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Possibility of violations of Born's rule in two dimensions?

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  1. Sep 20, 2015 #1
    Hi,

    Gleason's theorem fails if the dimension of the Hilbert space is two. Does this allow for violations of Born's rule in two-dimensional systems? Or can you somehow tensor the system with the (ever-present and infinite-dimensional) Hilbert space of the rest of the universe, apply Gleason's theorem and reduce to the system again to find Born's rule in the original 2D system?

    Have experiments been conducted to check for violations of Born's rule in 2D systems?
     
  2. jcsd
  3. Sep 21, 2015 #2

    bhobba

    Staff: Mentor

    No.

    The new version of Gleason based on POVM's rather than resolutions of the identity works in two dimensions - the assumption is just slightly stronger - in fact some would say more intuitive because its doesn't have the positive operators as disjoint which seems a bit unnatural when you think about it.

    Thanks
    Bill
     
  4. Sep 21, 2015 #3

    atyy

    User Avatar
    Science Advisor

    But if we weaken the assumption to the original Gleason's theorem, can a counterexample be produced showing that the theorem truly fails in 2D (as opposed to a proof not having yet been found)?
     
  5. Sep 21, 2015 #4

    bhobba

    Staff: Mentor

    Gleason fails in 2D in its original form because a counterexample exists showing its not true.

    And no I cant recall the counter example.

    Thanks
    Bill
     
  6. Sep 21, 2015 #5
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