Quantum states and generalized observables: a simple proof of Gleason's theorem

Phys Rev Lett. 2003 Sep 19;91(12):120403. doi: 10.1103/PhysRevLett.91.120403. Epub 2003 Sep 19.

Abstract

A quantum state can be understood in a loose sense as a map that assigns a value to every observable. Formalizing this characterization of states in terms of generalized probability distributions on the set of effects, we obtain a simple proof of the result, analogous to Gleason's theorem, that any quantum state is given by a density operator. As a corollary we obtain a von Neumann-type argument against noncontextual hidden variables. It follows that on an individual interpretation of quantum mechanics the values of effects are appropriately understood as propensities.