Foundations of Quantum Mechanics

The main goal of the course is to provide the students with the key theoretical tools to understand quantum mechanics as a probability theory, inherently different from the classical one.
After introducing the statistical formulation of quantum mechanics, we will investigate the most relevant features characterizing such a description. We will first introduce the Bell's inequalities and the related notion of non-locality, and then we will study contextuality, from the Kochen-Specker's theorem to the more recent developments. Moreover, we will study the possibility to detect the quantum nature of a system evolving in time, through the notion of measurement invasiveness and the Leggett-Garg's inequalities.

At the end of the course the student will be able to:
1. Use the mathematical formalism needed to provide a general description of quantum mechanics as a probability theory
2. Derive the Bell's inequality, Tsirelson's inequality and Fine's theorem in the case of two observers measuring two observables each
3. Detect the key role of (non-)locality and (non-)existence of the joint probability distribution in discriminating between the classical and quantum theories of probability
4. Distinguish different notions of non-contextuality and use them to derive the Kochen-Specker's theorem and noncontextuality inequalities
5. Present the main experimental tests of quantum contextuality, along with the most relevant applications
6. Describe and quantify the invasiveness of a quantum measurement, compare it with the notion of contextuality and use it to derive the Leggett-Garg's inequalities
7. Characterize the conditions for classical simulability of multi-time probabilities and discuss the possibility to exploit them for experimental tests of non-classicality