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Derivation of the CHSH inequality

  1. Sep 24, 2014 #1
    Bell's 1971 derivation
    The following is based on page 37 of Bell's Speakable and Unspeakable (Bell, 1971), the main change being to use the symbol ‘E’ instead of ‘P’ for the expected value of the quantum correlation. This avoids any implication that the quantum correlation is itself a probability.

    We start with the standard assumption of independence of the two sides, enabling us to obtain the joint probabilities of pairs of outcomes by multiplying the separate probabilities, for any selected value of the "hidden variable" λ. λ is assumed to be drawn from a fixed distribution of possible states of the source, the probability of the source being in the state λ for any particular trial being given by the density function ρ(λ), the integral of which over the complete hidden variable space is 1. We thus assume we can write:

    22652640b460de6399f0eb85fb5d75b5.png
    where A and B are the average values of the outcomes. Since the possible values of A and B are −1, 0 and +1, it follows that:

    d0823c0637b7ffae47eaa9232f8d0e97.png
    Then, if a, a′, b and b′ are alternative settings for the detectors,

    ba9e0adafbad6d0ef40507804a84790a.png
    ff17109dfef0cfa366ccc63279ce1c5c.png
    94f3fbf512dcca86f509d4b95b3c2aed.png
    Then, applying the triangle inequality to both sides, using (5) and the fact that 03acf6967abe09c58288c5a22e32bf1e.png as well as 8a8e5ce7571912da587c503c796d8446.png are non-negative we obtain

    ad55548f602bef99d7a667d8a5bdec53.png
    64b0c82401c602b83937c84ff35c5674.png
    or, using the fact that the integral of ρ(λ) is 1,

    26a56c0d3c312cf050f75a4e3ade5126.png
    which includes the CHSH inequality.
    --- END QUOTE
    http://en.wikipedia.org/wiki/CHSH_inequality#Derivation_of_the_CHSH_inequality


    1. "We start with the standard assumption of independence of the two sides, enabling us to obtain the joint probabilities of pairs of outcomes by multiplying the separate probabilities..."

    Obtain the joint probability of what particular event?


    2. We see the premise from step (5) applied in step (6) which adds that [itex]\pm 1[/itex] into the equation, but without it, what is the underlying relation between four expectation values the equation describes? Would it be this: [itex]E(a,b) - E(a,b') = E(a,b) * E(a',b') - E(a,b') * E(a',b)[/itex]?


    3. Step after (6), applying the triangle inequality to both sides. What is justification for this?
     
    Last edited by a moderator: Apr 18, 2017
  2. jcsd
  3. Sep 24, 2014 #2
    Step 6.
    There is E(a,b), E(a,b′), E(a′,b) and E(a′,b′), which I will call E1, E2, E3 and E4. Is that what you call A, B, C and D, so that: E1 = E2 + E3 + E4 - E4?
     
  4. Sep 24, 2014 #3

    atyy

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    The term I'm calling ##D## is ##\underline{A}(a,\lambda)\underline{B}(b,\lambda)\underline{A}(a',\lambda)\underline{B}(b',\lambda)##.

    The term in the square brackets in the integrand is:
    ##\underline{A}(a,\lambda)\underline{B}(b,\lambda) - \underline{A}(a,\lambda)\underline{B}(b',\lambda)##
    ##= \underline{A}(a,\lambda)\underline{B}(b,\lambda) - \underline{A}(a,\lambda)\underline{B}(b',\lambda) + \underline{A}(a,\lambda)\underline{B}(b,\lambda)\underline{A}(a',\lambda)\underline{B}(b',\lambda) - \underline{A}(a,\lambda)\underline{B}(b,\lambda)\underline{A}(a',\lambda)\underline{B}(b',\lambda)##
    ##= \underline{A}(a,\lambda)\underline{B}(b,\lambda) - \underline{A}(a,\lambda)\underline{B}(b',\lambda) + \underline{A}(a,\lambda)\underline{B}(b,\lambda)\underline{A}(a',\lambda)\underline{B}(b',\lambda) - \underline{A}(a,\lambda)\underline{B}(b',\lambda)\underline{A}(a',\lambda)\underline{B}(b,\lambda)##
    ##= \underline{A}(a,\lambda)\underline{B}(b,\lambda) + \underline{A}(a,\lambda)\underline{B}(b,\lambda)\underline{A}(a',\lambda)\underline{B}(b',\lambda) - \underline{A}(a,\lambda)\underline{B}(b',\lambda) - \underline{A}(a,\lambda)\underline{B}(b',\lambda)\underline{A}(a',\lambda)\underline{B}(b,\lambda)##
    ##= \underline{A}(a,\lambda)\underline{B}(b,\lambda)[1 + \underline{A}(a',\lambda)\underline{B}(b',\lambda)] - \underline{A}(a,\lambda)\underline{B}(b',\lambda)[1+ \underline{A}(a',\lambda)\underline{B}(b,\lambda)]##
     
    Last edited: Sep 24, 2014
  5. Sep 24, 2014 #4
    So looking just at the equation in the first two lines, and given:

    ##\int \underline{A}(a,\lambda)\underline{B}(b,\lambda) \rho( \lambda ) d\lambda = E(a,b) = E1##
    ##\int \underline{A}(a,\lambda)\underline{B}(b',\lambda) \rho( \lambda ) d\lambda = E(a,b') = E2##
    ##\int \underline{A}(a',\lambda)\underline{B}(b',\lambda) \rho( \lambda ) d\lambda = E(a',b') = E3##
    ##\int \underline{A}(a',\lambda)\underline{B}(b,\lambda) \rho( \lambda ) d\lambda = E(a',b) = E4##

    then in the original terms of expectation values it goes like this:

    ##E1 - E2 = E1-E2 + E1*E3 - E1*E4##

    Correct? The question is where did that come from, according to what premise or mathematical principle is it supposed to be true? What is it E1, E2, E3 and E4 have in common to justify such a statement about their shared relationship? How do you prove that equation is true? Is it supposed to be true only for QM, only classical physics, or both?
     
  6. Sep 25, 2014 #5

    atyy

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    No, that is not right. Don't think about the expectation values, just the term in the square brackets of the integrand.

    ##\underline{A}(a,\lambda)\underline{B}(b,\lambda) = D1##
    ##\underline{A}(a,\lambda)\underline{B}(b',\lambda) = D2##
    ##\underline{A}(a',\lambda)\underline{B}(b',\lambda) = D3##
    ##\underline{A}(a',\lambda)\underline{B}(b,\lambda) = D4##

    Then the original term can be rewritten:

    ##D1 - D2 = D1-D2 + D1*D3 - D1*D3##
     
  7. Sep 25, 2014 #6
    Paraphrasing Alien8, one could ask the same question about D1, D2, D3, and D4:

    The question is where did that come from, according to what premise or mathematical principle is it supposed to be true? What is it D1, D2, D3 and D4 have in common to justify such a statement about their shared relationship? How do you prove that equation is true? Is it supposed to be true only for QM, only classical physics, or both?
     
  8. Sep 25, 2014 #7
    We are back talking about relations between binary states instead of expectation values, even though they do not compare. Just like AB + AB' + A'B - A'B' = -2 or +2, that equation too is true according to pure algebra involving specifically numbers -1 and +1. There seem to be many combinations of arithmetic operations involving four variables with {-1,+1} limit that will yield equality, in which case the choice of that particular expression is simply arbitrary.

    The only thing they have in common is their limit {-1,+1}. But that is sufficient to construct numerous combinations of general algebraic equalities concerning four independent arbitrary variables with such a limit.

    I evaluated the equation several times, each time assigning arbitrarily different -1 or +1 values to D variables. The equation produced 0 = 0, -2 = -2, and 2 = 2 results.

    There seems to be an error in the third line though, which involves all four variables: D1-D2 = D1-D2 + D1*D3 - D2*D4 and it's true only for some combinations.

    It's true for every possible combination of four variables under condition each one can be only -1 or +1. It's general and purely mathematical statement about numbers, it has no more to do with QM than 1 + 1 = 2.
     
    Last edited: Sep 25, 2014
  9. Sep 25, 2014 #8

    atyy

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    Yes, if you assign the values D1,D2,D3,D4 with no relation between them (except that they have the same limits), then you will get an error in the third line. However, D1*D3 is not independent from D2*D4. In fact D1*D3 = D2*D4, because D1*D3 and D2*D4 are made up of the same "A" and "B" terms written in different orders.

    So if you want to assign values between -1 and +1 independently to check the equation, you should assign them to the "A" and "B" terms.
     
  10. Sep 25, 2014 #9
    If you lose the integral you don't have a number any more, but a binary state which is not a subject to arithmetic operations: dead & alive - alive & alive is undefined, it does not compute. That a photon goes one way or the other has no numerical value, it's an event or state. Assigning -1 and +1 labels to a binary state is very peculiar choice because it can obviously be misleading.

    In any case, those D variables do not represent numbers, but four possible binary states or events: ##(++), (--), (+-), (-+)##. We can't do arithmetic with that, we need to count occurrences of many such events in order to work out probabilities and expectation values, and then we get the numbers we can actually do arithmetic with.

    Relation between independently arbitrary binary states is not relevant to expectation values. That two coins can either flip heads or tails has nothing to with how often they will both flip the same side. The derivation never looses the integral, expectation values E1 - E2 always remain on the left hand side of the equation. The equation ought to able to be expressed only in terms of expectation values, which is what the derivation begins and ends up with anyway.
     
  11. Sep 25, 2014 #10

    Avodyne

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    In hidden variable theory, you do have a number that is subject to arithmetic operations. That is the whole point of hidden variable theory.

    If you deny that ##D1## is a definite number (which must be either ##+1## or ##-1##), then you are outside the framework of hidden variable theory, and the Bell inequalities cannot be derived.

    The Bell inequalities apply only to local hidden variable theories, which are defined to be those theories in which ##A(a,\lambda)##, etc., have definite values (either ##+1## or ##-1##) for each value of the hidden variable ##\lambda##.

    The Bell inequalities do not apply to any theory in which ##A(a,\lambda)## does not have a definite value (either ##+1## or ##-1##) for each value of the hidden variable ##\lambda##.

    One such theory is quantum mechanics.
     
  12. Sep 25, 2014 #11
    I don't deny that, I observe that for probabilities and expectation values it is irrelevant whether the four possible events will be labeled ++, --, +-, -+ or 11, 00, 10, 01 or HH, TT, HT, TH, or whatever other binary state label with the Boolean domain. It's probability of those events happening which has a definite decimal range from 0.0 to 1.0, and it's expectation values which have definite decimal range from -1.0 to +1.0.

    By the way, boolean logic operations do not directly translate to numbers arithmetic, and the outcome sample space always has the same boundary as input Boolean domain: {true, false}.

    The Bell inequality I quoted in the OP is not about binary states like: ##\underline{A}(a,\lambda)\underline{B}(b,\lambda)## , but about expectation values such as: ##\int \underline{A}(a,\lambda)\underline{B}(b,\lambda) \rho( \lambda ) d\lambda = E(a,b)##. Some people seems to think relations between binary state events are directly consequential to expectation values, but they are general and only define input domain limit, it's always the same for any theory you want to test the inequality against, and how the inequality will evaluate only depends on expectation value function or "hidden variable".
     
  13. Sep 25, 2014 #12

    Avodyne

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    Sure, but you prove the Bell inequality for ##E(a,b)## by using properties of ##\underline{A}(a,\lambda)##.

    Consider a particular combination of expectation values, ##E(a,b)+E(a,b')+E(a',b)-E(a',b')##. In local hidden variable theory, this combination can be written as

    ##E(a,b)+E(a,b')+E(a',b)-E(a',b')={}##
    ##\int\left[
    \underline{A}(a,\lambda)\underline{B}(b,\lambda)
    +\underline{A}(a,\lambda)\underline{B}(b',\lambda)
    +\underline{A}(a',\lambda)\underline{B}(b,\lambda)
    -\underline{A}(a',\lambda)\underline{B}(b',\lambda)\right]\!\rho(\lambda)d\lambda##

    where ##\underline{A}(a,\lambda)##, etc., are each equal to ##+1## or ##-1##. (This is simply how we are choosing to represent the two binary values. This choice implies ##-1\le E(a,b)\le +1##.)

    Do you agree with this, or not?
     
    Last edited: Sep 25, 2014
  14. Sep 25, 2014 #13
    ##\underline{A}(a,\lambda)## is unknown function with output sample space: {event 1, event 2}. This output is then only a part of the input for expectation value function. Event naming is arbitrary, probability only cares about the count or ratio of their occurrences.

    Expectation value is measure of probabilities between four possible events. Probabilities naturally range from 0.0 to 1.0, so the reason why expectation values range from -1.0 to +1.0 is because: ##E(a,b) = P_{++}(a,b) + P_{--}(a,b) − P_{+-}(a,b) − P_{-+}(a,b)##.
     
  15. Sep 26, 2014 #14

    atyy

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    You can use the concept of a random variable to assign numbers to the outcome.

    http://en.wikipedia.org/wiki/Random_variable
    http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm

    The expectation value of a function of a random variable ##f(x)## is ##E(f(x)) = \int f(x) p(x) dx##. In general expectation values do not range between -1 and 1. However, if ##f(x)## ranges between -1 and 1, then the expectation ##E(f(x)) = \int f(x) p(x) dx## also ranges between -1 and 1.

    http://mathworld.wolfram.com/ExpectationValue.html
     
  16. Sep 26, 2014 #15

    Avodyne

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    OK, then we can write

    [tex]P_{++}(a,b)=\int_{\underline{A}(a,\lambda)=+1,\;\underline{B}(b,\lambda)=+1}\rho(\lambda)d\lambda[/tex]

    and similarly for ##P_{+-}(a,b)##, etc. This notation means that we do the integral only over those values of ##\lambda## for which both ##\underline{A}(a,\lambda)=+1## and ##\underline{B}(b,\lambda)=+1##. Here I am adopting the convention that the two binary results are called ##+1## and ##-1##. This convention gives us a notation that is useful, in the following sense: the outcomes ##++## and ##--## occur if and only if ##\underline{A}(a,\lambda)\underline{B}(b,\lambda)=+1##, and that the outcomes ##+-## and ##-+## occur if and only if ##\underline{A}(a,\lambda)\underline{B}(b,\lambda)=-1##. Therefore

    [tex]P_{++}(a,b)+P_{--}(a,b) = \int_{\underline{A}(a,\lambda)\underline{B}(b,\lambda)=+1}\rho(\lambda)d\lambda[/tex]
    [tex]P_{+-}(a,b)+P_{-+}(a,b) = \int_{\underline{A}(a,\lambda)\underline{B}(b,\lambda)=-1}\rho(\lambda)d\lambda[/tex]

    Now, using ##E(a,b)=P_{++}(a,b)+P_{--}(a,b)-P_{+-}(a,b)-P_{-+}(a,b)##, we have

    [tex]E(a,b)=\int_{\underline{A}(a,\lambda)\underline{B}(b,\lambda)=+1}\rho(\lambda)d\lambda
    -\int_{\underline{A}(a,\lambda)\underline{B}(b,\lambda)=-1}\rho(\lambda)d\lambda[/tex]

    Now comes the magic trick:

    [tex]\int_{\underline{A}(a,\lambda)\underline{B}(b,\lambda)=+1}\rho(\lambda)d\lambda
    -\int_{\underline{A}(a,\lambda)\underline{B}(b,\lambda)=-1}\rho(\lambda)d\lambda
    =\int \underline{A}(a,\lambda)\underline{B}(b,\lambda)\rho(\lambda)d\lambda[/tex]

    You must stare at this until you understand it. It is the key to everything. The point is that the factor of ##\underline{A}(a,\lambda)\underline{B}(b,\lambda)## on the right-hand side takes on the values ##+1## or ##-1## only (because of the convention that we have adopted). When ##\underline{A}(a,\lambda)\underline{B}(b,\lambda)=+1##, we get the first term on the left-hand side, and when ##\underline{A}(a,\lambda)\underline{B}(b,\lambda)=-1##, we get the second term on the left-hand side.

    I'll pause again. I'm hoping that I have now convinced you that, if we adopt the convention that the two binary values are ##+1## and ##-1##, then we can write

    [tex]E(a,b)=\int \underline{A}(a,\lambda)\underline{B}(b,\lambda)\rho(\lambda)d\lambda[/tex]
     
  17. Sep 26, 2014 #16
    Binary state events have no range, it's either one or the other. To calculate probabilities and expectation values it is irrelevant whether detections on the two detectors are marked with -1 and +1, or heads and tails. P(+1 and +1) can mean the same thing as P(heads and heads) if we choose so. The things inside probability function brackets are not numbers, but letters. Numbers arithmetic does not directly translate to logic operations of boolean algebra and probabilities.

    Do you really mean to say if we decided to mark recordings of the two detectors with heads and tails instead of -1 and +1 the expectation value would range from heads to tails instead of from -1.0 to +1.0?
     
    Last edited: Sep 26, 2014
  18. Sep 26, 2014 #17
    Why do you think you can multiply "photon A went left" with "photon B went right"?

    The integral doesn't imply multiplication of the two terms, but pairing, enumeration and counting.
     
  19. Sep 26, 2014 #18

    Nugatory

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    Alien8, you are again starting to argue instead of trying to learn. What you should be taking away from the last few posts is that you will have to learn a bit more probability theory before you'll be ready to work through the CHSH derivation and proof.
     
  20. Sep 26, 2014 #19

    Nugatory

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    You are right that discrete ("binary" is a special case of "discrete") outcomes don't have a range. However, we're talking about the expectation value of the result of series of such measurements, and that does have a range. Indeed, that's how casinos stay in business - every spin of the roulette wheel produces a discrete win-lose result for each bet, but the casino knows the exact expectation value of their winnings over a large number of such events, and it approaches a continuous function as the number of events becomes large.

    In the Bell and CHSH experiments, the "correlation" values that appear in the formulas are all some form of ##(N_1-N_2)/(N_1+N_2)## where ##N_1## and ##N_2## are the number of trials in which both detectors gave the same result and the number of trials in which both detectors give different results. It should be clear that the expectation value of this quantity can take on values betwen -1 and 1, even though each individual trial has a binary result.
     
  21. Sep 26, 2014 #20
    I think the misunderstanding is about how the integral works and the meaning of the term ##\underline{A}(a,\lambda)\underline{B}(b,\lambda)## when separated out of this expression: ##
    E(a,b)=\int \underline{A}(a,\lambda)\underline{B}(b,\lambda)\rho(\lambda)d\lambda##.


    Alain Aspect paper:
    http://arxiv.org/abs/quant-ph/0402001

    Equation (5):
    ##E(a,b) = P_{++}(a,b) + P_{--}(a,b) − P_{+-}(a,b) − P_{-+}(a,b)##

    Equation (28):

    ##E(a,b) = \frac {N_{++}(a,b) + N_{--}(a,b) - N_{+-}(a,b) - N_{-+}(a,b)} {N_{++}(a,b) + N_{--}(a,b) + N_{+-}(a,b) + N_{-+}(a,b)}##


    I'd say these two equations make it pretty clear the two terms ##\underline{A}(a,\lambda)\underline{B}(b,\lambda)## are not multiplied under that integral, but paired and counted. It's not a pair of integers, it's not a pair of probabilities, it's a pair of events. Isn't that true?
     
    Last edited: Sep 26, 2014
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