations A and B and some change is made at A such that if we were now to solve things for the new boundary conditions we'd get a different EM field at B (it might only be a small difference, but a difference nevertheless). It's going to take time for that change made at A to propagate to B, isn't it?

I also thought Bell's result was really much more general. It simply assumes that there is an experiment performed that gives a binary outcome. So one of these experiments is set up at A and another is set up at B. Now something causes the detectors at A and B to 'click'. When the detector settings and results are compared we find the interesting correlations. I don't think there's any assumption made about what's causing those clicks to occur - particles, fields, furglesquiddles, or whatever. We can suppose there's some dependency on some unknown parameters - but those parameters can be variables or functions (or as Bell himself puts it, even wavefunctions) - the nature of those parameters is unimportant.

The locality assumption is that the probabilities of results 'here' do not depend in any way on settings 'there'. Whether we think of particles or fields there has to be something that, in effect, transmits the information about the settings (if we are to see a violation with a hidden variable model) - that has to be non-local. I think the particle/field issue is a red herring. If there is a classical field theory underpinning QM - then it can't be a local one.

Maybe I'm misunderstanding the arguments. I usually do :-)

 

[A. Neumaier]

Maybe I'm misunderstanding the arguments.

There are two kinds of nonlocality. Causal nonlocality is what you describe - in this sense the Maxwell equations are local. bell nonlocality is a different kind of nonloclaity that doesn't need a relativistic context to be meaningful, and is indeed usually discussed in a non-dynamical context where the speed of light doesn't enter the arguments at all. In this sense, the Maxwell equations are nonlocal. For details see the discussion at http://www.physicsoverflow.org/34140/

 

[Simon Phoenix]

Sorry Prof Neumaier,

I looked at the discussion and I'm afraid I still haven't the faintest idea what is meant by "Bell locality" as opposed to the normal locality in which stuff "here" takes a while to get "there". I thought this causal locality, expressed by the conditions on the conditional probabilities, was rather the whole point of Bell's analysis? Where in Bell's derivations does he assume (explicitly or implicitly) this different kind of 'locality' that you mention? I assume if we have something called Bell non-locality - then we must have something called 'Bell locality' - I just have no idea what this might be.

Could you perhaps explain more simply what you actually mean by Bell non-locality as opposed to everyday common-or-garden non-locality?

 

[A. Neumaier]

Where in Bell's derivations does he assume

Bell nonlocality is defined as what is revealed by violations of Bell type inequalities, hence what Bell calls nonlocality.

These assume nowhere the speed of light, and as there is no dynamics involved in their analysis, there cannot be a relation with how fast information moves. Thus it cannot be causal nonlocality that is captured through his analysis. In fact no information flows between the places where things are measured - the seeming dependence of the correlations comes through the common (entangled) past. It is therefore an illusion to think that something moves with superluminal speed through measurements on entangles states.

 

[stevendaryl]

There are two kinds of nonlocality. Causal nonlocality is what you describe - in this sense the Maxwell equations are local. bell nonlocality is a different kind of nonloclaity that doesn't need a relativistic context to be meaningful, and is indeed usually discussed in a non-dynamical context where the speed of light doesn't enter the arguments at all. In this sense, the Maxwell equations are nonlocal. For details see the discussion at http://www.physicsoverflow.org/34140/

That particular discussion doesn't provide much illumination. It seems to define "Bell nonlocality" in terms of violating Bell's inequality. That would make Bell's proof that every local theory satisfies his inequality to be completely circular.

To me, Bell's theorem doesn't have anything particular to do with particles. To me, he seems to be assuming the following:

1. There is such a thing as the state of the system, and a measurement of the system reveals some fact about that state.
2. The state of an extended system factors into the states of localized parts of the system. Roughly speaking, this means that if you have complete information about the state of region A, and you have complete information about the state of region B, then you have complete information about the union of regions A and B. Entanglement specifically violates this assumption, because there can be facts about pairs of distant particles that cannot factor into facts about each particle, separately. But the hope of Einstein and his colleagues Podolsky and Rosen was that entanglement is a matter of lack of information, in the same way that nonfactorable classical probabilities are due to lack of information. They hoped that QM probabilities were due to ignorance about an underlying physics that satisfied this type of factorability.
3. The state of any region evolves according to the speed of light limitation: The state of one region at one time can only be influenced by states of other regions in the backward lightcone.
   
4. It is possible to perform a measurement that has a discrete (yes/no) outcome.
5. The outcome of a measurement depends only on the state of the small region around the measuring event. Specifically, if you have complete information about the state of the region near the measurement event, then you have as much information as you can possibly have about possible outcomes. Your prediction about possible outcomes can't be made more accurate by learning information about distant regions.

 

[rubi]

Bell nonlocality is defined as what is revealed by violations of Bell type inequalities, hence what Bell calls nonlocality.

These assume nowhere the speed of light, and as there is no dynamics involved in their analysis, there cannot be a relation with how fast information moves. Thus it cannot be causal nonlocality that is captured through his analysis. In fact no information flows between the places where things are measured - the seeming dependence of the correlations comes through the common (entangled) past. It is therefore an illusion to think that something moves with superluminal speed through measurements on entangles states.

It's true that Bell's assumptions don't specifically invoke the causal structure of spacetime, but they are expected to hold for observables in spacelike separated regions and it wouldn't be surprising if they were violated for observables in causally connected regions.

 

[Simon Phoenix]

I'm so sorry (again) but I don't get your logic here Prof Neumaier.

The Bell inequality is derived by using the usual notion of locality - it basically states that any hidden variable theory, be that a theory of fields or particles or oojimaflips that obeys the constraints of this causal locality must lead to probabilities that satisfy this inequality.

Conversely it means that IF we find the inequality violated in an experiment then the system cannot be described by any local theory of fields, particles or oojimaflips.

So you're saying that just because Bell didn't mention dynamics or the speed of light it's a different kind of locality he's talking about? I really don't understand this perspective at all.

 

[stevendaryl]

Bell nonlocality is defined as what is revealed by violations of Bell type inequalities, hence what Bell calls nonlocality.

These assume nowhere the speed of light, and as there is no dynamics involved in their analysis, there cannot be a relation with how fast information moves.

That is not how I understand Bell's argument. The way I see it is that Bell, in his analysis of the EPR experment, is assuming that there are "hidden variables" V_A describing the state at Alice's measuring device, and variables V_B describing the state at Bob's measuring device. Let's separate the variables into three parts: V_A = (\lambda, \alpha, V_{other_A}), V_B = (\lambda, \beta, V_{other_B}), where \lambda is whatever state information is common to both Alice and Bob (due to the intersection of their backward lightcones), \alpha is Alice's device setting, \beta is Bob's device setting, V_{other_A} is other unknown variables that might be local to Alice's measurement, and V_{other_B} is other unknown variables that might be local to Bob's measurement. Bell is assuming that the probability of Alice getting an outcome A depends only on her state variables, and not Bob's. The probability of Bob getting outcome B depends only on his state variables. So mathematically:

P(A, B | \alpha, \beta, \lambda, V_{other_A}, V_{other_B}) = P_A(A | \alpha, V_{other_A}, \lambda) P_B(B | \beta, V_{other_B}, \lambda)

The lightspeed limitation of information propagation is captured in the assumption that the state information common to Alice and Bob, denoted by \lambda, includes only information about conditions in the intersection of their backward lightcones. If you don't make a speed of light assumption, then \lambda could include information about Bob or Alice or both. So Bell's conclusion depends on the lightspeed limitation.

 

[stevendaryl]

I'm so sorry (again) but I don't get your logic here Prof Neumaier.

The Bell inequality is derived by using the usual notion of locality - it basically states that any hidden variable theory, be that a theory of fields or particles or oojimaflips that obeys the constraints of this causal locality must lead to probabilities that satisfy this inequality.

Well, there is a second locality assumption involved in Bell's proof, which is the assumption that state variables are local. That is different from a lightspeed assumption.

Here's an example from classical probability theory that is nonlocal in this sense, even though it doesn't have anything to do with light speed: Suppose that you have a box containing a red ball and a black ball. You randomly select one ball and deliver it to Alice, and deliver the other to Bob.

As far as Alice's and Bob's knowledge about the situation, prior to examining the color of their ball, you would describe it by a probability:

P(X,Y) = probability that Alice's ball is X and Bob's ball is Y = 1/2 (\delta_{X, red} \delta_{Y, black} + \delta_{X, black} \delta_{Y,red})

where \delta_{x, y} = 1 if x=y and is zero, otherwise.

This probability distribution is nonlocal, in that it doesn't factor into independent probabilities for Alice and Bob. Classically, though, nonlocal (or nonfactorable) probability distributions always arise from lack of information.

 

[A. Neumaier]

The state of an extended system factors into the states of localized parts of the system. Roughly speaking, this means that if you have complete information about the state of region A, and you have complete information about the state of region B, then you have complete information about the union of regions A and B.

It is precisely this condition (augmented by the requirement that this information propagates independently if A and B are disjoint) that I call locality in Bell's sense. (I augmented my imprecise description at PhysicsOverflow accordingly.) It has nothing to do with dynamical considerations or the speed of light or light cones.

This condition is satisfied for classical point particles but not for classical coherent waves extending over the union of A and B. The Maxwell equations in vacuum provide examples of the latter, although they satisfy causal locality. Thus causal locality and Bell locality are two essentially different concepts.

 

[Simon Phoenix]

Thus causal locality and Bell locality are two essentially different concepts.

I'm still not really getting the distinction - but I'll think some more about it. Thanks for trying though.

The word local gets used in different contexts, but I remain convinced that the Bell inequalities demonstrate that no locally 'realistic' causal field theory can reproduce all the predictions of QM for spacelike separated measurement events. The essence of the argument for me is that for a hidden variable theory to reproduce the predictions of QM it's going to have to produce a correlation function that has a functional dependence on the relative angle of the detector settings. So in an appropriate frame if Alice changes her mind at the last moment about her measurement setting there can be no way this 'information' is transmitted to Bob's location before Bob's measurement - certainly not with a locally causal field. The role of the hidden variables is to make explicit the reasons for an observed correlation. So although the correlation happens because of some prior connection we can't apply the same reasoning to the last minute change of setting, which for want of a better word occurs pretty much in the 'present'. It's that potential change that must, somehow, be accommodated within our hidden variable description. How does that happen within a causal locally realistic description? I'm not seeing any possible physical explanation of that (within the context of a hidden variable theory).

 

[wle]

It has nothing to do with dynamical considerations or the speed of light or light cones.

I thought it was quite well known that Bell was thinking in terms of relativistic causality. Certainly the reasoning in at least two later works* by Bell are very explicit about being grounded in light cones and such:

• The theory of local beables (1975).
• La nouvelle cuisine (1990).

In both of these works, Bell also explicitly cites classical electromagnetism as an example of the type of model that his theorem applies to. Are you saying that Bell was wrong about the meaning or implications of his own theorem?*A scanned typewritten version of the "local beables" essay is available here. La nouvelle cuisine is available here (NB: behind a paywall). Both are reprinted as chapters in the second edition of the book Speakable and Unspeakable in Quantum Mechanics.

 

[A. Neumaier]

Bell was thinking in terms of relativistic causality.

While this may be true he did formalize something different, and his inequalities are based on that formalization, not on causality.

Moreover, he was clearly thinking in terms of particles, not fields. The propagation of fields violates the basic assumption of Bell-type arguments that systems in disjoint regions propagate independently once they are separated. in a classical relativistic field theory the value of a field at a position x at time t (in a fixed foliation defining observer time) depends on the values of the field at all points at position in the past light cone of x at any fixed earlier time. This allows Bell nonlocal behavior in a causally local field theory.

 

[A. Neumaier]

How does that happen within a causal locally realistic description?

I am not claiming that it does happen; i am just claiming that the assumptions used to derive Bell-type inequalities are not satisfied by classical fields. Hence Bell-type argument and the experimental verification of the inequalities rules out a theory satisfying the Bell locality assumption, But not a classical field theory that is local in the causal sense.

 

[A. Neumaier]

Bell also explicitly cites classical electromagnetism as an example of the type of model that his theorem applies to.

Did he prove that his assumptions were satisfy, or just do some handwaving?

 

[rubi]

I am not claiming that it does happen; i am just claiming that the assumptions used to derive Bell-type inequalities are not satisfied by classical fields. Hence Bell-type argument and the experimental verification of the inequalities rules out a theory satisfying the Bell locality assumption, But not a classical field theory that is local in the causal sense.

Maybe one could argue like this: While it might be possible that there are solutions to classical relativistic field equations that exhibit Bell non-local correlations, these solutions must be regarded as unphysical, since they can never be generated by local interactions. It can be shown that wave-fronts of fields that are initially localized in bounded regions of spacetime don't propagate faster than $c$. So while it might be possible to reproduce quantum non-locality using solutions to classical relativistic field equations, no physical process could ever generate such solutions, since interactions can only generate localized fields. (However, this argument might not apply to other observables constructed from the fields and I'm also not sure whether all conceivable non-local field interactions must necessarily violate Lorentz invariance. I can imagine that some integro-differential equations might work.)

 

[Nugatory]

I thought it was quite well known that Bell was thinking in terms of relativistic causality. Certainly the reasoning in at least two later works* by Bell are very explicit about being grounded in light cones and such:

There's some history here. You are correct that Bell's later writings strongly emphasized relativistic causality, but this is much less true of the earlier presentations (just a few sentences in the original paper). This shift was to a great extent driven by the success of Bell's initial work.

If the detection events are not spacelike separated, then the hypothesis that some causal influence passes from one detector to the other is not dead on arrival - but it still lacks any plausible mechanism so is deeply distasteful. This distaste was behind much of the early hunger for a hidden variable theory that would explain the results at a detector in terms of the state at that detector and the state of the detected particle without considering the other detector and the other particle. From this point of view relativistic causality is a digression - the goal of the early hidden variable program was to get rid of that causal influence altogether, not just the superluminal causal effects.

But then Bell showed that that could not be done... And then the question of how to reconcile this result with relativity becomes unavoidable.

 

[morrobay]

In a good experiment, Alice and Bob choose their measurement angles freely. Moreover, the time elapsed between choice of angle and registration of measurement outcome (on each side of the experiment) is so short (relative to the distance between the two measurement locations), that there is no way that Alice's angle could be known at Bob's location before Bob's measurement outcome is fixed.

I understand your point: Distance between detectors A and B is 20 units.
Distance between emission source and A is 8 units. Source to B is 12 units. 4/c less than 20/c
In the model above post # 45 the elapsed time/space like separation between measurements with any relative angle does not apply.
For every angle Alice sets at detector A there is a pre existing / hidden variable correlated outcome relative to any angle at detector B.