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Questioning assumptions behind Bell's and related theorems

  1. Dec 9, 2013 #1
    I have trouble understanding the mathematical arguments behind this view but I thought I would post it, in case anybody has any information/understanding/insights. The basic idea is that the mathematical assumptions on which the validity of Bell's inequality depends are that all the random variables are defined on a single probability space. These authors then go on to question this assumption using a Bohrian-type argument which they refer to as "the chameleon model". Note that this has nothing to do with questioning loopholes, etc. as they are suggesting that Bell’s argument fails even before the issue of these loopholes. Also, note, that they are not basing their arguments on the contextuality as per Kochen-Specker theorem, as they question the assumptions behind this theorem also. There are a number papers/books taking this perspective:

    Locality and Bell's inequality
    Chameleon effect, the range of values hypothesis and reproducing the EPR-Bohm correlations

    Is the Contextuality Loophole Fatal for the Derivation of Bell Inequalities?

    Non-Kolmogorovian Approach to the Context-Dependent Systems Breaking the Classical Probability Law

    For anyone who has some understanding of Probability theory, do these Non-Kolmogorovian approaches/axioms seem reasonable/make sense?
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  3. Dec 9, 2013 #2


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    I always laugh when people create examples which do not map to Bell's Theorem, and then go on to disprove them. Bell Theorem is:

    "No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics."

    Note that the ballot box and chameleon examples in the first paper are nothing like the quantum spin examples because spin can be any mixture of axes. When they connect their example to a suitable quantum prediction, I think they will have something more meaningful.

    On the other hand, I mostly agree with the author through the point where he cites Bell's "vital" assumption: "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 essentially a nod to the EPR assumption that Alice's reality cannot be dependent on Bob's choice of measurement setting. That in turn is a restatement of the EPR assumption that elements of reality should not need to be simultaneously demonstrated to exist.

    The rest of the paper mainly argues that "physicists ... use the same symbol to denote the results of different mutually incompatible experiments... (etc)". This argument has been around in numerous variations for some time, and has failed to gain traction. Primarily because it goes directly against the EPR assumption (prior paragraph) regarding simultaneous elements of reality. In other words: if you reject that EPR assumption (as Accardi essentially does after about 10 pages) then you don't get the Bell result. That is already generally accepted, hence nothing really new in this line of reasoning. To quote EPR:

    "Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted . ... No reasonable definition of reality could be expected to permit this."

    If you accept that at face value, you arrive at Bell's result. If you reject it, you arrive at Accardi's.
  4. Dec 9, 2013 #3
    Let me put the argument in a different way: The terms in Bell's inequality are functions A(λ), B(λ), C(λ). After some algebra of these functions, Bell obtains relations/inequalities which contain those functions. The above argument is essentially equivalent to the statement that any such relations between functions is meaningless unless the functions have the same domain. ie, the set of all λ must be exactly the same for each function.

    In simple terms, it may appear trivial that A(λ) - A(λ) = 0, but if the first term was obtained in a situation in which λ = {1,2,3} and the second term was obtained in a situation in which λ = {4,5,6}, then the expression does not make much sense and can be violated, because A(λ = {1,2,3}) is a different random variable from A(λ = {4,5,6}) (aka. are defined on two different probability spaces)

    Looking at page 406 of Bell's original paper(http://www.drchinese.com/David/Bell_Compact.pdf), the algebra leading up to equation 15 makes this clear.

    Bell starts with A(λ)B(λ) - A(λ)C(λ) then factors out A(λ)B(λ) to obtain [1- B(λ)C(λ)] which makes it clear that all three expectation values E(A,B), E(A,C) and E(B,C) are calculated from the three functions A(λ), B(λ), C(λ) simply recombined in pairs. Which means any expression/relation which Bell obtained between those three expectation values, such as the one in equation 15, is meaningless unless those terms have the exact same domain for λ.

    Every derivation of a Bell-like inequality including the CHSH has this "hidden" assumption at a crucial step. Just verify what is being factored out. See for example http://en.wikipedia.org/wiki/Bell's_theorem

    If you think about what Bell was trying to prove, it becomes obvious that you can not claim to prove that A, B, C are/aren't simultaneously real for a given particle, if you measure them on different particles.
    Last edited: Dec 9, 2013
  5. Dec 9, 2013 #4


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    Wrong as usual, Bill.

    That they are individually real WAS the EPR argument, and that point was not in dispute per se. That was done using TWO different particles. They then ASSUME that those elements should be simultaneously real as well, as I quoted verbatim above. Bell takes their assumption and shows it to be impossible in conjunction with the other EPR assumptions (locality, correctness of QM).

    So do we really need to go through this time-wasting process again? As always, you can expect to have your speculative personal opinions flagged. Please re-read the rules and comply.
  6. Dec 9, 2013 #5
    With respect to Accardi et al, and the assumptions behind Bell's they write:
    Note on the EPR-Chameleon Experiment
    http://ics.org.ru/doc?pdf=855&dir=e [Broken]

    Anyway, I was under the impression that these models do rely on a subquantum theory and do make different predictions than QM. I know this is the case with Khrennikov.
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  7. Dec 9, 2013 #6


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    A relatively old paper (2003) to reference, don't you think, considering it claims to refute Bell? And there is no experiment actually presented despite the title. There is discussion of a computer simulation.

    I really don't see how we can go too far down this path without running into rampant speculation about the subject. If you were discussing Bell, that would be one thing. You are really attempting to discuss the pros and cons of an anti-Bell argument as if this has equal footing with pro-Bell arguments. As you well know, this is not the place for such debate. It is well accepted that there are several assumptions going into the Bell result. If you reject any of those (locality, realism, accuracy of QM), you will be at a different spot - and essentially that is your right.

    But it is flat out incorrect to say that your choice of assumptions must be accepted by everyone else. They won't be. The entire point of the EPR/Bell assumptions is that they are reasonable. That is why the Bell result is so widely accepted and why it is so important.
    Last edited by a moderator: May 6, 2017
  8. Dec 9, 2013 #7
    I don't think it's really refuting Bell. It's questioning whether Bell's applies to QM; that is, it's questioning whether the mathematical formalism of QM has some features of non-Kolmogorovian probability theory as is the case with some phenomena outside QM. And there are 2013 papers (like the one by Khrennikov) I linked above. One of the assumptions of Bell's is the acceptance of Kolmogorovian axiomatics. If one is willing to accept a non-Kolmogorovian probabilistic model, then one can have both locality and "realism". And again, I'm not competent to evaluate the mathematical arguments but what I wanted to know/understand is whether this is, in fact, even possible? As the author wrote in the 2013 paper:
  9. Dec 9, 2013 #8


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    Well sure it is refuting Bell. Bell, as summarized above:

    "No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics."

    And the paper of Accardi et al says:

    ... the claim that the experimental validity of the correlation is incompatible with a local realistic interpretation of quantum mechanics, is definitively unwarranted both for theoretical and for experimental reasons.

    This is not really the place to dissect an old paper (or line of reasoning, regardless of how it is warmed over) that is diametrically opposite of accepted thinking, and always has been. You and Bill should take this off-line, I am sure he will have plenty of comments, as this is almost verbatim his reasoning as to why Bell is wrong.
  10. Dec 9, 2013 #9


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    If random variables are defined on different probability spaces then you have either unfair source, unfair detection or unfair something else. So it's already taken into consideration.
  11. Dec 10, 2013 #10


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    More generally, Bell's inequality applies to a very broad class of possible alternatives to QM. Bell's inequality does not apply to every possible alternative to QM, because some input assumptions are needed.

    One question (that several of these papers seem to be addressing) is this: Can an alternative to QM that does not obey Bell's inequality be "locally realistic"?

    To me, this question is completely uninteresting. It's just semantics, how we choose to define the phrase "locally realistic". Exactly how this phrase is defined has nothing to do with physics.

    A much more interesting question is whether an alternative to QM can be constructed that (1) agrees with all tests of QM to date, and (2) makes a prediction different from QM for some experiment yet to be performed.

    Without this, there is really nothing in these papers to attract the attention of most physicists.
  12. Dec 10, 2013 #11
    This an interesting thread and if someone can explain to me why the equation 4 in this link holds I would be externally grateful. I just cannot see the the right hand side is not squared? If the physics is encoded in the wavefunction and the probability is that multiplied by the complex conjugate why would you expect correlations to go as cos rather than as cos squared ? Many thanks.
  13. Dec 10, 2013 #12
    I do not follow. Taken into consideration in what way? I do not see how it is taken into account in the derivation of Bell's inequality, or in the proof of Bell's theorem. Do you mean in the interpretation of experimental results? The papers by Accardi are arguing that even with 100% detection you can still have a local realistic situation in which a joint probability distribution is not possible.

  14. Dec 10, 2013 #13
    Accardi's argument seems pretty straight-forward and correct to me:

    They reduce the assumptions required to obtain the inequalities down to only 2
    (I) Random variables with values [-1, +1]
    (II) Random variables from the same probability space
    With those two assumptions only, they get the inequalities. Violation of the inequalities means one of those two assumptions is false, which is obviously the second one. It is not difficult to see why (II) is false in QM and in the experiments.

    If as you say, Bell's inequality applies to a very broad class of possible alternatives to QM, then this class must be the one for which those two assumptions are true. The authors argue convincingly in my opinion that the second assumption is not a reasonable one. Specifically they discuss the class of systems for which the second assumption fails -- those that are adaptive, i.e. we measure an interaction between particle and apparatus (color of chameleon on leaf) and not per-existing properties independent of observation (color of ball in the box). They even construct a counter-example along the same lines which reproduces the EPR correlations with 100% detection efficiency. They conclude that QM systems are of the color of chameleon on a leaf kind, which they term "Chameleon Reality".

    This should be interesting to physicists interested in QM alternatives, or the interpretation of QM.
  15. Dec 10, 2013 #14
    A recent Sept. 2013 video presentation of Accardi's arguments can also be found here:

    Urns and Chameleons: two metaphors for two different types of measurements
  16. Dec 10, 2013 #15


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    Doesn't matter if it's "reasonable" or not (whatever that might mean).

    What matters is whether anyone can construct a theory that agrees with all experiments done to date, and either (1) makes a prediction that differs from QM that can be tested (e.g., Penrose's objective collapse), or (2) is equivalent to QM in terms of experimental predictions, but allows for new methods of calculation (e.g., Feynman's path-integral formulation of QM). Nothing less than either (1) or (2) would be interesting to the overwhelming majority of physicists.
  17. Dec 11, 2013 #16


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    With all due respect, it looks like Prof. Luigi Accardi is making a terrible mistake. Bell's theorem is not mainly about probabilities, but the key factor is the correlations between Alice & Bob.

    Let’s say that Prof. Accardi discovers and proves a new kind of non-Kolmogorovian probability theory, that proves that in EPR-Bell we are not measuring the predefined LHV in the "Urn" (which is basically what Bell/QM has already proven) but "Labile Flying Chameleons", that interacts with measuring apparatus to gain their final value (which is what QM has been saying all along).

    Would this change anything??

    No, nothing. If we discover that there are not only Green and Brown in the properties of the "Labile Flying Chameleons", but also Purple, it would not change anything.

    The correlations regarding 'the old' Green/Brown (+1/-1), that are 100% incompatible with LHV, would obviously still be there (unless someone is claiming that a theory can change the outcome of physical experiments!), and I think most would agree that the new Purple property can’t change this fact, no matter what it’ll do, correlated or uncorrelated.

    Hope it helped.

    Regards DA
  18. Dec 11, 2013 #17


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    Accardi says that you can have local realistic situation with 100% efficient detectors not 100% efficient detection.

    later in the same paper he says:
    "Computer 1 computes the position of particle 1 using the deterministic dynamics and sends back S(1)a(sigma j)(=1 or -1) if the particle is inside the apparatus. It sends back nothing if the particle is outside the apparatus."

    This is detection loophole even with 100% efficient detectors. Detection efficiency is found from proportion between coincidence count and single count (for whole experimental setup).
  19. Dec 11, 2013 #18
    I think Richard Gill gives a similar argument:
    But then Gill goes on to argue that:
    The chaotic chameleon

    Personally, I tend to think that the "memory loophole" has the most promise since such an effect has been seen in quantum analogues in Couder experiments:
    Last edited: Dec 11, 2013
  20. Dec 11, 2013 #19


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    Gill in no way believes that Accardi's conclusion (ie local realistic models are compatible with QM) is correct. As a reminder, the detection loophole (mentioned in Gill's paper in regard to Accardi's ideas) is already closed. And was when Accardi's paper was written.

    So I think my point is that there is still nothing to make this thread - regardless of level of interest - make sense within our rules. One cannot start with Accardi's non-standard scientific conclusion* and debate its merits. What are you going to accomplish, overturn accepted science? This is not the place for that. And there is really nothing being explored here other than various mathematical ideas regarding the nature of reality. Well, as mentioned, each of us is entitled to reject EPR realism - which is well-defined and well-accepted. If you do, there's your answer - you will reject Bell too.

    *See my post #8. Instead, Gill would agree that "No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics."
  21. Dec 11, 2013 #20


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    Short answer? No

    I try to cut down on the Buchstabensalat, it has a bad influence on my poor little green brain... but I made a short exception and skimmed the paper to see if there is any physical meaning of the EPR-chameleon experiment. Same answer, no meaning, at all.

    As you can see they talk only about EPR experiments, not EPR-Bell, and that’s because the whole thing breaks down as soon as you go beyond measuring 2 out of 3 possible fixed settings (i.e. DrC’s 0°, 120°, 240°) that between them has the same ratio (i.e. 120°), which is based primarily on the dilemma of 3 predetermined counterfactual values that will destroy the party, without knowledge of the twin partners destiny.

    This setup is quite vulnerable to the detection loophole. I know; I hacked a simple little JavaScript that crushes "DrC’s Proof" without mercy... and it didn’t required a professor title or extended papers to accomplish - the math is basically arithmetic on the level of primary school.

    So, if you are a little bit woolly about the detection loophole, mixing EPR & EPR-Bell without discretion, and never proceed to the tougher Bell settings (that don’t have the same ratio between them, i.e. 0°, 22.5°, 45°), I guess you can sleep well at night – claiming that there indeed is a physical meaning to the EPR-chameleon experiment...

    [bolding mine]

    And of course, if we run the entire "Chameleon Rainbow" – from 0° to 360° – I guess those little creatures would start looking like "Glowing Crackpots" more than anything else. :wink:
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  22. Dec 11, 2013 #21
    I think they do in fact challenge the notion of 100% detection that you imply here.
    In fact the efficiency is measured by the ratio of the number of detected particles over the number of particles which have interacted with the apparatus.
    It would be totally meaningless to take into account, in the determination of the efficiency, also those particles whose space trajectory has brought them so far from the apparatus that no physical interaction between them is conceivable.

    Why should the apparatus be expected produce an outcome at a given instant in time, when there is no particle inside? You could think about it in this way: each apparatus simply produces a list of time-stamped outcomes for every particle that comes into it. One list for Alice and another for Bob. Once the source has done it's thing and the apparata have recorded their lists, the experiment is done. Now where is the detection loophole. All the particles emitted were detected.

    So then what you call "detection efficiency" is really a statement about how well you can match the list of results from Alice's side to the list at Bob's side during the data analysis. Probably closer to the coincidence loophole than detection efficiency. And I do not see how this loophole can be eliminated unless anyone doing data analysis is absolutely sure they have matched the list correctly. But how can anyone be sure?

    This is why Gill and Larsson conclude in their 2003 paper on the "coincidence loophole" that:
    The results underline the importance of eliminating post-selection in future experiments.
  23. Dec 11, 2013 #22
    In case, there's any misunderstanding, the point of the thread was to get input from others so I can better understand these arguments, as I had trouble with them. So don't jump all over on me...please. Moreover, I just had a chance to briefly look at Richard D. Gill's critical paper on all these attempts and although he does acknowledge that he hasn't had yet (as of 2003) studied Khrennikov's p-adic probability model he's skeptical. I'm not sure if he's published any thing more recently. Still, I found this comment by Gill interesting:
    Time, Finite Statistics, and Bell’s Fifth Position

    I was wondering, does anybody see any connection between this point and the path memory or pseudo non-locality seen in the Couder experiments:
    Information stored in Faraday waves: the origin of a path memory

    Classically quantum
    Last edited: Dec 12, 2013
  24. Dec 11, 2013 #23
    The memory loophole is just another way to avoid having a single probability space for the random variables, just as Accardi argued. In short the gist of Accardi's argument is more general, and in summary says that anything that guarantees a different probability space for the random variables, can violate the inequalities, in other words, violation of the inequalities proves that you do not have a single probability space for the random variables. Which is exactly what Boole had proved more than a century ago.

    If by memory loophole you are referring to the model by De Raedt, then this paper, just published might interest you http://jpsj.ipap.jp/link?JPSJ/82/034004/ [Broken], it is an experimental test which claims to have falsified it. Of course the response from De Raedt was published in the same journal. You can find both on arxiv.
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  25. Dec 12, 2013 #24


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    The experiment:
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  26. Dec 12, 2013 #25
    I'm still having trouble understanding this. I mean, contextualism is also a necessary feature of other models like Bohmian mechanics. What Bohmians refer to as "contextual realism" (e.g. spin, etc.) seems similar to what Accardi calls adaptive realism/chameleon effect. The difference though is that Bohmians accept non-locality as a necessary feature of their model on top of the contextuality. But Accardi is arguing that there's a difference between quantum contextuality vs probabilistic contextuaity. So while the former necessitates Bell's and non-locality the latter doesn't. He writes:
    Chameleon effect, the range of values hypothesis and reproducing the EPR-Bohm correlations

    I don't understand this and I think you are making the same point, but I don't think it can be that simple, which is why I was hoping someone could explain it to me, as I can't follow the math.
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