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

While therefore the condition on quasi-states is much weaker than that for states, Gleason's theorem asserts that if A=B(H)A = B(H) for dimH>2dim H \gt 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 AA – 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 (