nLab
quasi-state

Idea

In quantum mechanics, a qausi-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 $A=B(H)$ for $\dim H>2$, then all quasi-states are in fact 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 $A$. 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.

Related concepts

• state, state on an operator algebra