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A Paradox: Do LHV Theories Need the HUP?

  1. Oct 6, 2005 #1

    DrChinese

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    Here is one I am having trouble following. Can anyone help me through my confusion?

    Our setup is a normal Bell test using entangled photons created using spontaneous parametric down conversion (PDC). Such a setup uses 2 BBO crystals oriented a 90 degrees relative to each other. See for example Dehlinger and Mitchell's "...Bell Inequalities in the Undergraduate Lab".

    1. Say we have Alice and Bob set their polarizers at identical settings, at +45 degrees relative to the vertical. Once the individual results of Alice and Bob are examined, it will be seen (in the ideal case) that they always match (either ++ or --). According to the local realist or local hidden variables (LHV) advocate, this is "easily" explained: if you measure the same attribute of two separated particles sharing such a common origin, you will naturally always get the same answer. There is no continuing entanglement or spooky action at a distance, and conservation rules are sufficient to provide a suitable explanation. I.e. in LHV theories there is no continuing connection between spacelike separated particles that interacted in the past. The results will be 100% correlation.

    But that explanation does not seem reasonable to me, even in the case above in which Alice and Bob have identical settings. Here is the paradox as I see it. The source of the photon pairs is the 2 crystals. They achieve an EPR entangled state for testing by preparing a superposition of states as follows:

    [tex] |\psi_e_p_r\rangle = \frac {1} {\sqrt{2}} (|V\rangle _s|V\rangle _i + |H\rangle _s|H\rangle _i) [/tex]

    This is the standard description per QM. We already know this leads to the [tex] cos^2 \theta [/tex] relationship and the results will be 100% correlation.

    The local realist presumably would not accept this description as accurate because it is not complete, and violates the basic premise of any LHV theory. He has an alternate explanation, and the Heisenberg Uncertainty Principle (HUP) is not part of it. So now it appears that our experimental results are compatible with the expectations of both QM and LHV (at least when Alice and Bob have matching settings); however, they have different ways of obtaining identical predictions. But let's look deeper, because I think there is a paradox in the LHV side.

    2. Suppose I remove one of the BBO crystals, say the one which produces pairs that are horizontally polarized. I have removed an element of uncertainty of the output stream, as we will now know which crystal was the source of the photon pair. Now the results of Alice and Bob no longer match in all cases, and such is predicted by the application of QM: Alice and Bob will now have matched results only 50% of the time. This follows because the resulting photon pairs emerge from the remaining BBO crystal with a vertical orientation. Each photon has a 50-50 chance of passing through the polarizer at Alice and Bob. But since there is no longer a superposition of states, Alice and Bob do not end up with correlated results.

    But what about our LHV theory? We should still get matching results for Alice and Bob because we are still measuring the same attribute on both photons and the conservation rule remains in effect! Yet the actual results are now matches only 50% of the time, no better than even odds. What happened to our explanation that "measuring the same attribute" gives identical results? It seems to me that the only way for a LHV to avoid the paradox is to incorporate the HUP - and maybe the projection postulate too - as a fundamental part of the theory so that it can give the same predictions as QM.

    I mean, if the LHV advocate denies there is superposition in case 1 (such denial is essentially a requirement of any LHV, right?), how does the greater knowledge of the state change anything in case 2?
     
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  3. Oct 6, 2005 #2

    ZapperZ

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    Hum... but is this really a sufficient set of measurements? The whole purpose of measuring the observable at various angles is to distinguish between QM and LHV, per Bell's theorem. This is especially crucial based on the EPR paper of what happens with regards to the non-commuting observable, which in this case, is the "perpendicular" component of the detected angular momentum.

    The superposition aspect of QM is what makes this different than, let's say, the fragmentation of an object with zero angular momentum into 2 pieces going in opposite direction, where one knows immediately the angular momentum of the other piece upon the measurement of the angular momentum of the first piece. This is pure conservation law, but without any superposition of angular momentum before a measurement.

    Zz.
     
  4. Oct 6, 2005 #3

    vanesch

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    The LR adept will probably tell you that this changed the setup of the source, and as such, the "measuring the same attribute" result can change.
    I think you have to make a distinction between two classes of LR theories: Maxwellian field theories (which are of course LR) in which we have a more or less clear model of the physics that is happening (+ detector model) - eventually with modifications, such as SED, and "abstract LR theories" where no specific model is given (the kind of thing that is considered by Bell).
    I guess that the first category is what you are adressing your criticism against. This kind of theory has extreme difficulties explaining the 100% correlation that happens when the two polarizers are parallel, but make an angle with the polarization of the beam. I'm not expert enough in SED with the added noise and "detector background subtraction" to know whether it can talk itself out of this, but *classical* optics certainly can't.
    On the other hand, the abstract kind of LR theories, where photons walk around with entire books of prescriptions of what to do when they encounter which detector have the easy way out of saying that you simply change the contents of the books when you change the setup of the source.

    cheers,
    Patrick.
     
  5. Oct 6, 2005 #4

    DrChinese

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    Ya, we have the advantage today of Bell's Theorem. And that is quite an advantage. IF Einstein had that, well... who knows.

    But Bell figured out that something about the EPR argument didn't make sense. In retrospect, there must be more things about the local realistic argument that don't make sense. One thing I notice very clearly in dealing with the "crackpots" (you know who you are) is that they never come up with their own predictions! You can't pin them down to any particular formula. I have concluded that in addition to Bell's Inequality, there must actually be a number of critical relationships that hold true to constrain LHV theories. I would like to document these. I think if we documented them, we would see that constructing LHV theories that are even remotely close to experiment is impossible. Even more impossible than the Bell Inequalities might otherwise imply.
     
  6. Oct 6, 2005 #5

    DrChinese

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    Yes, and of course you know how I like Bell. But the LR folks keep trying to attack Bell tests by arguing that there is experimental error present. I, of course, don't agree with that logic and I don't believe there are any more loopholes in Bell tests than I believe there are loopholes in measuring the value of c.

    But I believe that if one advances any particular mechanism for local reality, one will quickly see that it fails on more fronts than a violation of a Bell Inequality. You can't make sense of any LR theory that makes specific predictions as regards a PDC setup. There are always angles that can't be explained.

    The norm is to say that QM and LR agree at 0 and 90 degrees. But I don't think that an LR theory can be constructed that works at those angles AND makes consistent sense within the context of PDC optics. And I think "naive" LR theories that match the Bell Inequality (i.e. as close as you can come to QM without crossing the line) become circular immediately. I.e. you have to assume that which you are trying to prove.
     
  7. Oct 7, 2005 #6

    vanesch

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    Well, you should check out nightlight's posts on s.p.r. where he claims that Stochastic Electrodynamics (by Santos and co) do exactly that. There are indeed some peer-reviewed articles by him on the subject, and Santos claims that he has constructed a model of PDC which has identical predictions with quantum optics in all cases (up to some redefinition of what is a detection event). I have to say, I tried to read the paper, I think I understand more or less the gist of it, but I'm not enough of a quantum optician to be able to critically read it and understand it.
    What I understand of it is the following: the EM field is classical, except that the field in vacuum is not empty, but has 1/2 hbar omega intensity in each mode (and this is then responsable for all "zero point energy" in quantum theory, also for electrons and so on). When we have a photodetector, it DOES respond to this classical noise, and does "click" a lot, only, we call that noise, or background, and subtract it off the "real" signal.
    The PDC receives the pump laser, but also the "noise", and the outgoing beams (classical beams) also have "noise". He then claims that this system gives exactly the same correlation function as quantum optics.
    But you will see that the paper is quite complicated and I just couldn't bring myself in spending weeks in digging out all the details.

    cheers,
    Patrick.
     
  8. Oct 7, 2005 #7

    DrChinese

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    Thanks for the reference, and I will check it out. What I am looking for is if the Santos theory can explain the results with both crystals, and just one as well. I realize that his approach must fail to Bell's Inequality anyway, but it seems to me that there must be other serious flaws in any LR theory being put forth as well. There are too many references to optics experiments nowadays for it not to fail victim to one or the other!
     
  9. Oct 7, 2005 #8

    vanesch

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    Also have a look at:

    http://www.qols.ph.ic.ac.uk/~kinsle/e-docs/kinsler-1996-arXiv.pdf

    cheers,
    Patrick.
     
  10. Oct 7, 2005 #9

    ZapperZ

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    Hey DrChinese, I'm not sure if this will address your question, but I should have pointed this out to you. You MAY want to read it:

    Research on hidden variable theories: A review of recent progresses, M. Genovese, Phys. Rep. v.413, p.319 (2005).

    It is 77 pages long! However, I think this is a worthwhile read.

    Zz.
     
  11. Oct 7, 2005 #10

    DrChinese

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  12. Oct 7, 2005 #11

    DrChinese

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    Looks great, I guess it may keep me busy as with Vanesch's ref. Good stuff guys!
     
  13. Oct 14, 2005 #12

    ttn

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    No, no, no. The EPR argument is not the same thing as the kind of theory that that argument argued for. The *argument* was that, if locality is true, there must exist a certain kind of hidden variables which determine the outcomes on each side. That argument was, is, and always will be completely valid.
    Now a separate point: Einstein (and others) believed that the premise of the EPR argument (namely, locality) was true. Hence they thought, based on the argument, that QM wasn't complete (since it didn't contain those outcome-determining variables) and that we should be looking for a hidden variable theory to complete it.
    Now what did Bell prove? He proved that no local hidden variable theory can agree with the QM predictions / experiment. That is, he proved that the kind of theory Einstein was hoping for (on the basis of (1) his belief in locality and (2) the EPR type argument) can't exist.
    That is simply not the same thing as proving that the EPR argument was wrong, that "something about it didn't make sense." It makes perfect sense, and Bell was one of the few people who has been able to rise above all the Copenhagenish obfuscation and see this clearly. If you read Bell's papers, you'll find that there is no room for doubt on this. Bell himself was 100% convinced that the EPR argument was perfectly valid.
    I'm not one of the LHV people you (rightly) criticize. LHV theories are ruled out by experiment, by any reasonable standard. As you say in another post, there are no more "loopholes" in the Bell tests than there are in such mundane experiments as measuring the speed of light. Yes, in principle there are loopholes, but you basically have to be a nut at this point to think those loopholes are of any relevance.
    My point is: one doesn't have to be a nut to insist on getting the logic of EPR and Bell straight. It's really not that complicated, and it undercuts your position when you say things like "Bell figured out that the EPR argument didn't make sense."
     
  14. Oct 14, 2005 #13

    DrChinese

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    Well, I agree with some of what you say but disagree with other parts.

    I like what you have to say about "if locality is true, there must exist a certain kind of hidden variables which determine the outcomes on each side". That is a great way to frame the argument about local reality. And I agree about the importance of getting EPR and Bell straight, which is why I try to stay close to their words on the subject where possible.

    But I disagree about Bell not finding something wrong with EPR. There is something wrong with EPR, and Bell showed it to us! EPR said both of the following:

    a) Either QM is incomplete, or there is not simultaneous reality to non-commuting observables.
    b) They believed that there IS simultaneous reality to non-commuting observables because a more complete specification of the system is possible.

    a) was supported by the logic presented. Clearly, b) was not rigorously supported and was an ad hoc assumption. Some people never accepted b) anyway, so it may not be material to them. Maybe that is your opinion too. On the other hand, some people did accept b) - but Bell saw a problem with that. He formulated his paper ("On the EPR Paradox") mathematicially assuming there WAS simultaneous reality to such observables, and found that was incompatible with QM itself. Hardly a result that EPR envisioned.

    I would definitely say that EPR took locality as an axiom. Bell definitely did not, as he was explicit in this regard.

    FYI: I do not believe people who believe in local reality are nuts. I think any scientist is a nut who says there is no scientific evidence for something when there is a lot of scientific evidence for it.
     
  15. Oct 14, 2005 #14

    DrChinese

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    Yeah, that is what I was wanting to say but somehow you said it a lot better. :smile: That is, that the superposition of states is central to QM and is not a feature of LHV theories. And in fact, the results are radically different when you look at the H and V streams independently than when you look at them combined. I don't think you can make any sense of the results if you postulate hidden mechanisms. (I.e. How can you get more matches at +45 degrees with the HV combined stream than with the H or V streams independently unless the superposition of states is real?)
     
  16. Oct 15, 2005 #15
    My current understanding is that the EPR and Bell arguments and experimental results, taken together, aren't necessarily telling us anything about locality. That is, locality-nonlocality isn't *necessarily* an issue. Rather, the issue might have to do with the way hidden variables can be modelled (which is what DrChinese seems to be getting at) which has to do with what can be ascertained about an *underlying reality* (ie., what it is that is incident on, say, the polarizers in an optical Bell test) from experimental results.

    Might one conclude that the *observables* are not in one to one correspondence with the underlying reality, and therefore that the qm form and experimental tests aren't revealing that nonlocal phenomena exist (or that they don't exist)?
     
  17. Oct 15, 2005 #16

    ZapperZ

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    No, one might not. By making such a statement, you are already making a HUGE assumption that there is (i) an underlying reality and that (ii) it is inaccessible via ANY measurement since, after all, it is, then we would have detected a deviation from QM's predictions.

    This isn't obvious, nor automatic.

    Zz.
     
  18. Oct 15, 2005 #17

    vanesch

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    Sherlock, I never understood fully what you wanted to say when you wrote about Bell.
    Bell's theorem tells us that it is impossible to construct a theory which:
    1) has a deterministic mechanism in it that determines (not that is 1-1, but that determines) all the outcomes of potential experiments (whether or not we perform them!)
    2) that this deterministic mechanism obeys local dynamics
    3) can generate exactly the same probabilities for outcomes as quantum theory does (the only randomness being a distribution of certain uncontrolled parameters in the source and in the detectors which have statistically INDEPENDENT distributions)
     
  19. Oct 15, 2005 #18
    The main assumptions are that reality exists independent of measurement, and that the results of probings of reality so far don't allow a complete assessment of what reality *is*.

    I would think that a physicist would embrace this view, if only for reasons of job security. :-)

    By 'underlying reality' I'm just referring to the exact qualitative form of what (which presumably came from the same oscillator) is incident on the polarizers during a coincidence interval . Or, for that matter, what is incident on a polarizer in an individual measurement. (One might assume that there is *no* qualitative physical form, but that doesn't make sense to me.)

    Obviously *some* aspect of this isn't inaccessible to measurement. But it isn't (to me at least) readily apparent what the form of the *incident* disturbance is. The polarization model that would seem to fit ok with individual measurements does not fit joint measurements.

    So, either the polarization model is not in one to one correspondence with what the incident disturbance is, or joint measurements are measuring something different than individual measurements, or both.

    In any case, the individual measurements are unpredictable, which indicates that the models involved are not in one to one correspondence with reality. That is, the results of polarization measurements don't necessarily tell us anything about the polarization of the incident disturbance prior to filtration or, in the case of joint measurements of entangled particles, if polarization is the applicable model.
     
    Last edited: Oct 15, 2005
  20. Oct 15, 2005 #19
    I'm assuming that there is a physical mechanism which determines, qualitatively, the outcomes of Bell tests -- and that it is so far unknown.

    Since quantum theory doesn't deal explicitly with this mechanism, then the form of qm calculations can't be taken to mean that there exist nonlocal phenomena in nature. It is of course certain that there exist, in some more artificial sense, nonlocal 'phenomena' in qm.
     
  21. Oct 15, 2005 #20

    ZapperZ

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    Which again emphasized my previous message, that you ARE making an a priori assumption of something beyond that can be measured. I can easily make a similar assumption, that there's no such thing, and you can't dispute that since I will only put validity on what I can measure. Everything else is speculation.

    Zz.
     
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