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Bell's assumption

  1. Jun 20, 2013 #1
    The recent activity on Bells theorem in PF has triggered my interest in the subject and I have added Bell to the list of things I might look at in greater detail. Unfortunately I cannot see the justification in one of the assumptions apparently made by Bell.

    As far as I understand it, Bell considered the assumptions made in the EPR paper and showed that these lead to predictions which are contradicted by the observations. So it seemed that there was something wrong with EPR. Fair enough.
    From this it was assumed that not just EPR but all theories of hidden variables are incorrect :

    NO PHYSICAL THEORY OF LOCAL HIDDEN VARIABLES CAN EVER REPRODUCE ALL OF THE PREDICTIONS OF QUANTUM MECHANICS.

    How can such a sweeping generalistion, which is based on EPR only, be made about all potential theories? If any theory is developed then to be a good theory it must conform to the observations. EPR failed but that doesn't mean that other theories will fail.
    What am I missing?
     
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  3. Jun 20, 2013 #2

    DrChinese

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    You are actually correct, the idea is that theories following in the footsteps of the EPR definitions are ruled out. So let's examine the 2 keys assumptions:

    a) There are no FTL influences (locality): I think most people intuitively understand what is meant by this. Bell used the idea that a measurement setting here cannot affect a measurement outcome there, and vice versa. Some call this separability. At any rate, if you are a believer in locality, then logically entanglement of 2 particles which are separated CANNOT include a physical connection between the 2. So here is a big leap for all local realistic theories: entanglement must be represented by independently evolving particles that have some set of attributes/properties that were originally correlated in some manner. These properties then give rise to the correlated results, independently of decisions the observer might choose to make as to what is to be measured. Alice's decision does not affect what Bob sees, although they both may be dealing with particles that have closely correlated states.

    Does the above make sense?

    b) The realism assumption is the one that seems to be a sticking point for many people. To EPR, it meant that the so-called "perfect correlations"* of entangled pairs implies that measurement outcomes must actually be predetermined. That means, in essence, that the observer plays no role in the outcome other than to select which predetermined properties are being revealed. That is because if the observer played any significant role, the results would NOT be perfectly correlated! Keep in mind that there are plenty of Local Hidden Variable theories in which the observer plays a role. In those, however, you don't have perfect correlations in these cases so the theory fails a basic test.

    Does the above make sense? So really, your question comes down to: what OTHER definitions or representations of realism are there in which perfect correlations are preserved?

    *which means that the results are 100% correlated at identical angle settings.
     
  4. Jun 20, 2013 #3

    kith

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    If we put it in a simple logical statement, the EPR argument goes like this:
    locality, realism => QM does not posess the property of completeness

    Now Bell showed that:
    locality, realism => contradiction with predictions of QM => there are no complete theories in the sense of EPR which reproduce the predictions of QM
     
  5. Jun 20, 2013 #4

    Nugatory

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    I'll suggest a cheapie analogy about how very sweeping statements can be true: When I say "No odd number can be the sum of two even numbers", that's an equally sweeping generalization - there are an infinite number of odd numbers, and I can't possibly have examined them all. How can I be sure that somewhere out there, among the infinity of odd numbers that I haven't looked at, there's not one that is the sum of two even numbers?
    Of course the answer to this question is obvious - an odd number is defined in such a way that it cannot be the sum of two even numbers, and therefore I can be confident that all the numbers I find that are the sum of two even numbers cannot be odd numbers.

    Bell's theorem follows a similar logic, except that instead of working with a class of numbers (the odd numbers) he works with a class of theories, namely those that assume local hidden variables. He demonstrates that if local hidden variables (he has a more rigorous formulation, of course) exist, then certain results must follow. Therefore, any theory that assumes the existence of local hidden variables must also predict those results.

    If the theory does not predict those results then, just as any number that is the sum of two even numbers is necessarily not an odd number, that theory is necessarily not a local hidden variable theory.
     
  6. Jun 21, 2013 #5
    Thank you all for your replies. I need time to digest the comments and I need time to refer to different sources on the subject. My personal feeling is that hidden variables as envisaged by EPR do not exist and that even if they do they are irrelevant. Of course personal feelings can change as one looks more deeply into a subject.
     
  7. Jun 21, 2013 #6
    .........sweeping conclusions.


    there are Non-local Hidden Variables Models.



    .
     
    Last edited: Jun 21, 2013
  8. Jun 22, 2013 #7
    As has been pointed out in early discussions, (see posts #48, 49 here: https://www.physicsforums.com/showthread.php?t=664394&highlight=counterfactual&page=3), Bell's argument relies among other things on counterfactual definiteness, and that appears to be a bit tricky.

    I came across the following discussion based on Tomasz F. Bigaj, Non-locality and Possible Worlds: A Counterfactual Perspective on Quantum Entanglement:

    "it is not entirely clear how to handle counterfactual reasoning in an indeterministic context. Suppose that while you are flipping a coin (which we will suppose to be a fundamentally indeterministic event for this discussion -- ex hypothesi, nothing in the actual world is causally sufficient to determine the result of the flip), I hum a bar of Ode to Joy. My humming (again, ex hypothesi) has no causal influence on your coin-flipping. You get heads. If I had not been humming, would you still have gotten heads?
    Logical intuitions seem to differ on this point. Some argue that, because your flip was indifferent to my humming, you would still have gotten heads if I had not been humming. My humming, or lack of it, could not have affected the outcome. Others argue that we cannot affirm that the flip sans humming would have resulted in heads, because the result is in fact not determined by anything -- it was completely indeterministic. The (imagined) trial flip sans humming must be considered to be another, independent flip of the coin, the result of which we cannot predict. [..] Yet another 'intuition' is that the counterfactual in question is itself indeterminate in truth-value."
    - http://ndpr.nd.edu/news/23047-non-l...rfactual-perspective-on-quantum-entanglement/

    I came across several of that type of discussions; and I have not yet made up my mind. I'm certainly not convinced that counterfactual definiteness should hold in a "realistic" world, that is, according to concepts of "realists" which are not necessarily limited to definitions of EPR and Bell.

    In an earlier thread about another topic, Lugita asked me some questions concerning the topic under discussion here, and so I'll partly reply here:

    See above. Usually such tests are done with the help of random generators. As we are clueless about how stochastic processes work, IMHO we have no theory to support or reject the possibility to predict the outcome of a stochastic process that has not happened because another stochastic process did not select it.

    Apparently different people mean different things with "proving a negative"... Cantor discussed known sets, he did not try to prove a negative in the way Bell did. Bell presumably showed that all kinds of not yet imagined theories cannot match a known model. And that much wider sounding claim is the topic here.

    Sure, I exaggerated a little. A model is composed of assumptions of that which is modeled. In other words, if we make assumptions about something (as EPR and Bell did), that already consists a model. More assumptions provides us with a more detailed model.
    Thus, in agreement with the OP and DrChinese: Bell's theorem necessarily applies to the class of theories that match the EPR definitions of terms and Bell's assumptions about such theories. That is less general than Bell's theorem as cited in the first post sounds (and I think, also less general than Bell intended).
     
  9. Jun 22, 2013 #8
    Thank you audioloop and Harrylin. Before I can go any deeper into this I need to understand more about the basics. I will be grateful if anyone could answer the following question about entangled particles:

    Suppose that Bob makes a measurement on one of the particles and as a result observes that a certain property, eg spin, has a definite value. Does quantum mechanics assume anything about that property before the measurement is made?
    My understanding at present is that before the measurement is made the particle is in a superposition state and has all possible values of the property simultaneously. Is that correct or does quantum mechanics make different assumptions?

    (I am aware that a measured property of one of the particles results in a correlated value for the other particle)
     
  10. Jun 22, 2013 #9

    stevendaryl

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    It seems to me that counterfactual definiteness is a consequence of his notion of local hidden variables, together with facts about quantum correlations, not an assumption. His basic assumption is that distant correlations are "implemented" in terms of local correlations.

    Suppose you have two distant experiments:

    • Measurement [itex]M_1[/itex] is performed at space-time region [itex]r_1[/itex]. The outcome [itex]o_1[/itex] is given by some probability distribution [itex]P_1(o_1)[/itex]
    • Measurement [itex]M_2[/itex] is performed at space-time region [itex]r_2[/itex]. The outcome [itex]o_2[/itex] is given by some other probability distribution [itex]P_2(o_2)[/itex]
    • [itex]r_1[/itex] and [itex]r_2[/itex] are spacelike separated (so according to SR, no causal influence can travel from one to the other).

    The outcomes are correlated if
    [itex]P(o_1 \wedge o_2) \neq P_1(o_1) P_2(o_2)[/itex]

    Bell's notion of local hidden variables assumes that all such correlations are explainable in terms of shared histories. That is:

    • There is some set of facts [itex]f_1, f_2, ...[/itex]
    • These facts refer to events or conditions in the common causal past of [itex]r_1[/itex] and [itex]r_2[/itex] (that is, these facts refer to the intersection of the backwards light cones of the two regions of spacetime).
    • If these facts were known, then the correlations would disappear:
      [itex]P(o_1 \wedge o_2 | f_1, f_2, ...) = P(o_1 | f_1, f_2, ...) P(o_2 | f_1, f_2, ...)[/itex]

    These assumptions by themselves don't imply counterfactual definiteness, and don't assume it. But if the correlations are perfect (which they are in an EPR-type experiment when both experimenters choose the same orientation), then counterfactual definiteness follows.
     
  11. Jun 22, 2013 #10

    Nugatory

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    We're getting dangerously close to interpretations of quantum mechanics... And although many sensible conversations have gone down that rabbit hole, few have come back... :smile:
    I'm going to answer your question using the "shut up and calculate" interpretation, which feels right for a discussion about how QM results match experiments, wisely avoids (stay away from that rabbit hole!) any questions about what might be "really happening".

    Quantum mechanics says that before the measurement the particle's state is described by a mathematical construct (called, depending on which mathematical formalism you are using, things like "state vector" or a "ket" or a "wave function"). By subjecting this mathematical construct to various mathematical manipulations, we can derive results like "if I were to measure in a vertical direction there is an x% chance of getting spin-up, y% of getting spin-down, x+y=100%, and although we aren't going to go there now, the measurement will change the state".

    A hidden variable theory would say that the particle is really spin up or spin down all along, so that if we knew enough about its state and how spin worked inside of particles, we'd be able to say "it is spin-up" or "it is spin-down"; and the x%, y% prediction of QM is just because we don't have this knowledge. It would be as if I flipped a coin, and then without looking at it, said "50% heads, 50% tails" - that doesn't tell us anything about the coin, which is really either 100% heads or 100% tails, it just tells us something about what we know about the coin.
     
    Last edited: Jun 22, 2013
  12. Jun 22, 2013 #11
    Thank you Nugatory. I agree (with some reservations that I am unable to pin down at the moment) that a spin property is revealed when a measurement is made. I also see that hidden variable assumes that such properties are carried by the particles before the observations are made.
    What bothers me is that there are other properties that are assumed to exist before the measurements are made.For a photon these properties include :

    1. Photons travel at the speed of light
    2. When displaying wave properties a photon has a certain frequency.

    Is it so that the existence of properties such as one and two are assumed to exist before observation for the different interpretations of quantum mechanics and even for "shut up and calculate". If so what is special about these properties and properties like spin which need an observation to display some sort of reality?
     
  13. Jun 22, 2013 #12

    DrChinese

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    Photons have position, momentum, etc and all of these observables can be entangled. They all have constraints imposed by the Uncertainty Principle.
     
  14. Jun 22, 2013 #13

    Nugatory

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    Get away from that rabbit hole! Stop asking questions and start calculating - what part of "shut up and calculate" don't you understand? :smile:

    OK, seriously, kidding aside... The answer to your question is "no", those observables that we think of as fixed properties of a system do not have any special philosophical status in the formalism. We often treat them as known going into a problem, but if you dig into it, you'll find that somewhere in the past we did something to the system ("preparation" is the term often used - see also my weasel words about "although we aren't going to go there now" in #10) to make it so that an observation of that quantity would necessarily give us the value we're assuming.

    In writing the previous paragraph I found myself trying to decide whether the right word would be "philosophical" or"ontological". And if you're thinking about stuff like that, you're doing something interesting and important but it's not physics. See how easy it is to start down into the rabbit hole - you had me going too.

    Fortunately, we don't need to go there to understand the significance of Bell's theorem. All we need is the quantum mechanical prediction for the outcome of certain measurements performed on certain particles after they have undergone a particular preparation procedure.
     
    Last edited: Jun 22, 2013
  15. Jun 22, 2013 #14
    Thank you DrChinese. I'm a bit familiar with the uncertainty principle but I don't see how it puts constraints on properties such as the speed of light.

    If you have the time I would be interested to hear your take on the question I posed in post eight.
     
  16. Jun 22, 2013 #15
    I have a lot to take in here so with thanks to all I'm leaving it for a bit and going off to do some non thinking but absolutely necessary tasks such as having a nice cold pear cider.
     
  17. Jun 22, 2013 #16

    stevendaryl

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    This is perhaps an overly picky point, but a hidden variables theory does NOT by itself say that the particle is really spin up or spin down all along. Instead, it allows for the spin outcome to be a possibly nondeterministic function of the state of the detector and the state of the particle being detected. But the existence of perfect correlations (in the case of distant detectors measuring along the same axis) implies that a hidden variables explanation can't possibly agree with experiment unless that outcome is a deterministic function of the hidden variable and the detector orientation. So the claim that the spin had a definite value all along is a conclusion from the local hidden variables assumption, it's not assumed.
     
  18. Jun 22, 2013 #17

    Nugatory

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    Picky but not "overly" picky.... good point, thx.
     
  19. Jun 22, 2013 #18
    Bold mine.


    http://www.tau.ac.il/~vaidman/lvhp/m105.pdf
    "the core of the controversy is that quantum counterfactuals about the results of measurements of observables, and especially “elements of reality” are understood as attributing values to observables which are not observed. But this is completely foreign to quantum mechanics. Unperformed experiments have no results! “Element of reality” is just a shorthand for describing a situation in which we know with certainty the outcome of a measurement if it is to be performed, which in turn helps us to know how weakly coupled particles are influenced by the system. Having “elements of reality” does not mean having values for observables. The semantics are misleading since “elements of reality” are not “real” in the ontological sense"


    .
     
  20. Jun 23, 2013 #19
    Thank you. Many posts here have me thinking and in some cases reaching for the dictionary. I will "shut up and calculate",but before this there are some details I need in order for me to understand more about the system upon which the calculations are based.If these details are considered as irrelevant I would like to see for myself why.Let me try to explain:

    Suppose there was a source which emitted photons. I think that there is an accepted view that each photon has certain properties and in some cases the nature of those properties is accepted as being known without the necessity of making further observations.
    Each photon has a:
    1.known speed
    2.known mass
    3.frequency ( which is known for some methods of photon emission)
    4.spin

    I think it is accepted that, amongst other things, photons possess properties 1,2 and 3, even before observation, but is it accepted that photons have the property of spin, whether they be entangled or not? I know that an observation may be needed to detect the nature of the spin and that what is observed depends, in part, on the nature of the observation.
    My main question which is a reiteration of a reiteration is:

    Is it assumed that photons have the property of spin before appropriate observations are made on that photon?

    I'm not wishing to attribute values to any spin I just want to know if it is assumed that the photon has a spin or not. I have my own take on this and I just want to know whether there is a generally accepted answer to the question. If any of the old guard come in with an answer thank you for your patience.
     
    Last edited: Jun 23, 2013
  21. Jun 23, 2013 #20

    DrChinese

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    The accepted version is that NON-commuting observables can be known in advance (ie prepared). Most particles have multiple spin components, and generally these components do not commute. Therefore, it is not meaning to make assertions about those that do not commute.

    For example: if you know a photon's linear polarization at 0 degrees, its polarization at 45 degrees is completely uncertain.
     
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