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Bell's inequality to CHSH

  1. Oct 13, 2007 #1
    1. The problem statement, all variables and given/known data

    I need to get the CHSH inequality from Bell's inequality

    2. Relevant equations

    |C(a, b) - C(a, b')| + |C(a', b) - C(a', b')| <= 2

    to

    -2 <= C(a, b) - C(a, b') + C(a', b) + C(a', b') <= 2


    3. The attempt at a solution

    I know the CHSH allows for no correlation of 0 but I can't get anywhere. I think my math with inequalities just sucks.
     
  2. jcsd
  3. Jan 15, 2008 #2
    The above looks a bit confused. How about this trick?

    (1) (a + a')b + (a - a')b' = +/-2;

    since (a + a') = 0 AND (a - a') = +/-2, XOR (a + a') = +/-2 AND (a - a') = 0.

    So, expanding (1):

    (2) ab + a'b + ab' - a'b' = +/-2.

    Then, ensemble-averaging:

    (3) <ab> + <a'b> + <ab'> - <a'b'> </= +/-2.

    Or:

    (4) |<ab> + <a'b> + <ab'> - <a'b'>| </= 2;

    where <ab> = C(ab), etc. (4) is known as the CHSH Inequality; and the real trick is to see why EPR-Bell tests cannot (in general) satisfy (4).
     
  4. Mar 31, 2010 #3
    Very nice answer, sir. Now I'd like to ask what you lead to: why cannot the EPR state satisfy the Bell inequality generally?
     
  5. Apr 4, 2010 #4
    Good question, but it's a homework question at our school. So please follow the PF homework rules.
     
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