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

[greypilgrim]

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?

 

[bhobba]

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?

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

 

[atyy]

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

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)?

 

[bhobba]

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)?

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

And no I can't recall the counter example.

Thanks
Bill

 

[Heinera]

http://ncatlab.org/nlab/show/Gleason's+theorem#counterexample_for_dimension_two