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.