# Contents

## Idea

In quantum mechanics, a quasi-state on an algebra of observables $A$ is a function $\rho :A\to ℂ$ that is required to satisfy the axioms of a genuine state (linearity and positivity) only on the poset of commutative subalgebras of $A$.

While therefore the condition on quasi-states is much weaker than that for states, Gleason's theorem asserts that if $A=B\left(H\right)$ for $\mathrm{dim}H>2$, then all quasi-states are in fact already genuine quantum states.

Notice that a quasi-state is naturally regarded as an actual state, but internal to the ringed topos over the poset of commutative subalgebras of $A$ – the “Bohr topos”. Therefore Gleason's theorem is one of the motivations for regarding this ringed topos as the quantum phase space (“Bohrification”.) The other is the Kochen-Specker theorem.

Revised on October 3, 2013 00:33:36 by Urs Schreiber (82.169.114.243)