# Requirements of All Hidden Variable Theories

1. Nov 28, 2005

### DrChinese

I came across this article, which was published in the February 2004 issue of Physical Review A.

"All quantum observables in a hidden-variables model must commute simultaneously" by James Malley

Abstract: Under a standard set of assumptions for a hidden-variables model for quantum events, we show that all observables must commute simultaneously. And, despite Bell's complaint that a key condition of von Neumann's was quite unrealistic, we show these these conditions are entirely equivalent to those later introduced by Bell, Kochen and Specker. As is it known that these conditions are also equivalent to those under which the Bell-Clauser-Horne inequalities are derived, we see that any experimental violations of the inequalities demonstrate only that quantum observables do not commute.The same conclusion applies to the collection of elegant inequality-free no-go proofs of Greenberger-Horne-Zeilinger, Mermin and Peres.

In essence, this paper states that all hidden variable theories - local or non-local - must necessarily require that observables commute - which is empirically false. This conclusion makes sense to me, if for no other reason than it is another way of saying that reality is observer dependent (and which hidden variable theories are designed to negate).

Von Neumann's famous HV no-go proof of 1932, which was torn down by Bell, is also vindicated in a certain sense. (Although I would not call it a ironclad defense.)

It seems to me that as time passes, it will become clear that no Hidden Variable theory can be made which is both self-consistent and consistent with basic predictions of QM.

2. Nov 28, 2005

### Tez

What troubles me about that paper (its been on my list to think about for a while!) is that one can clearly augement classical mechanics in such a way that some observables don't commute. Namely, one can simply bring in a disturbance principle. So I generally think of non-commutativity as not being "as much" of a unique feature as Bell violation. A more sophisticated line of thinking is to look at a subset of quantum states and measurements for which the wigner function is always positive - these will have an LHV description, but the observables won't necessarily commute. I know those guys know that, so they must be saying something deeper...

3. Nov 28, 2005

### RandallB

I’ve found Bell's writings on von Newmann's points as "absurd" convincing. But I'm not able to follow how Malley has Bell as being "equivalent" to von Newmann's. I'd be concerned that if this link was convincing it would indicate that Bell's would be insufficient to contradict LR by his own argument against von Newmann.

I think for the moment I'll question trying to redeem von Newmann’s as equivalent to Bell.

4. Nov 28, 2005

### ttn

What does that even mean? What's given empirically is information about things that are observed. "Observables" (which I guess you're thinking of as operators given the talk of their status vis a vis commutation) are an element of a particular *theory*, namely orthodox QM. How in the world could it possibly be proved that "all observables must commute simultaneously" for theories which might not even *include* the idea of "operators as observables"?

Case in point: Bohmian Mechanics, which can be formualted in terms of a wave-function obeying Schroedinger's equation and particles following definite trajectories *and that's it*.

So the question I have is this: how could this guy's claim possibly be true in regard to Bohmian Mechanics? What does/would it even *mean* when applied to that theory? And, if the claim doesn't apply to Bohm, what in the heck is it supposed to apply to?

Sigh. Quit worrying abstractly about what hidden variable theories can and can't do, and just study up on Bohm's theory already! Then you won't have to wring your brains trying to figure out what is and isn't possible -- you'll just have an explicit *example* of how a hidden variable theory can be made to actually *work* (and hence a counter-example to most of the silly claims that they can't be made to work).

5. Nov 28, 2005

### DrChinese

I interpret his statement as follows: Bohmian Mechanics cannot be a self-consistent non-local Hidden Variable theory because:

a) BM predicts $$pq\ne qp$$
b) BM is a HV theory
c) All HV theories must predict $$pq=qp$$ (this is the part Malley added)

I realize that those who accept BM as a viable alternative to oQM will not accept his conclusion. However, I think it is worthy of consideration, especially given where it was published.

Specifically, I find merit in the concept that - BY DEFINITION - any theory that claims to be "Bell realistic" (i.e. Hidden Variable theories) cannot have pq=qp. If they did, then they would be an observer dependent theory. And that is precisely why Hidden Variable theories exist: so there is NO OBSERVER DEPENDENCE for reality. What is the point of having Hidden Variables if this fundamental tenet of QM is preserved?

---------

In other words, you SAY you have a "super" theory in your hand that solves every problem - but BM avails itself of certain criticisms and you certainly must be aware of these given your stated views. Why do you have such difficulty acknowledging the obvious? You don't have to agree with a criticism to acknowledge it. So what if Malley's work is aimed squarely at BM? You are always free to write a rebuttal to Malley and get it published.

This above part is not meant to start a debate, but rather to get us back to the substance of this thread. Which is about Malley's work. So I would simply ask if Malley's ideas could be discussed here. I will start a separate thread to discuss the merits of Bohmian Mechanics. I would invite you to participate in that discussion and would be most interested in learning to see BM in its best light.

Last edited: Nov 28, 2005
6. Nov 29, 2005

### Tez

This is where you need to be careful. Take a hidden variable theory which interprets quantum states as epistemic (analogous to Liouville distributions in classical mechanics - i.e. they correspond to some probability distribution over some hidden variables. This is the sort of thing Einstein wanted, in analogy with statistical mechanics and thermodynamics). Then that HVT for QM is very observer dependent - collapse of the wavefunction corresponds to "collapse" of the observers probability distribution when they acquire information after a measurement for example. So features of an HVT may well be observer dependent without being unnatural.

Irrelevant aside: Personally I prefer to call HV theories "ontological models of quantum mechanics" since it is more neutral as to the status of underlying variables...

7. Nov 29, 2005

### DrChinese

I actually prefer the term "realistic" in place of "hidden variables". I think these have very different implications even though they are often thrown around as identical.

Hidden variables are inputs to a black box set of functions. Bell collectively calls these lambda. A Hidden Variable theory hypothesizes lambda exists. This somewhat follows his (1) but lambda is actually discarded in the wash when it comes down to it.

A realistic theory is one in which particle observables have simultaneous definite values independent of the act of observation. This corresponds to the idea of the moon being there when we aren't looking at in, in Einstein's terms. Bell has a completely different definition for these: in his terms, they are A, B and C (actually he calls them a-hat, b-hat, c-hat and these are the hypothetical measurement settings). In the original Bell, this definition comes in several spots, but the critical one is just after his (14): "It follows that c is another unit vector..." (in addition to A and B). You never go anywhere in Bell without having C in addition to A and B, and I think this point is generally overlooked because it is so casually introduced.

So if you can agree with my thinking (probably a dangerous thing to do), then Bell Reality can be defined as the hypothesis that A, B and C exist independently of observation. In my definition of a realistic theory, there can be simultaneous "reality" for observables even if there is no determinism present that explains the values. This is beneficial because you don't waste time trying to determine the mechanical relationship of hypothetical input variables (inputs) to the outcomes (outputs). So skip lambda altogether as you don't need it for anything anyway. You don't need it to prove Bell.

So I believe that a Bell Realistic theory says that a particle has attributes independent of their observation. This view has absolutely nothing to do with entanglement or Bell tests. I do not believe a Bell Inequality will apply to a theory if you don't agree with this definition.

(Admittedly, there are a lot of ways to make your definitions when it comes to Bell and everyone has their own little twists. I am trying to stick with what I personally believe is Bell's exact mathematical definition - which is why I call it Bell Reality. Bell Reality does not exactly map to EPR's "elements of reality", which have a stricter definition than I think is necessary.)

8. Nov 29, 2005

### ttn

My question was what this statement even *means*. Since Bohmian Mechanics isn't formulated in terms of operators, I literally have no idea what you are talking about when you say "BM predicts [that certain operators don't commute]."

I just looked very briefly at Malley's paper, and then at another more recent one (quant-ph/0505016, co-authored with Arthur Fine). This latter paper ends with the following statement: "Finally, the sharp conflict with noncommutativity presented here makes essential use of the inner product machinery of Hilbert space and does not apply to models for quantum theory that do not, with the Bohm model being a primary case in point." In other words, this allegedly-general proof that "Under a standard set of assumptions for a hidden-variables model for quantum events, we show that all observables must commute simultaneously" doesn't even apply to the one really-existing example of a hidden variable theory! At very least this means the proof isn't *nearly* as generally applicable (i.e., as interesting) as Malley tries to suggest. I'm inclined to think this makes the proof entirely *uninteresting*, but maybe that's just me.

As we have discussed before, according to your definition of "Bell realistic", Bohmian Mechanics isn't a Bell realistic theory. Yet it's clearly a hidden variable theory if anything is. So I think your terminology ("Bell realism") is just confusing and misleading.

I don't think I ever said that. But I will say one thing: if one wants to understand whether there might be a viable hidden variable alternative to OQM, one would be better-served by studying Bohm's theory than by studying "impossibility proofs". Study the example first, then you'll be able to see with hardly any effort what's *wrong* with (nearly) all the alleged proofs that it can't work. The fact is, it does work.

Sure, I'm aware of some possible objections to BM. But I'm also aware that 99% of the objections that are usually given against BM, are completely bogus or completely false or complete nonsense. Goldstein does a nice job of listing these bogus objections in the Stanford article you linked to on the other thread.

Different things are obvious to different people, I guess. You seem to think that it's obvious that Malley's proof is general and right, and that this therefore constitutes some kind of problem for BM. To me it's obvious that his proof is (at best) simply inapplicable to (at least) some hidden variable theories. It seems in this case I was right, given what Malley confesses at the end of that other paper.

Yes, you're absolutely right; my intention isn't to hijack this thread. I think there's plenty of value in analyzing Malley's paper. I just think the best way to do that is to know going in that it doesn't apply to BM. Just like, if there were a thread here dedicated to studying von Neumann's anti-hidden-variables proof, I'd interject that a hidden variable theory actually *exists*, so we know going in that the proof is *wrong*, which means our analysis should be in the nature of a post-mortem.

9. Nov 29, 2005

### DrChinese

Apparently, there is an operational difference between "Bell Realism" and "Hidden Variables". Bell Realism is the requirement that observables have well-defined values independent of the act of observation. That is a component of Bell's Theorem, and I see that as simply a statement of fact. If that is true, then the question is does BM qualify as a Bell Realistic theory? Perhaps it doesn't - even if it is truly a hidden variable theory. I do not purport to know enough about BM to be aware of these differences.

But that is certainly relevant to Malley's paper.

EPR argued that QM is incomplete because there are "elements of reality" which map to my definition of Bell Realism (i.e. that observables have well-defined values independent of the act of observation, or in EPR's terms that there is not simultaneous reality to non-commuting observables). Bell essentially shows that conclusion as incorrect. Malley is addressing this too.

But as I follow your argument: BM does not qualify as a Bell realistic theory even if it is a good Hidden Variable theory. The reason is that the distinction between the observer and that being observed is not being asserted by BM (just as it is not in oQM) - and it would need to be in order to qualify. If that were the case, then it would become clear to me that the distinction in terms is more relevant than I would have previously suspected. It certainly never occurred to me that BM could fit in between such cracks!

10. Nov 29, 2005

### ttn

Which ones? All of them? Or just some? BM has particle positions that exist independent of the act of observation (and which observation simply reveals). Does that make it "Bell Realistic"? Or maybe it isn't Bell Realistic, because Bell Realism requires that *all* observables have pre-measurement values that measurement reveals. But then it is *trivial* to see that no "Bell Realistic" theory can reproduce QM predictions. For example, $$L_x = \pm 1/2$$, and same for $$L_y$$ and $$L_z$$. But we know $$L^2 = L_x^2+L_y^2+L_z^2 = 3/4$$. But clearly there are no ways to assign values to the components that will give the right value for $$L^2$$.

In other words, your definition of "Bell Reality" amounts to the same sort of criterion von Neumann and all the others used to prove that you can't have a realistic theory that reproduces the QM predictions. And we know that all of those "proofs" are wrong, because there exists an explicit counter-example. So I will simply repeat my earlier statement that you'd be better served by looking at how a *real*, *viable* hidden variable theory actually *works*, than by arbitrarily dreaming up plausible-sounding criteria for how they "should" work and then worrying about who can prove what theorem about whether in fact they can be made to work that way.

EPR argued that the assumption of locality *requires* the existence of these "elements of reality". Bell didn't prove that this *argument* (from locality to hidden variables) is incorrect. He showed that the *conclusion* of that argument entails something that is contradicted by experiment. And that means the premise of that original argument (locality) must be wrong. That, at least, is how Bell interpreted what he did.

If you think Bell disproved the possibility of hidden variable theories (or you think Bell thought that's what he did) then it's rather a mystery why Bell was the #1 champion of Bohmian Mechanics for about 20 years!

Honestly, I have no idea. You haven't defined "Bell realistic" precisely enough (yet?) to make it possible to decide if a given theory is or isn't "Bell realistic."

Not that I'm encouraging you to do that. As I said above, I think it's rather pointless. This thread started with your praise (well, your mention) of a paper that claims to show that hidden variables are impossible (b/c all hvt's have to endorse commuting observables which is, allegedly, empirically falsified). My point is simply that this seems like a backwards and contorted strategy. If you want to know whether it's possible to have a realistic quantum theory, don't waste your time getting tangled up in these abstract impossibility proofs. It's a waste of time. Just look at Bohm's theory and understand how it works, because it *does* work, and that means all the "impossibility proofs" are *wrong*. Bohm *disproves* the impossibility proofs, by explicit counterexample.

This makes it sound like BM lives in some very dark, shady nether-region of the space of all possible hidden variable theories. It makes it sound like, by virtue of the fact that it eludes all the alleged impossibility proofs, Bohmian mechanics is somehow suspect. But I think that's a very misleading perspective. Bohmian Mechanics is the *obvious* way of adding hidden variables to quantum theory to reproduce the QM predictions (but eliminate the measurement problem, i.e., QM's "unprofessional vagueness and ambiguity"). And (in a certain sense) it's the only serious example of a hvt that really exists and is known to work. So if anything's shady, it's all the other (unspecified, non-existent) theories. All of these weird non-existent (or only very vaguely specified) theories (like the ones you seem to call "Bell realistic") are the ones that hide in the cracks. Bohmian Mechanics is right out there in the direct sunlight for all to see -- so let's all take advantage of that fact and look!

11. Nov 29, 2005

### Blackforest

With all respect to you; did you read the paper of Aharonov and Gruss: Two-time interpretation of quantum mechanics ; arXiv :quant-ph/0507269 v1 28 Jul 2005? There is an interesting exposé of the actual situation concerning the question you are indirectly asking; I believe. I didn't red the other interventions on this thread; so I apologize if my suggestion is in someway a repetition of some other made here. The classification proposed by Aharonov and Gruss pages 1 + 2 seems to be clear enought to me.

12. Nov 29, 2005

### Blackforest

So my intervention does not concern the question of commutation but only the properties owned by the different types of theories (HLV, BM, local, non local, deterministic or not, ...). Perhaps it can help to reduce the number of questions and eliminate some theoretical possibilities inside your investigation (e.g: a non local theory defies the relativistic covariance...)

13. Nov 29, 2005

### DrChinese

1. I am trying to map Bell Reality to the exact definition Bell uses, so I think that is pretty specific. Assume that a particle has simultaneous well-defined values for observables independent of a measurement. For Bell that was 3 different angle settings for a SG magnet which he calls a, b and c. These are NOT hidden variables in the sense that they are somehow deterministic!! The purpose of the EPR argument was not to restore determinism to the theory, but rather to restore the concept of an objective reality independent of the role of the observer.

2. You are putting words in my mouth on this one, there was nothing sinister intended by my comment. The cracks are the distance between 2 definitions that are almost always used interchangeably - those of "Hidden Variable" and "Bell Realistic". It has nothing to do with BM. And I am as guilty of using those terms interchangeably as anyone anyway.

The point is that I learned something I didn't know before: BM is a hidden variable theory which apparently does not qualify as a realistic theory by Bell's standards.

14. Nov 29, 2005

### ttn

OK, sure. Really he assumes even more than that -- namely, that there exist *functions* like A(a,L) which specify Alice's outcome as a function of the particle state (L) and Alice's magnet setting (a). But the existence of this function obviously entails what you specifically mention -- the values of A for three different values of a.

I don't follow you. In the paper you have in mind (Bell's first paper on this stuff) he does assume deterministic hidden variables -- i.e., the functions A(a,L) and B(b,L) that I mentioned above. These are *functions*, right? You tell me the state of the particles and the orientation of your polarizer, and the function spits out what the outcome is going to be -- and it's either +1 or -1, nothing wishy-washy or superpositiony or stochastic. One can derive Bell-type inequalities without the assumption of determinism, too... but then, determinism is *required* by the fact of perfect correlations (when Alice's a and Bob's b are the same). Any randomness will have to show up as a chance for imperfect correlations, so (assuming we want to reproduce this particular aspect of the QM predictions) we can dispense with non-deterministic local theories right away.

But I don't really understand why you said what you said about determinism, so maybe I'm missing your point. But I think it's nevertheless clear that Bell's hidden variables -- the functions A(a,L) and B(b,L) -- are deterministic. The outcomes are determined by the setting and the state, without any randomness.

The purpose of the EPR argument was to prove exactly what it proved -- that locality *requires* the existence of local hidden variables... i.e., that the locality requirement *conflicts* with the completeness assumption. If you want locality, you have to have hidden variables. If you want completeness, you have to jettison locality.

I'm still not clear on the definition of "bell reality". But one thing is for sure: BM is a hidden variable theory which does not qualify as a *local* theory by Bell's standards. BM violates Bell Locality. That's just a formal and rigorous way of saying something that's rather plainly obvious anyway: BM is a non-local theory. And *that* is the main reason why Bell's Theorem doesn't apply to it. Bell's Theorem proves that *local* hidden variable theories have to disagree with the QM predictions (and, it seems, experiment). Nonlocal theories (like Bohm's or like OQM) can of course agree with experiment.

I know this is old ground that we've been over painfully many times, but I think it's a crucial point that people seem to repeatedly miss. Bell's Theorem shows that non-locality is inevitable. It simply doesn't speak to the issue of completeness (i.e., hidden variables) at all. Post-Bell, it is entirely possible to have a viable theory either with or without hidden variables -- OQM and Bohm being the obvious two examples. What we now know you *can't* have is a local (specifically, "Bell Local") theory.

So why encumber QM with hidden variables if you're stuck with non-locality either way? To solve the measurement problem. Which Bohm's theory does, clearly and unambiguously. And it provides a wonderfully simple, intelligible ontology to boot. Who could ask for anything more?

15. Nov 29, 2005

### NateTG

Actually, and the good doctor has been repeatedly pointing this out, Bell's theorem makes multiple assumptions, and (stipulating that oQM makes accurate predictions) only eliminates local and realistic theories.
For example, it's entirely plausible to have a local, hidden, non-realistic theory, and, from what I know, an example might be the Many Worlds interpretation where Bell's $\lambda$ isn't necessarily well-defined until measurement occurs, another possibility might be a non-standard notion of probability.
Moreover, it's not always clear which assumption to consider as invalid: It's not necessarily pretty, but it seems like 'tiny wormholes' could also be a viable QM interpretation. In this case, Bell's theorem does not apply because there is no space-like seperation.
Naturally, there are limitations on what assumptions we're willing to question.

16. Nov 29, 2005

### DrChinese

OK, we going to agree to disagree about Bell proving non-locality. And this discussion doesn't need that branch since we have another thread for that.

But I am going to point out that the hidden variable functions "i.e., the functions A(a,L) and B(b,L)" are mentioned in Bell DISAPPEAR completely by the end of his (21). In fact, you absolutely need no reference to the hidden variables to make his proof. So I am saying unambiguously that the hidden variables of (1) and (2) play no part in the proof of Bell's Theorem. The only relevance is that we need to make explicit that:

"The vital assumption is that the result B for particle 2 does not depend on the setting a of the magnet for particle 1, nor A on b."

This is the Bell Locality condition, plain and simple. You need this and the Bell Reality condition to get Bell's Theorem. You don't need Hidden Variables EVEN though Bell refers to them as if they were important to the final result.

It is instead the Bell Realistic requirement (in his (14), "It follows that c is another unit vector"). If there is no a, b and c, then there is no proof. You only need a, b and c with specific angle settings and the cos^2 rule to get a truth table with "impossible" values. You can see my version this proof for yourself and see that there are no hidden variables - only simultanous non-negative values for the 8 hypothetical outcome permutations.

Assuming there are hidden variables - local or non-local - that actually determine the outcomes is not needed. The last question is whether the setting at Alice affects the outcome at Bob and therefore skews the experiment itself. You handle this by assumption - if you think it is needed. If you do incorporate it, as Bell did, then the scope of the theorem is made to apply to local realistic theories rather than all realistic theories.

And again, Bell's definition of Realism is intended to match up to EPR's elements of reality. Despite their name, these are observable properties and are NOT hidden variables. Determinism is NOT required to explain the perfect correlations. Otherwise oQM would have been ruled out a long time ago. The Born rule says it all.

17. Nov 29, 2005

### NateTG

I don't see how non-local hidden variable theories can be categorically invalidated since they're not necessarily causal.

18. Nov 29, 2005

### ttn

So... let me get this straight. You're saying that you can derive a Bell Inequality without assuming hidden variables? You only need to assume... what? Bell Locality?

I think you're wrong that this is possible, though of course I'd love to see how the proof goes if you want to show me. But even leaving that aside, I don't understand how this relates to what I took you to be saying before. If you can derive a Bell Inequality without any assumption of hidden variables, then it follows (from the fact that the Bell Inequality is violated in experiments) that Bell Locality (or whatever you *did* assume in the derivation) is false. And that sounds alot like the conclusion I've been pulling my hair out to convince you of (though by a somewhat more complicated argument). So I'm now quite baffled.

Yeah, that's Bell Locality. I agree. But what exactly is this "Bell Reality" condition we also need to get the inequality? You said before that you wanted to identify "Bell Reality" with the idea of hidden variables, but then you said above we don't have to postulate any hidden variables to get the inequality. Ack.

OK, so you need "Bell Reality" but not "Hidden Variables". You'll have to define these terms, though, or else this is really vague. Usually people use the term "hidden variables" to refer to any extra variables beyond the wave function. So Bell's outcome functions A(a,L) (etc.) are hidden variables... at least the way people I know have always used that term.

This is a standard derivation -- same as the one in, e.g. Sakurai's text, and I think it goes back originally to Wigner. But regardless, you're loopy if you think there are no hidden variables in this derivation. These 8 different categories you're talking about are the 8 possible states that could explain the perfect correlations (when A and B measure along the same axis) -- each state attributing a definite possible outcome to the measurements on each side (and, crucially, all 3 measurements even though only one can in fact be performed!). Each row of your table is one of the 8 possible "hidden variable states", one of the 8 possible "values" of "lambda" in this situation.

Now maybe you want to play pointless semantic games and say that your 8 different possibilites aren't "hidden variables" but are instead "Bell realities" or something. But please don't. Each row of your table is supposed to be a possible state of the particles, and each row attributes more definite identity to the state than is contained in the wf (namely, it attributes definite values for the 3 possible spin outcomes). So it has hidden variables.

Then why do you do it in your proof?

Are you saying you can get a Bell Inequality without the locality assumption either! :rofl:

You're confusing yourself with terminology. Of course the hidden variables are in some sense observable. If you think particles carry around instruction sets that tell them how to react for spin measurements along different axes, you can obviously "observe" (at least) part of the instruction set by making some measurement or other. But the point is that QM doesn't attribute any such instruction set, and what "hidden variables" *means* is: anything additional to the wf that's supposed to describe the state of a quantum system. If your point is just that this is a stupid name, since (as is the case with Bohm's theory) the "hidden variables" are often anything but hidden, I agree. But then that's a very different point.

Of course not. OQM is non-deterministic and explains the perfect correlations just fine. But then it's not Bell Local. My actual claim was that, once you insist on Bell Locality, determinism is required to explain the perfect correlations.

19. Nov 29, 2005

### DrChinese

There are groups of no-go theorems out there that are nipping at the non-local HV theories. The basic idea, as I read them, is that they must make assumptions that result in an internal inconsistency somewhere even if that inconsistency is not otherwise obvious. So even a non-deterministic one might make assumptions are problematic. Honestly, it comes back to how the theory defines "hidden variable" or "realistic".

Mind you, I am not sure they are good proofs. However, I have to wonder aloud** how we can have Bell non-locality and yet have signal locality. So that is what leads me to the idea that the proofs *might* have some merit.

I am still trying to evaluate Malley's proof. It is pretty involved, probably more than I can handle. But try I will.

**It's a rhetorical question, because obviously you can construct such a theory as ttn has demonstrated.

Last edited: Nov 29, 2005
20. Nov 29, 2005

### ttn

I know, I know, it was a rhetorical question. But it's a good one, so it deserves to be answered anyway. So...

There are at least two ways. Either the non-local causation is *uncontrollable* so we can't use it to send a signal (this is the case with OQM -- if you could make the wave function collapse one way or the other by controlling which of the two possible outcomes occurs locally, you could send a signal). Or the non-local causation is "in principle controllable" but, in fact, washed out by ineliminable uncertainty over the initial conditions (this is the case with Bohm's theory -- if you knew the initial positions of both particles in the pair, you could control the trajectory of one particle by adjusting the magnetic fields exerted on the other, and thereby send a signal, but since knowledge of the initial positions is limited by the QEH, you can't send a signal).

There might be other ways, too, for a theory to be Bell Nonlocal but still Signal Local. But those seem to be the most obvious two ways, and it's interesting that the two existing theories utilize them.