The Theory of Statistical Comparison: From Majorization to the Quantum Blackwell Theorem and Beyond

For problems that allow statistical interpretation, such as parameter estimation or information transmission, whenever different strategies are possible, it is natural to compare them on the basis of their statistical performance with respect to the problem at hand. The theory of statistical comparison provides a rigorous formalization of this scenario. In this colloquium, I will introduce the basic ideas of the classical theory, including the theory of majorization and the Blackwell-Shermal-Stein theorem, present their extensions to the quantum setting, and discuss applications in quantum measurement theory and quantum statistical mechanics.