Does Bell's Theorem apply to non-local HV theories?

DrChinese

Does Bell's Theorem apply to non-local HV theories?

I know that Bell says it doesn't, and he has included logic intended to make separability a requirement. But consider this argument:

What does it mean to say there are hidden variables? Einstein said: "I think that a particle must have a separate reality independent of the measurements. That is: an electron has spin, location and so forth even when it is not being measured. I like to think that the moon is there even if I am not looking at it." In Bell's original paper, he says it is when there are well-defined values for the results of measurements that are NOT made. (Bell: "It follows that c is another unit vector...") This is expressed mathematically by imagining that there is an A measurement, a B measurement, and a C measurement. Only 2 of these are actually made (one on each of 2 entangled particles), the 3rd is hypothetical (assumed). We want to determine if all 3 exist simultaneously.

Thus: It doesn't matter under what mechanism or set of determining factors/variables the outcomes are calculated or determined, the only assumption is that the outcomes have a likelihood of occurance in the range 0 to 1. Thus, if we measure at A=0 degrees, B=67.5 degrees and C=45 degrees, and are looking for + or - as possible results, then there are 8 permutations:

[1]A+ B+ C+
[2]A+ B+ C-
[3]A+ B- C+
[4]A+ B- C-
[5]A- B+ C+
[6]A- B+ C-
[7]A- B- C+
[8]A- B- C-

So far, we have followed Bell's argument without inserting any condition relating to separability of the hidden variable functions of A, B or C.

However, in this case the combined QM predicted likelihood of the [2] and [7] cases is less than zero. (For the derivation of this figure, see Bell's Theorem and Negative Probabilities.) In fact, it is -.1036 which is nonsensical, therefore indicating that our original assumption that A, B and C are well defined simultaneously is incorrect IF we are to have results compatible with QM.

It didn't matter to my proof that the polarizers are in communication with each other or not. I don't care if there is FTL signalling or guide waves or similar. I don't care how the various cases are calculated or determined, or if the various results are influenced by space-like separated polarizer settings. I don't care if the [2] and [7] cases are rare. The fact is, if there are 8 permutations, then 2 of them cannot have negative expectation values.

By my thinking, this argument should apply to any hidden variable theory - local or not. The only assumption I make is that there is a definite value (either + or -) for an observable that is not actually observed. So I believe that even non-local theories must sport an observer dependent reality. And by observer dependent, I mean that there is not simultaneous reality to non-commuting observables. I.e. the moon is not there when you are not looking at it. (Please note that this does not literally mean the moon is not there when you are not looking at it. It is just a metaphor.)

Your thoughts are invited. Thanks in advance.

vanesch

In fact, we did. We include the list of all possible outcomes in each of the particles only to obtain the 100% correlation when the polarizers have identical settings ; this is the only way if they cannot communicate.
But when they can communicate, you could equally well limit the particle's settings just to A and B, say (and a common random number). If you attempt to measure A and/or B, then you just do as usual, however if you attempt to measure C, this is going to be determined from whether the state is A or B, and the common random number, and IF C IS MEASURED UP THEN THIS CHANGES IMMEDIATELY THE RANDOM NUMBER IN THE OTHER STATE, such that perfect correlation is possible for the C-direction. So you do not need to establish the list A,B,C in advance.
The state would be A or B, and if you happen to try to measure C, this will not measure the pre-existing state (which was A or B), but something derived from the state, and which is immediately influencing the state of the other particle.
cheers,
Patrick.

DrChinese

I know it must be there somewhere, but try as I might I just don't see it. In the perfect correlated state where A=B=0 degrees, you can make the table work out. But not in the case A=0, B=67.5, C=45.

Out of 100 tries
[1]A+ B+ C+ : 4
[2]A+ B+ C- : 4
[3]A+ B- C+ : 21
[4]A+ B- C- : 21
[5]A- B+ C+ : 21
[6]A- B+ C- : 21
[7]A- B- C+ : 4
[8]A- B- C- : 4

No matter what you put in for the A/B/C I gave, you won't get the right table unless the [2] + [7] cases add to less than zero. I.e. The values I gave above won't match experiment.

In other words, suppose for the sake of argument that you allow 3 people, Alice, Bob and Charlie, to talk to each other by telephone. They will agree to give a + or a - according to a scheme in which:

i) Alice and Bob match 15% of the time
ii) Bob and Charlie match 85% of the time
iii) Alice and Charlie match 50% of the time

So this matches your scenario, where one changes to match the correlation percentage. BUT... when you repeat this over and over, they will be unable to comply because satisfying two conditions prevents satisfying the third. Yet these are the statistics that would agree with QM and are consistent with experiment.

No matter how I try, I don't see that it is possible to conclude that C exists independently of observation. It seems to me that is falsified when you get negative odds. Help!!

DrChinese

In case I wasn't clear with the prior post:

In the case A=0, B=67.5, C=45, the QM predicted results would be (rounded slightly):

Out of 100 tries
[1]A+ B+ C+ : 12.5
[2]A+ B+ C- : -5.5
[3]A+ B- C+ : 12.5
[4]A+ B- C- : 30.5
[5]A- B+ C+ : 30.5
[6]A- B+ C- : 12.5
[7]A- B- C+ : -5.5
[8]A- B- C- : 12.5

This satisfies the following rules (again rounded slightly):

i) A and B match 14% of the time
ii) B and C match 86% of the time
iii) A and C match 50% of the time

vanesch

Ah, but you are right, C doesn't exist of course ! But the point of a non-local HV theory is just that it does not HAVE to exist, that the measurement can be "garbled" by the settings of the OTHER measurement.
That's why I only proposed A and B to really exist as states, and C to be just an apparent result that doesn't describe the *state* of the local systems, but that can generate outcomes which are compatible with QM if the other measurement is allowed to change immediately the state.
But of course, you'll never be able to write down the ABC table with positive probabilities. The point I tried to make is that you don't NEED to do that to mimick quantum results if you allow for action at a distance. You can have a "smaller state" travelling, and generate the QM correlations by MODIFYING that state at a distance.
So in a way, yes, you're right that the measurements, in that case, do not measure the "independent state" of the local thing that arrives. But if it were, you wouldn't need the action at a distance, and you would have a LR theory. You measure only a disturbed quantity, with a disturbance coming from the the action at a distance from the other measurement.
If you look at Bohmian mechanics, that's exactly what happens.

ttn

And if you look at orthodox QM, that's also exactly what happens!

vanesch

Yes, except that there, you don't even have an individual state of each particle which is to be "disturbed".

ttn

Semantics. You have a state, and it's definitely "disturbed". Initially, the distant particle is part of a superposition -- which means (among other things, and according to the standard interpretation) that it has no definite value for spin along any direction, that spin measurements along any direction are equally likely to come out up or down, etc. Then the nearby measurement occurs and the state of the distant particle is different: now it's in a particular spin state, with a definite value for the spin along some particular direction, and with definite probabilities for outcomes along general directions that differ from what those probabilities were before the measurement.

In short, the state of the distant particle changes. The collapse postulate violates "no action at a distance" if you interpret the wf as a complete description. Or an even cleaner argument: orthodox QM violates Bell Locality.

DrChinese

Then what would be the point of saying there is a non-local HIDDEN VARIABLE theory? Why not simply say there is a non-local theory that is not a hidden variable theory? If it is a HV theory, you must be saying that there is reality independent of observation. The moon is there even when it is not being observed.

I think...

NateTG

Um, doesn't assigning well-defined probabilities to those conditions correspond to violating the HUP?

vanesch

Yes, the "hidden-ness" of the hidden variable is of course another point to debate. I would say, that if you want to know what such a theory looks like, simply look at Bohmian mechanics.

At each moment, in Bohmian mechanics, particles DO have a well-defined position and momentum, but the trick is that we don't know how they are initially distributed. Only, their dynamics is influenced by all OTHER particles, that may be lightyears away, through the "quantum potential" ; only, this influence is minor in most cases, except in Bell-like setups; this follows from the expression of the quantum potential which is only significantly contributing in this case. The "hidden variable" here is not so much the position as it is the quantum state (wave function in position configuration space).
Spin does not exist as a particle property, but is just something which pulls and pushes on the position of particles through the quantum potential.

DrChinese

(Sorry, I didn't mean to emphasize the "hidden" part of the hidden variable issue.)

1. But I still don't see how BM yields an explanation which allows for the simultaneous existence of our A, B and C. If C existed when we measured A and B, then we must get the negative probabilities anyway. Even if the influence is from particles light years away, I don't see how that removes this requirement (i.e. that probabilities must be non-negative).

2. And this is also not clear to me: I use the term "hidden variable" to mean something which is "observable" but which is not actually observed or measured. In my mind, this is fully consistent with Bell's description (such as "It follows that c is another unit vector..."). I feel it is also consistent with EPR's description of an element of reality and Einstein's "a particle must have a separate reality independent of the measurements". The well-defined value of the observable is the hidden variable and it is hidden simply because we did not actually measure it. So a hidden variable is also a state of a particle (or entangled particle ensemble).

I realize that it can also be taken to mean what Bell calls lambda, or a set of parameters/variables/functions which determine the value of an observable.

In parlance, both of these defintions are loosely interchangeable; but I am using it here to denote the former definition. In other words, I am asking if there is simultaneous reality to particle observables (i.e. an A, B and C)independent of measurement. I believe there is not, and therefore there are no hidden variables. But maybe this should be described a better way?

ttn

I'm still not entirely sure what you mean by these A's, B's, and C's, but if they mean spin-component values, Bohm's theory doesn't have them! The "hidden variable" in BM is the definite particle positions (even when the wf isn't a position eigenstate), and that's it -- there are no additional hv's for spin properties because, as it turns out, just letting the particles have definite positions is enough to explain all of the results of SG type measurements. (The wave function is a spinor, of course, but the particle is just a particle.)
So maybe you're wondering: if the only serious existing hidden variable theory doesn't include definite spin components (A, B, C, etc.) why does anyone care about these things? Why did Bell spend so much time worrying about them? The answer is EPR: According to the EPR argument (or better, the various clarifications of that argument that Einstein gave before and after 1935 and substituting Bohm's 1951 transfering of that argument to the example with two spin 1/2 particles) the *only* way to explain *locally* certain correlations predicted by QM is for each particle to carry these pre-measurement spin components A, B, and C. Locality plus the fact of perfect correlation *requires* these things to exist.
And *that's* why Bell's theorem (which examines this possibility further) is so important. He showed that even with the A's, B's, and C's, you still can't have a local theory that agrees with experiment. So there is no possible way of having a local theory that agrees with experiment. So locality is false.
(...or the experiments are somehow misleading us and really the QM predictions are wrong.... or the experiments are *really* misleading us and they don't even have definite outcomes the way we think they do!)

DrChinese

By A, B and C, I am referring to any particle attributes that can be measured separately, but according to QM do not simultaneous real values. In EPR, they are referred to as non-commuting operators but really they would count as anything restricted by the HUP in some way.

In BM there may not be spin components, but, according to EPR, these components correspond to an "element of reality". Therefore, according to the concepts of EPR, they should be explained by a complete theory (presumably because they can be predicted with certainty).

Of course, I absolutely agree with you that the correlations cannot be explained by any local theory in which A, B and C exist simultaneously. By my definition, a theory in which A, B and C are required to exist simultaneously (real well-defined values) is a hidden variable theory. So that is why Bell's Theorem applies to all local HV theories. I think your description of the "hidden variable" in BM matches my definition closely enough - it is what is hypothesized to be real and definite even if we cannot measure it.

But ANY theory which requires spin components A, B and C to exist simultaneously falls to Bell's Theorem - if it maps to something more complete than the HUP allows. (That is simply because there are no values of A, B and C which avoid the negative probabilities and the factorizing requirement - usually given as the requirement of locality - is not required to arrive at that conclusion.)

Don't get me wrong, I have always believed that Bell's Theorem applied ONLY to local realistic theories. But recently I have developed a couple of web pages for my site that provide very simple versions of the math surrounding Bell... and I cannot figure out why I didn't need the factorizing requirement to make it work out. The first approach was the "negative probabilities" shown in the OP above, which is pretty much exactly following Bell. The second was an approach which follows Mermin's arguments. Neither needed a locality requirement to work (although in both I mention the locality requirement so as to match common accepted practice).

The problem, I think, is that there are 2 different requirements associated with a Bell test. The first is explaining the perfect correlations when the angles match. And I admit this seems to involve the locality requirement. But that is not the core of Bell's argument. He doesn't even mention the perfect correlation case. Bell's argument is about the A/B/C components and their simultaneous existence. And I don't see the locality requirement itself coming in to play on that. Why am I so blind? Where is it?

ttn

OK. Just to make sure we're on the same page, are A, B, and C operators, or are they the results of measurements? QM associates one with the other, so the distinction might not seem important, but some other theory might, in principle, not have such an association (or might not even involve "operators"). So it's good to be clear.

Also, perhaps everyone already knows this, but I just want to make sure we're on the same page: Bell uses the capital letters A, B to denote the outcomes of measurements -- A referring to the outcome of Alice's measurement, B referring to the outcome of Bob's measurement. So then it's not entirely clear to me what C is supposed to denote. I'm worried that you're confusing the capital letters (A and B) with the lowercase unit vectors like "a hat", "b hat", etc. These refer to spatial directions -- specifically, the directions along which the spin of the particles are measured. So for example A(a-hat) refers to the result of Alice's experiment (either +1 or -1) when she measures the spin of her particle along direction a-hat. "b-hat" and "c-hat" are then just some other directions in space, some other axes along which Alice or Bob might measure the spin of their particles.

I'm trying to be clear about this because you keep referring to this phrase from Bell's paper ("It follows that if 'c' is another unit vector...") as if he's introducing a new hidden variable which you seem to want to call C. No new hidden variable is being introduced there. He's just playing with the already-established formula for the correlations between the outcomes -- in particular, writing down the *difference* in the correlations for the case where Alice and Bob measure (respectively) along directions a-hat and b-hat, and the case where they measure respectively along some other directions, a-hat and c-hat.

How would we assert the real simultaneous existence of different spin components in Bell's terminology? By saying that A(a_1), A(a_2), and A(a_3) all exist. Or if you like, A(a), A(b), and A(c). Or we could just assert that there is a *function* A(a) which has a definite value for *any* direction "a" -- this commits us to the real existence of spin components not just along three particular directions but along all directions. Heisenberg would hit the ceiling.

According to EPR, a local theory (which explains the perfect correlation when Alice and Bob both measure along the same direction) will have to include these hidden variables. EPR believed in locality, so they believed that these hidden variables really existed, i.e., were elements of reality.

But that doesn't mean every theory has to have them -- a theory could be non-local and hence explain the correlations without these variables. Bohmian Mechanics and orthodox QM are two obvious examples of such non-local theories.

That's true but only half the story. The correlations cannot be explained by any local theory *at all* -- even one in which the A, B and C *don't* exist simultaneously. Again, BM and oQM are two examples. If you have a counterexample, I'd love to hear about it. (Just to be clear: a counterexample would be any local theory which explains the correlations. And just to keep vanesch off my back, when I say "explains the correlations" I mean "explains the particular sequence of definite outcomes from which we later calculate a correlation coefficient.")

If A, B, and C are definite spin components that exist simultaneously, then, no doubt, a theory which includes those is a hvt. But that doesn't mean all hvt's have to include those particular elements!

Bell's Theorem applies to all local HV theories because, as shown by EPR, any such theory has to include those local elements of reality, the spin components. And that's the only assumption (well, other than re-asserting Bell Locality) Bell makes in deriving the inequality. So it applies to all of them.

Yes, though, for the record, it's a ridiculous bit of historical silliness that the particle positions in Bohm's theory are called "hidden variables." The fact is, we can measure them. Every time you do an experiment and find out where a particle is, you're discovering the value of the "hidden variable." If anything is hidden, it's the wave function. If someone hands you a particle that they have prepared in a certain quantum state, there is no experiment you can do to measure the quantum state. That means, for BM, it's the wf that is hidden and the particle positions that aren't. And it means, for orthodox QM, *everything* is "hidden" -- the thing that is supposed to compose a complete description of the state can never be measured! So the terminology is really stupid. But it's entrenched so we might as well stick with it, ... sigh...

No, you could have a theory where three different spin components all exist simultaneously, but where there values were affected non-locally by other particles to which the one in question is entangled. The only thing that is ruled out by Bell's theorem (and the associated experiments) is local theories. You can *always* make correlations be as strong as you want by introducing a non-local mechanism.

I haven't (yet) looked at your webpages, but it's clear enough where Bell brings in the locality assumption. First off, the perfect correlations (when Alice and Bob measure along the same axis) are expressed in Equation 13 of "On the EPR paradox". A(a,L) = -B(a,L) just means Alice's outcome has to be the opposite of Bob's outcome when they both measure along the same direction ("a"). And what's normally called "Bell Locality" is hiding in Equation 2 of that same paper. Generally, the correlation coefficient should be defined as
[latex]
P(a,b) = \int d\lambda \; \rho(\lambda) \; AB(a,b,\lambda)
[/latex]
where I have defined a new symbol "AB" to mean the "joint" outcome, the product of the two outcome values, which in principle might depend on both of the magnet settings a and b. But according to Bell Locality, this joint outcome has to factorize into the product of the two individual outcomes -- each of which can depend on the local magnet setting *only*. That is Bell Locality.

I think it's reasonable to say that Bell himself wasn't 100% clear on all of this when he wrote this first paper. If you really want to understand it, read his later papers, where it is absolutely crystal clear. Especiall "Bertlmann's Socks and the Nature of Reality" and "La Nouvelle Cuisine" (the latter appearing only in the 2nd edition of Speakable and Unspeakable).

DrChinese

Thanks for taking the time for your reply.

I am going to split up a few comments and questions into a couple of posts because there is plenty to discuss.

a-hat, b-hat and c-hat in Bell's paper map to possible settings of the SG device. I use A, B and C to map to these with the idea that they supply the answer to the question that is posed by a particular SG setting. You can say that A, B and C correspond to elements of reality because they can be predicted in advance if tested individually (the perfect match case). But the question is whether they all exist simultaneously. QM says they don't, only 2 exist (or maybe 1 depending on how you apply the collapse).

Any hidden variable theory - by definition - says that particle attributes exist whether or not they are measured.

When Bell says "It follows that if 'c' is another unit vector..." he is absolutely hypothesizing that a, b and c could have been measured simultaneously if there were a more complete specification of the system. If there is no c along with a and b, then Bell's derivation stops there.

Certainly we agree that the inequality deals with the 3 pairs of settings ab, ac and bc which can actually be measured. Each such measurement gives us definite values for what I call measured variables (AB, AC or BC). The hidden variable is the third not measured.

So if QM is correct in its predictions, its predictions must hold for correlations of AB, AC and BC. If there is a hidden variable/observable - Bell hypothesizes just one by my viewpoint, and again I am not talking about lambda which I realize is a set - then the correlations for AB, AC and BC must all fit simultaneously.

I realize that any number of input parameters - lambda - might actually determine the values of A, B or C and that these could even be sets of functions and also that in BM these do not need to be completely independent. So the A, B and C variables I am referring to are outcomes IF a measurement at settings a, b and c are performed.

So for me, the critical phrase is "It follows that if 'c' is another unit vector..." because without that there is nothing. That is the embodiment of the hidden variable requirement in Bell's Theorem.

ttn

OK, but then it's confusing whether you're talking about the outcome of the experiment on Alice's side, or Bob's. But I don't care about terminology -- so long as we all understand that, in Bell's paper, A and B refer to the outcome of Alice's and Bob's experiments (respectively) and that the hidden variables assumption is the assumption that these exist as *functions* of a-hat. To talk of A(a-hat) is to assume that there exists some definite outcome for a spin measurement along *any* direction (any a-hat).


QM says *none* of these exist! I mean, for one of the particles in a spin singlet state, the particle is not in an eigenstate of sigma-dot-a-hat for *any* a-hat. According to orthodox QM it doesn't have a definite spin along any direction -- not an infinite number, or three, or two, or even one.


A hvt is simply something that adds more structure than exists in QM, i.e., which supplements the wf description with something else. These might be definite spin components (as is assumed in Bell's theorem), or particle positions (as in Bohm's theory), or definite momenta (as in some version of the modal interpretations), or whatever. It's dangerous to *define* a hvt as something that includes some particular element of reality like spin components -- then you've defined Bohm's theory out of existence... but it exists!


No, nothing physically interesting is happening at that point in the derivation. All the physics has already happened. For example, the assumption that these functions A(a-hat) *exist* (which as noted above is a *massive* assumption that all kinds of hidden variables exist) has already been made. Nothing new is being added here -- he's just comparing the correlation functions for different pairs of angles.

I don't follow this last sentence at all. And that makes me think we don't agree at all about the previous sentences. Yes, 3 pairs of settings are involved in the inequality. And it's indeed possible to measure the correlations with any such pair of directions. And indeed (we assume that) each such measurement gives some definite outcomes. But in any given run of the experiment -- or better, for any given emitted pair, you only measure two things. Alice measures her particle along some particular direction, and Bob measures his particle along some other particular direction. The hidden variables have nothing to do with some other property that isn't measured.


I can't follow this.



I don't follow the first part, but the last sentence is the standard orthodox view. We shouldn't talk about these functions A(a-hat) (etc.) because "only actually performed measurements have definite outcomes" or whatever.

This is exactly why people say that Bell's theorem refutes local hidden variable theories, but doesn't say anything about orthodox QM as such. And that's true. But to say that is to forget about EPR -- it's to forget that orthodox QM is nonlocal. It's to forget that the hidden variables Bell assumes for his theorem *have* to exist if we want to eliminate the non-locality of orthodox QM. So when you put those two pieces together, you have an argument against locality and nothing else. You can have an empirically viable theory in which the wf alone describes reality completely, and you can have an empirically viable theory in which the wf is supplemented with some other variables. Both work, and both exist. But both are non-local, and you can never have a local theory that is empirically viable.


No, I think you're really missing the important action much earlier in the paper. I've said this now 3 times, but maybe this one'll be the charm: if you want to understand this stuff, you *must* read Bell's other papers: Bertlmann's Socks and La Nouvelle Cuisine in particular.