Statistical Structure of Quantum Mechanics

We explore the statistical structure of quantum theory providing a construction of the general notion of state, and by duality of observable, in the quantum setting. This will lead us to consider states as statistical operators, and observables as positive operator-valued measures. The connection of these notions with state vectors and self-adjoint operators introduced in the standard formulation of quantum mechanics is considered, pointing to the novel aspects inherent to this more realistic description of the outcomes of statistical experiments.