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Bell's Inequalities and Double Dependencies on Hidden Variables

  1. May 7, 2010 #1
    I've been trying to wrap my head around the issues with Bell's Inequalities (while following the three related threads in this forum) and I think I finally have it figured out well enough to ask a question that's been bothering me.

    In particular, I'll start with http://drchinese.com/David/Bell_Theorem_Easy_Math.htm" [Broken] for complete background)

    My understanding of Bell's Inequalities for this case is that it states that any local realistic hidden variable theories must have a minimum probability of 0.3333 which is higher than the prediction of QM of 0.25. Finally, actual experiment verifies the QM prediction rather than the "realistic" prediction of 0.333.

    First, do I have this basically right?

    Second, consider the following potential scenario:

    1) There are two separate local hidden variables for a photon, let's call them polarization angle and phase.

    2) Detection of the photon through a measurement apparatus is a function of both polarization angle with respect to the measurement apparatus (an angle we'll call θ), and phase with respect to the measurement apparatus.

    3) The polarization angle and the phase of the photon interacts with the quantum state of the measurement apparatus. Thus without knowledge of the quantum state of the measurement apparatus, it cannot be known what a given measurement will be. If you knew what the quantum state of the measurement apparatus would be at the time of measurement and you knew of the values of the hidden variables for the photon, then you could determine a priori whether or not the photon would be detected. Thus there are four total hidden variables for the system, two for the photon and two for the measurement apparatus (one with respect to the polarization angle and one with respect to the phase).

    4) Fortunately, it is possible to understand the statistical characteristics of the system to model the probability of a measurement even when the quantum state of the measurement apparatus remains hidden.

    5) Lets call the function which defines the probability of a photon passing through the measurement apparatus for the polarization hidden variable Alpha. Alpha is a function of θ.

    6) Let's call the function that defines the statistical probability for the phase hidden variable Beta. Beta is also a function of θ.

    So the probability of a detection at a given angle is defined by:

    ρ = Alpha(θ)Beta(θ)

    The probability is doubly dependent on θ. Is there anything about this scenario which is not realistic? Not local? Could a theory with these characteristics not be a local hidden variable theory?

    Now what if:

    Alpha(θ) = Cos(θ)


    Beta(θ) = Cos(θ)

    Do we not get a combined probability function which matches QM? Namely:

    ρ = Cos^2(θ)

    Returning to the 0.3333 probability of Dr. Chinese's example. It seems to me that the potential for a double-dependency on θ, makes the Bell's Inequality not work out. You'd have to multiply the minimum probability (i.e. 0.333 * 0.333 = 0.111). With this new Bell's Inequality of > 0.1111 we don't get a violation with a measurement of 0.25.

    What Bell seems to be missing (at least to my naive mind) is the idea that there could be hidden variables lurking in the measurement apparatus in addition to those in the photon or particle being measured in such a way as to make it possible that there are double dependencies on the same hidden variable.

    Clearly, lots of people have been studying Bell's Inequalities and their relation to QM for almost 50 years, so I must be missing something obvious. What am I missing?
    Last edited by a moderator: May 4, 2017
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  3. May 7, 2010 #2


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    A quick note: My page is tanked right now as I am switching hosts this week. I should have this back up shortly... My apologies, sorry you picked today to post above.
  4. May 7, 2010 #3


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  5. May 7, 2010 #4


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    inflector, your post doesn't seem to specifically address the issue of how you want to explain the fact that both experimenters always get the same result if they choose the same detector setting (same polarization angle for both detectors), which is a necessary part of the proof that the correlation should be greater than or equal to 0.333... for different settings--are you suggesting that in any case where they choose the same detector setting, the two hidden variables associated with each photon are identical, and the two hidden variables associated with their two measurement apparatuses are also identical? It would seem a little weird to call the variables associated with the measurement apparatus "hidden" if they are always completely determined by the choice of observable measurement angle.
    Last edited: May 7, 2010
  6. May 7, 2010 #5


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    There are a couple of issues we need to agree upon here. First, the photons are detected with a lot greater efficiency than you give credit for. The reason is that usually Polarizing Beam Splitters are used rather than Polarizing lenses. This means that you know affirmatively which path the photon takes.

    Second, you may in fact have a "bias" which makes the sample collected appear to violate the Bell Inequality, when the full universe does not. That is called the Fair Sampling Assumption. Now, I strongly recommend that before we go further, we suspend consideratiion of this particular issue for the time being. The reason for that is that Bell assumes that sample = universe. So this doesn't YET apply (although we still have to consider it later when actual experiments occur).

    So the question then returns to our .333 example. A local HV theory must respect the .3333 while QM says
    .250. If we agree on this portion, then we can move on to consider the Fair Sampling.
  7. May 7, 2010 #6
    I was under the impression that we haven't been able to experimentally verify this yet, i.e. that we haven't been able to run tests on single photon pairs by themselves reliably.

    I thought our tests (Aspect and the newer confirmations) have only been of the form of large samples of pairs.

    Is my thinking here incorrect? Can you point me to some papers which describe single-pair photon tests?

    Yes, I've been following that thread along with the other two current ones which describe Bell's Inequalities and LHV theories, i.e. "Is action at a distance possible as envisaged by the EPR Paradox," "Local realism ruled out? (was: Photon entanglement and...)," as well as "Trying to Understand Bell's reasoning."

    They don't address my scenario at all.

    I don't think my reasoning depended on detector efficiency. I had been assuming that a photon would either pass or not, that seems to be equivalent to take path A or B, a simple binary choice which is still dependent on the angle. Is it not?
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  8. May 7, 2010 #7


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    Well, what do you mean by "pairs by themselves"? A single pair can't tell you what the probability is that a pair measured at the same angle will always give the same result (or always give opposite results depending on the type of entanglement), you need a large sample for that. No experiment can prove that this works perfectly (there may be stray non-entangled photons so I think you always have to rely on coincidence counts) but they can verify it to a high degree of precision...anyway, the point is that the proof of the Bell inequality which says the probabilities should be greater than or equal to 0.333... when different settings are used is dependent on the assumption that they will be correlated with probability 1 when the same setting is used. There are other Bell inequalities which aren't reliant on such perfect correlations though, see the CHSH inequality which I discussed in the second half of post #2 on this thread (and I also gave my own explanation of the 0.333... inequality at the beginning, maybe it'll help explain why it depends on the assumption of getting the same results with probability 1 when the same measurement is performed)
    Last edited: May 7, 2010
  9. May 7, 2010 #8
    I mean by "pairs by themselves" that the detectors can detect single photons with sufficient efficiency that we can measure the correlation of individual entangled pairs reliably. So, for example, if a test was setup with the same measurement angle, that the photons would indeed give identical results.

    I was under the impression that detectors couldn't do this efficiently yet, so we had to rely on statistical methods to determine the coincidence probabilities. Perhaps that's what you mean by "you always have to rely on coincidence counts?"

    Okay, I think that makes sense.
  10. May 7, 2010 #9


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    "The Big Picture"
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