Quantum Statistical Inference

The first part of the course is a mathematical introduction to the probabilistic and statistical structure of quantum theory. This will in particular cover density operators, positive operator-valued measures, and completely positive maps. The second part then applies the formalism and combines it with other techniques, especially from data analysis, in order to address problems of hypothesis testing, parameter estimation, tomography, learning, and predictive inference.

After successful completion of the module, students are able to analyze, describe and design quantum statistical experiments on abstract mathematical grounds. They master in particular the use of data analysis tools that are tailored to infer properties of an underlying quantum system. Moreover, they know and understand the basic operational concepts, opportunities, and limitations of quantum statistical inference.

Analysis 1, Analysis 2, Linear Algebra 1, Linear 2, any course that contains an introduction to Hilbert spaces and linear operators (e.g. MA0003 Analysis 3 or MA3001 Functional Analysis, but much less is required),
any course that introduces basic notions of statistics and probability theory (e.g. MA0009).
Helpful but not required: Introductory course on Quantum Theory or Quantum Information Theory.

The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should animate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress.

blackboard

Lecture notes and further literature will be provided.

The exam will be in written (60 minutes) or oral (25 minutes) form, depending on the number of participants. Students demonstrate that they have gained a deeper knowledge of definitions and main mathematical tools and results concerning the mathematics of quantum statistical inference. The students are expected to be able to derive and explain basic methods, concepts, and properties and to apply them to specific examples.