Violation of Bell inequalities for classical fields?

[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" [itex]V_A[/itex] describing the state at Alice's measuring device, and variables [itex]V_B[/itex] describing the state at Bob's measuring device. Let's separate the variables into three parts: [itex]V_A = (\lambda, \alpha, V_{other_A})[/itex], [itex]V_B = (\lambda, \beta, V_{other_B})[/itex], where [itex]\lambda[/itex] is whatever state information is common to both Alice and Bob (due to the intersection of their backward lightcones), [itex]\alpha[/itex] is Alice's device setting, [itex]\beta[/itex] is Bob's device setting, [itex]V_{other_A}[/itex] is other unknown variables that might be local to Alice's measurement, and [itex]V_{other_B}[/itex] is other unknown variables that might be local to Bob's measurement. Bell is assuming that the probability of Alice getting an outcome [itex]A[/itex] depends only on her state variables, and not Bob's. The probability of Bob getting outcome [itex]B[/itex] depends only on his state variables. So mathematically:

[itex]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)[/itex]

The lightspeed limitation of information propagation is captured in the assumption that the state information common to Alice and Bob, denoted by [itex]\lambda[/itex], includes only information about conditions in the intersection of their backward lightcones. If you don't make a speed of light assumption, then [itex]\lambda[/itex] 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:

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

where [itex]\delta_{x, y} = 1[/itex] if [itex]x=y[/itex] 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.

 

[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.

 

[Ilja]

No, there are no point-like trajectories in the argument. There are two decisions, a and b, and two measurement results, A and B, and there is the assumption that a does not influence B and b does not influence A. No point-like trajectories necessary. Only the hidden variable, which also influences the measurement outcomes, [itex]A(a,\lambda), B(b,\lambda)[/itex].

 

[A. Neumaier]

there are no point-like trajectories in the argument.

These appear silently, in the arguments connecting the formal claims. What goes in there is rarely spelled out completely.
There must be some causal connection between what was prepared and what is measured, and this causal connection is usually described rather informally, using point particle intuition. For waves, the causal connection is complicated, hence there don't seem to be simple arguments.

 

[Ilja]

All what one needs is that the general dependence [itex]A(a,b,\lambda)[/itex] can be reduced to [itex]A(a,\lambda)[/itex], because one assumes that the free decision b what to measure at B cannot causally influence the measurement result A. There is nothing implicit here, this is, completely explicit, Einstein causality. Once this assumption about the non-existence of a causal influence is made, we can continue and prove Bell's inequality.

 

[wle]

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.

How so? What you say about classical relativistic fields here fits in exactly with the locality assumption that Bell was working with. The exact statement in La nouvelle cuisine for instance is:

A theory will be said to be locally causal if the probabilities attached to values of local beables in a space-time region 1 are unaltered by specification of values of local beables in a space-like separated region 2, when what happens in the backward light cone of 1 is already sufficiently specified, for example by a full specification of local beables in a space-time region 3 (Fig. 4).

(Fig. 4 is this Minkowski diagram, with the caption "Full specification of what happens in 3 makes events in 2 irrelevant for predictions about 1 in a locally causal theory.") Bell goes on to explain how this (together with the standard "no superdeterminism/retrocausality" assumption) implies that correlations observed in a Bell-type experiment should admit a factorisation of the form $$P(ab \mid xy) = \int \rho(\lambda) P_{\mathrm{A}}(a \mid x; \lambda) P_{\mathrm{B}}(b \mid y; \lambda) \,, \qquad (*)$$ which in turn implies the Bell inequalities.

Intuitively, (*) roughly expresses the idea that correlations between the outcomes ##a## and ##b## should be explicable in terms of initial conditions ##\lambda## somewhere in the union of their past light cones. This assumes nothing about the underlying dynamics, so I see no reason that a classical relativistic field theory like electromagnetism should be an exception.

 

[zonde]

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.

No, I can't do that. But you are arguing that there is important difference between "locality" in Bell sense and "causal locality" of field model. Here:

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

So I can point you to older Eberhard paper that is using different locality assumption for derivation of inequality. But this derivation of inequality relies on detection efficiency model that is particle based so it can't be applied to field model straight away.

 

[stevendaryl]

No, I can't do that. But you are arguing that there is important difference between "locality" in Bell sense and "causal locality" of field model.

As I said, I think Bell's notion of locality is the assumption that probability distributions factor as follows:

[itex]P(A \wedge B | \lambda, \alpha, \beta) = P(A|\lambda, \alpha) P(B|\lambda, \beta)[/itex]

where [itex]A[/itex] is some yes/no measurement at one location, [itex]B[/itex] is some yes/no measurement at another location, [itex]\alpha[/itex] is the state, or situation, at the first location, [itex]\beta[/itex] is the state, or situation at the second location, and [itex]\lambda[/itex] is state information shared by both measurements. This factorizability assumption isn't the same thing as Einstein causality (or signal locality), because QM is not factorable in Bell's sense, but satisfies Einstein causality (no FTL signals).

 

[wle]

As I said, I think Bell's notion of locality is the assumption that probability distributions factor as follows:

[itex]P(A \wedge B | \lambda, \alpha, \beta) = P(A|\lambda, \alpha) P(B|\lambda, \beta)[/itex]

where [itex]A[/itex] is some yes/no measurement at one location, [itex]B[/itex] is some yes/no measurement at another location, [itex]\alpha[/itex] is the state, or situation, at the first location, [itex]\beta[/itex] is the state, or situation at the second location, and [itex]\lambda[/itex] is state information shared by both measurements.

If you're following Bell then, strictly speaking, the definition of locality he works with is what I quoted in my previous post. In your notation: that, given sufficiently specified information ##\lambda## in a suitable part of the past light cone of (e.g.) ##A##, knowledge of ##B## and ##\beta## should be redundant for making predictions about ##A##, or $$P(A \mid B, \alpha, \beta, \lambda) = P(A \mid \alpha, \lambda) \,.$$ The factorisation condition you wrote follows from this and the definition of conditional probability: $$\begin{eqnarray*}
P(A, B \mid \alpha, \beta, \lambda) &=& P(A \mid B, \alpha, \beta, \lambda) P(B \mid \alpha, \beta, \lambda) \\
&=& P(A \mid \alpha, \lambda) P(B \mid \beta, \lambda) \,.
\end{eqnarray*}$$

 

[zonde]

As I said, I think Bell's notion of locality is the assumption that probability distributions factor as follows

Yes (also wle made a good point)

This factorizability assumption isn't the same thing as Einstein causality (or signal locality), because QM is not factorable in Bell's sense, but satisfies Einstein causality (no FTL signals).

There is distinction between "can't be used to produce FTL signals" and "do not need FTL signals to explain phenomena".
As I understand you define "Einstein causality" as the former. But I don't see much point in such a utilitarian definition.

 

[stevendaryl]

Yes (also wle made a good point)

There is distinction between "can't be used to produce FTL signals" and "do not need FTL signals to explain phenomena".
As I understand you define "Einstein causality" as the former. But I don't see much point in such a utilitarian definition.

Well, to me the fact that QM nonlocality can't be used to send signals is pretty important. I'm not exactly sure what it means, but my feeling is that if there were truly something faster-than-light going on, there should be a way to expose it to experiment.

 

[zonde]

Well, to me the fact that QM nonlocality can't be used to send signals is pretty important. I'm not exactly sure what it means, but my feeling is that if there were truly something faster-than-light going on, there should be a way to expose it to experiment.

Being skeptical is important part of doing science.