What is the significance of the Gleason theorem in Quantum Logic?

[thankqwerty]

Hi all, I'm doing a modcule on Quantum Logic and there is this Gleason theorem but the lecturer didnt explain it clear enough, can somebody help me out please?

sorry that i really have no idea what it is about, all I've got is a heading "gleason theorem" in my note, then it started going on talking about logic of all hilbert subspaces... then go to introduce pure quantum states and convex combination of pure states...

thank you

 

[selfAdjoint]

I think you will find what you want in this paper. I found it by googling on gleason's theorem.

 

[slyboy]

Gleason's theorem

Gleason's theorem is a derivation of the quantum probability rule from the structure of observables in quantum theory. It has two assumptions:

1. Assume that observable quantities are represented by Hermitian observables and that the possible outcomes are represented by the projectors in the spectral decomposition of such operators.

2. Assume that the probability is a function of the projectors only, i.e. it does not depend on which observable the projector came from. This is called non-contextuality.

Conclusion: There is a density operator representing the quantum state, with measurement probabilities given by the standard probability rule.

The theorem holds in Hilbert sapces of dimension 3 or larger, so interestingly it does not hold for the spin of a spin 1/2 particle. However, a POVM version of the theorem has been proved which does hold for these cases.

 

[thankqwerty]

thank you very much
that's very helpful