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Need help with some aspects of Bell’s theorem

  1. Oct 10, 2014 #1
    Looking for the help with a conceptual framework of the Bell’s theorem and would appreciate any assistance.

    The Bell theorem has been verified in numerous experiments (lets assume that those experiments are free form loopholes).

    As I understand the essence of those proves are based on comparing correlations predicted by QM with the correlations derived from the hidden variables theories and after ruling them out, the QM (non-locality) is the only option.

    If the above makes a sense, should these hidden variables theories have some prerequisites to be used in the Bell theorem? Should they be in compliance with QM equations and yield the same experimental results as predicted by QM. If they don’t complies with QM, they are not real hidden variables (QM) theories but surrogatess only.

    If these theories indeed comply with QM prerequisites , why their correlations aren’t calculated based on the QM prediction, but the ‘classical’ approach is used instead?
     
  2. jcsd
  3. Oct 10, 2014 #2

    PeterDonis

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    Yes.

    No; that's the whole point of Bell's theorem. Bell's theorem says that it is impossible to have a local hidden variable theory that makes the same predictions for experimental results as QM. That's why the experimental proofs rule out hidden variable theories: the experiments show that the predictions of QM are correct, so any theory that makes different predictions--and Bell's theorem says any local hidden variable theory must make different predictions--is ruled out by experiment.

    In your terminology, Bell's theorem says that it is impossible to have a local hidden variable theory that "complies with QM prerequisites".
     
  4. Oct 10, 2014 #3

    DrChinese

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    miosim, I will give you a short reply but will otherwise bow out to allow others to answer you. I don't believe I have been effective in getting the message across on this subject in earlier threads. Hopefully, the following will help a little.
    ----------------------------------------------------------

    With Bell, you MUST distinguish between LOCAL hidden variable theories and NON-LOCAL ones. You didn't do that above, and it makes it nearly impossible to answer your questions. Here is the summary:

    1. LOCAL HV theories are incompatible with QM and Bell tests.
    2. Bell tests do NOT prove NON-LOCALITY. They prove what is call QUANTUM NON-LOCALITY.
    3. The reason for this different term (with Quantum added) is that there are LOCAL NON-REALISTIC theories that are QUANTUM NON-LOCAL. That may seem like a contradiction, but it is not.
    4. There are many interpretations of QM that display Quantum Non-locality. They include: Copenhagen, Bohmian, Many Worlds, Time Symmetric, etc. All of these predict as per orthodox QM. So all of these are either NON-LOCAL or NON-REALISTIC (or both).
    5. The observed correlations in Bell tests are not as predicted by classical statistics, so that approach is discarded completely when calculating the predictions. As you can see from the graph you presented previously, a classical approach to the statistics does not even come close to the actual results. The other approach is what is sometimes called semi-classical. That approach is closer to actual, but is still ruled out.

    Good luck,

    -DrC
     
  5. Oct 10, 2014 #4
    This is exactly what I can't grasp, doesn't matter how references I read.

    I thought that Bell (and other) select some generic candidate for hidden variable theory and applying some sort of classical approach (foreign for the QM system) calculate correlation.
    As I understand the REAL hidden variable theory should have the same predictions as orthodox QM but only having different interpretation of the observable results. Therefore it may not be fair to calculate correlation not deriving from QM.
     
  6. Oct 10, 2014 #5

    PeterDonis

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    First, as DrChinese noted, you need to carefully distinguish between local hidden variable theories and nonlocal hidden variable theories. Bell's Theorem only applies to local ones.

    What Bell did was to show that any local hidden variable theory must have certain properties that restrict the correlations it can predict; and that the correlations predicted by QM are outside those restrictions. He didn't pick any particular local hidden variable theory, not even a "generic candidate"; he just used general properties that any local hidden variable theory must have.

    It's perfectly possible for a nonlocal hidden variable theory to make the same predictions as QM. An example of such a theory is the De Broglie-Bohm pilot wave theory. But it's not possible for a local hidden variable theory to make the same predictions as QM. So one way of stating the conclusion of Bell's Theorem is that, if you want the "real" hidden variable theory to make the same predictions as QM, it has to be nonlocal.
     
  7. Oct 10, 2014 #6
    Thank you for both of you, it helps to focus .

    Dr. DrChinese

    I am back reading tutorials on your site ...

    http://drchinese.com/David/Bell_Theorem_Easy_Math.htm

    ... but stumble on the following sentence:

    “…Suppose we consider a single particle (photon) of light. We ask a simple question: does it have a definite polarization at the following three angles: 0 degrees (A), 120 degrees (B), and 240 degrees (C)? According to the EPR paper, its polarization at these 3 angles correspond to actual elements of reality…”

    My question is: How a single photon may have “definite polarization at the THREE angles”?
     
  8. Oct 10, 2014 #7
    I gess you mean that according to EPR a single photon have definite polarization regardless of the angle of polarizer.
     
  9. Oct 10, 2014 #8

    DrChinese

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    Exactly! In other words: your car (if you have one :) ) has a length, a height, and a width at all times. Does a photon? Obviously photons do not have a length, a height, or a width. But do they have a polarization at 0, 120, 240 degrees at one point in time (simultaneously) ?

    The EPR ipremise was that they did. (And you could call that a "semi-classical" idea because it would match the perfect correlations they expected for entangled systems. Please note that "semi-classical" has somewhat different meanings to different people in QM concepts.) In other words, EPR did not deny entanglement as phenomenon. They envisioned a day in which such a theory could be formulated (it never was of course).

    So EPR ASSUMED realism (elements of reality) but could not prove it rigorously. Bell formulated a challenge to that assumption.
     
  10. Oct 10, 2014 #9

    jtbell

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    For a given experimental setup, this restriction can usually be described using an inequality relation for the quantity being measured. This is often called a Bell inequality.

    That is, the predictions of QM violate the Bell inequality for the experiment in question; whereas the predictions of any local realistic theory must satisfy that inequality. To date all experiments of this type have violated their Bell inequality.
     
  11. Oct 10, 2014 #10
    I think this sentence may confuse reader, because I read it as the 'same photon may have 3 different polarizations (simultaneously) prior interacting with polarizer'.

    I have other difficulties to read explanations on your site, but I don't want to "bug" this thread with my silly questions about every sentence.

    I understand all comments provided so far, but I feel I need go back to reading and than I may have a better question.

    Thank you all
     
  12. Oct 10, 2014 #11

    DrChinese

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    You don't think it strange that any ordinary object has a length/width/height, but have a hard time imagining polarizations at 3 angles? EPR imagined the extra attributes existed (of course I follow QM which does not support that).

    Different sites use different words, and that may help you too.
     
  13. Oct 10, 2014 #12

    Nugatory

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    You should read that as "the photon may be in a state such that if I measure the polarization at a given angle I will get a particular result". A three-dimensional object works that way - I can say that if I measure its height I'll get some number, if I measure its width I'll get some number, if I measure its length I'll get some number. We call these three numbers the height, width, and length respectively, and we have no problem saying that the object has definite height, width, and length whether we make the measurement or not.

    Quantum mechanical measurements don't work that way. The formalism of QM says that if we don't measure something then it doesn't have a definite value (not "it has a value but we don't know what it is because we didn't measure it"). Bell's theorem says that any assignment of definite values to these unmeasured quantities will, under some circumstances, lead to results that differ from the quantum mechanical prediction.
     
    Last edited: Oct 10, 2014
  14. Oct 10, 2014 #13

    morrobay

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    EPR assumed realism but the conclusion for the inequality violation is usually non - locality. Why not non realism alone ?
    What would be a real (complete) local non realism hidden variable model that would make same predictions as QM ?
    Besides the many worlds interpretation.
     
  15. Oct 10, 2014 #14

    PeterDonis

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    That's because Bell used the assumption of locality to derive his theorem; he didn't use any assumption of "realism". AFAIK the MWI, since it's just an "interpretation" of QM and makes all the same predictions, violates the locality assumption as well, at least as Bell formulated it.
     
  16. Oct 11, 2014 #15
    I am trying to understand why it happens. What are the KEY physical characteristics of any deterministic LV theories that case a lower correlation compare to correlations derived from QM?

    I wonder if the difference is as follow: in the deterministic model the spin/polarization are not affected by position of detectors, but according to QM the position of detectors influences (positively?) the result of measurement thus causing a higher correlation.
    (see below Bell's article "On the E...P...R... paradox (1964) describing the deterministic LV model: "The vital assumption for this model is that the result of measurement B doesn't depend on setting of detectors."

    View attachment 74296


    It seams that Broglie–Bohm theory (see quote from Wiki below) has the same prediction as QM because according to this theory the measurement device is also influences the particle's property (spin, polarization, etc.) and thus increases (compare to deterministic model) correlation. Does it make sense?

    http://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory
    Thus, for the de Broglie–Bohm theory, the particle's spin is not an intrinsic property of the particle—instead spin is, so to speak, in the wave function of the particle in relation to the particular device being used to measure the spin.
     
  17. Oct 11, 2014 #16

    Nugatory

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    Yes, that's pretty much it. Bell's proof proceeds from the assumption that the probability distribution of the measurement at a detector can be written as a function of the initial state of the entangled pair and the setting of that detector, but not the setting of the other detector. That's pretty much the operational definition of a local theory, and altogether consistent with EPR's sense of locality.

    The dBB theory is indeed non-local in this sense, and thus is capable of reproducing the QM predictions and experimental results. It's a matter of personal taste and an endless interpretational rathole whether it is strictly correct that the measuring device "influences the particle's property"; the most that you can say without slipping in some additional assumptions is that the settings of the measuring devices influence the correlation of the results of the measurements.
     
  18. Oct 11, 2014 #17
    Do you mean that per QM (and not per deterministic model) the measurement at a closest detector can be written as a function of the initial state of the entangled pair and the setting of that detector, while the second (more distal) detector interacts with the particle that lost entanglements?
     
  19. Oct 11, 2014 #18

    Nugatory

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    No I do not mean that. It's easy to set up thought experiments in which the distances to the detectors and hence the relative ordering of the two observations is different for different observers, so there is no meaningful way of saying that one particle interacts before the other. The quantum mechanical prediction is that the after you've done both measurements, in either order, they will be correlated; bell's theorem says this correlation will have statistical properties that cannot be reproduced by any theory in which the probability distribution of results at one detector is not a function of, among other things, the setting of the other detector.
     
  20. Oct 11, 2014 #19
    Say we set up a thought experimentscase of entangled photons, in which the first detectors is just few meters away from the experimenter, while the second detectors is on Mars. Would the experimenter receive the results of measurement from the first detector in a few nanoseconds instead of few minutes (time that light travel to Mars)?
    This question is not about Bell correlation but about QM in general. Is the photon that will reach the detector on Mars will be still entangled?
     
  21. Oct 11, 2014 #20

    DrChinese

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    Don't mean to be contradictory, but...

    Bell did not identify the spot where he introduced realism... but he did introduce it and I can show you where. Look at his original paper. His (2) is the locality condition, and then after his (14) he makes the leap to a THIRD element of reality (in addition to a and b): "It follows that c is another unit vector..." This is referenced in EPR when they say that they assume that elements of reality that individually must exist (height/length/width are our examples of simultaneous realism in this thread) must also exist simultaneously. Bell is making the mathematical implementation of that statement by mixing a, b and c in the next few equations. (QM of course would reject all this, either (2) or (14+) or both.)
     
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