AA 2024-25, Francesco Buscemi (Nagoya University, Department of Mathematical Informatics)

For problems that lend themselves to 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 formalisation of this scenario. The aim of the course is to present the basic ideas of the classical theory, including the theory of majorization and the Blackwell-Sherman-Stein theorem, their generalisations to the case of non-commuting operators, and their applications to several cutting-edge research problems in quantum information theory and quantum statistical mechanics.

Prerequisites: basic knowledge of linear algebra and functional analysis, an introductory course on quantum mechanics.

The course consists of nine lectures of three hours each. We begin by reviewing the theory of majorization for probability distributions, generalizing it to the case of statistical models, for which we prove the Blackwell-Sherman-Stein theorem. We then consider the special case of statistical dichotomies and tackle the problem of relative majorization. We will then look at applications in classical statistics. We will then move on to the quantum case and introduce/review the basics of quantum theory and quantum estimation, in particular quantum states, quantum statistical models, quantum measurements and quantum channels. We will then define the idea of statistical sufficiency, quantum statistical morphisms, and prove the quantum version of Blackwell’s Theorem. The course will end with some applications of the theory of quantum statistical comparison in quantum physics and quantum information theory.

The course is organised as follows.

• Majorization of discrete probability distributions
• Statistical models and the Blackwell-Sherman-Stein theorem
• Statistical comparison of dichotomies and relative majorization
• Applications to channels comparison and statistical mechanics
• Introduction to quantum statistical estimation: quantum statistical models, quantum measurements, and quantum channels
• Quantum statistical sufficiency and quantum statistical morphisms
• The “Quantum Blackwell Theorem”
• Applications to quantum statistical mechanics and open quantum systems dynamics
• Applications to quantum information theory

1. R. Bhatia, Matrix Analysis, Springer (1997)
2. A. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, Edizioni della Normale (2011)
3. F. Buscemi, Comparison of quantum statistical models, Comm. Math. Phys. 310, 625 (2012)