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

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Discussion Overview

The discussion centers on the implications of Gleason's theorem in two-dimensional Hilbert spaces and its relationship to Born's rule. Participants explore whether violations of Born's rule are possible in such systems and consider the validity of counterexamples to Gleason's theorem.

Discussion Character

  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants propose that Gleason's theorem fails in two dimensions, raising the question of whether this allows for violations of Born's rule.
  • Others argue that a new version of Gleason's theorem, based on Positive Operator-Valued Measures (POVMs), still holds in two dimensions, suggesting that the assumptions are slightly stronger and potentially more intuitive.
  • A participant questions whether a counterexample can be produced if the assumptions of the original Gleason's theorem are weakened, indicating a desire to explore the nature of the theorem's failure in 2D.
  • Another participant asserts that a counterexample exists demonstrating that the original form of Gleason's theorem does not hold in two dimensions, although they cannot recall the specifics of the counterexample.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of Gleason's theorem in two dimensions and its implications for Born's rule. There is no consensus on whether violations of Born's rule are possible or on the existence of a specific counterexample to the original theorem.

Contextual Notes

The discussion highlights the complexity of the assumptions underlying Gleason's theorem and the potential for varying interpretations of its implications in lower-dimensional systems. Specific counterexamples and their details remain unresolved.

greypilgrim
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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?
 
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greypilgrim said:
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
 
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bhobba said:
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)?
 
atyy said:
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
 
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