Violation of Bell inequalities for classical fields?

[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.

 

[Point Conception]

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.

 

[Truecrimson]

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.

That sounds like superdeterminism which can never be refuted experimentally, hence is not a very useful model to have.

 

[zonde]

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.

There are different derivations of Bell-type inequalities that use different assumptions. Your claim does not sound serious if it seems like you are unaware of these other ways of deriving Bell-type inequalities.

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.

What is "local in the causal sense"? Is it this:

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]

What is "local in the causal sense"?

Causal means respecting the causality relation induced by the Minkowski metric. For a classical field theory,
it means a family of Poincare invariant hyperbolic differential equations, leading to an upper bound of c for the signal velocity.
Hyperbolicity and Poincare invariance imply the statement you had highlighted.

unaware of these other ways of deriving Bell-type inequalities

I probably have seen all until about 2007, when I started to lose interest since it didn't cover field theories. Maybe you can point me to a more recent paper where I can find a set of assumptions that covers causal fields.

 

[Truecrimson]

Professor Neumaier, in the example in your slides, why can't I use a classical field as a hidden variable?

I'm perplexed, and I want to understand this, because you said basically the same thing Bell said in La nouvelle cuisine.

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 is what Bell said as well. But he then went on to say that, given all the information in the past light cone of x, any information in the region Y space-like separated from x should be irrelevant for any prediction that could be made about a measurement at x. This allows Bell to factorize the probability. But this is not the case in quantum mechanics because a measurement result in Y can provide information that is not available elsewhere that predicts a future outcome at x. So Bell violation should also rule out hidden variables that are classical fields.

 

[A. Neumaier]

given all the information in the past light cone of x, any information in the region Y space-like separated from x should be irrelevant for any prediction that could be made about a measurement at x.

Given all the information in the past light cone of x, any other information is irrelevant, no matter what the separation.

So Bell violation should also rule out hidden variables that are classical fields.

But the past light cone of x will intersect the past light cone of y in a very big region (as the experiment started with a joint preparation),
whereas the usual arguments silently assume that once the trajectories of the particles are separate (i.e., already after an infinitesimal time), their dynamics is independent. This is the case for point particles but not for extended objects such as waves.

 

[Truecrimson]

Thank you for your quick reply. Yes, a complete information from any slice of time t in the past light cone completely specifies what happens at x. But the dynamics from Maxwell's equations is local. So can you clarify what the problem with intersecting past light cones is? I fail to see how classical fields can behave in any way like the situation with entangled states that I described above.

 

[A. Neumaier]

I fail to see how classical fields can behave in any way like the situation with entangled states that I described above.

I don't see how they can behave like that either.

But I see that all arguments I know to derive testable inequalities (I studied this thoroughly until about 2007) make use of assumptions that are inspired by a point particle picture and do not apply to waves.

This just means that in order to rule out classical local hidden field theories one needs more careful arguments; maybe these exist but I haven't seen any. The situation is complicated since the analysis must be dynamical, unlike the simple arguments used for Bell-type inequalities.

 

[stevendaryl]

That sounds like superdeterminism which can never be refuted experimentally, hence is not a very useful model to have.

I don't know why people keep saying that. You can't refute the claim "Everything is determined ahead of time". But you can certainly refute a superdeterministic theory that makes definite predictions. If your superdeterministic theory predicts that "Bob and Alice will always choose the same detector setting", and they DON'T choose the same detector setting, then that particular superdeterministic model is refuted.

 

[Ilja]

But I see that all arguments I know to derive testable inequalities (I studied this thoroughly until about 2007) make use of assumptions that are inspired by a point particle picture and do not apply to waves.

Wrong. There is nothing in the hidden variables which makes any assumption somehow "inspired" by point particles. The hidden variables may be anything, they may even live in 26 dimensional spaces or somewhere else, the other ingredients are the decisions of the experimenters what to measure and the results of the measurements, above are, obviously, macroscopic items, thus, in no way assume anything about microscopic theory.

 

[A. Neumaier]

The hidden variables may be anything

Of course, but in addition to the hidden variables there are the point-like trajectories. I am talking about the latter - these influence the arguments.