Comparison of quantum statistical models: a "Quantum Blackwell Theorem"

A theorem by Blackwell, providing criteria for one statistical experiment being more informative than another, plays a central role in the theory of comparison of statistical experiments in classical decision theory. In this paper we extend some of these ideas by constructing a general framework for the comparison of (discrete and finite-dimensional) statistical models, valid in both classical and quantum decision theory. We do this by introducing the concept of morphism between state spaces, which is analogous to the classical notion of stochastic transformation and generalizes the quantum notion of completely positive trace-preserving map. The main result is a comparison theorem that strengthens and unifies results that previously were independent, like Blackwell's theorem and its analogues for quantum states and quantum channels recently proved by Shmaya and Chefles, respectively.