Foundations of Quantum Mechanics
A.Y. 2024/2025
Learning objectives
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.
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.
Expected learning outcomes
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
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
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
1. Quantum mechanics as statistical theory
· Definition of statistical theory: states, measurements, probabilities and joint measurements
· Classical statistical theory: phase space, random variables, fuzzy measurements and Kolmogorov theorem
· The statistical model of quantum mechanics: statistical operators and positive operator-valued measurements (POVM)
· A first comparison among quantum and classical statistical theories: dispersion free states, Gleason's theorem and compatible and incompatible POVMs
2. Bell's inequalities
· EPR paradox: bipartire systems in quantum mechanics and EPR paradox for two 1/2 spins
· The hypotheses and the derivation of the CHSH Bell's inequality
· Violation of the inequalities in quantum mechanics and Tsirelson's bound
· Experimental tests of the violation of the inequalities
· Toward contextuality: Fine's theorem
· Bell's polytope: geometric structure of classical statistical theories
· Hidden-variable models (introduction)
3. Quantum mechanics as a contextual theory
· Contextuality as logical impossibility (and its representation via graphs)
· Kochen-Specker's theorem: original formulation and proof
· Examples in simpler scenarios
· Recent developments on contextuality: noncontextuality inequalities and operational definitions
· Comparison among the two definitions of contextuality
· Contextuality as quantum resource: experimental tests and and example applications
· Sequential measurements: invasiveness of measurements and contextuality in time
· Derivation of Leggett-Garg's inequalities
· Classical simulability of multi-time probabilities
· Definition of statistical theory: states, measurements, probabilities and joint measurements
· Classical statistical theory: phase space, random variables, fuzzy measurements and Kolmogorov theorem
· The statistical model of quantum mechanics: statistical operators and positive operator-valued measurements (POVM)
· A first comparison among quantum and classical statistical theories: dispersion free states, Gleason's theorem and compatible and incompatible POVMs
2. Bell's inequalities
· EPR paradox: bipartire systems in quantum mechanics and EPR paradox for two 1/2 spins
· The hypotheses and the derivation of the CHSH Bell's inequality
· Violation of the inequalities in quantum mechanics and Tsirelson's bound
· Experimental tests of the violation of the inequalities
· Toward contextuality: Fine's theorem
· Bell's polytope: geometric structure of classical statistical theories
· Hidden-variable models (introduction)
3. Quantum mechanics as a contextual theory
· Contextuality as logical impossibility (and its representation via graphs)
· Kochen-Specker's theorem: original formulation and proof
· Examples in simpler scenarios
· Recent developments on contextuality: noncontextuality inequalities and operational definitions
· Comparison among the two definitions of contextuality
· Contextuality as quantum resource: experimental tests and and example applications
· Sequential measurements: invasiveness of measurements and contextuality in time
· Derivation of Leggett-Garg's inequalities
· Classical simulability of multi-time probabilities
Prerequisites for admission
The course is structured to be self-consistent and assumes that students have the basics notions of the classical theory of probability and quantum mechanics.
Teaching methods
The adopted didactic method is based on lectures with the use of the blackboard, including both theoretical explanations and exercises.
Teaching Resources
· Lecture notes (available on the online platform Ariel)
· T. Heinosaari e M. Ziman, The Mathematical Language of Quantum Theory, Cambridge, 2012
· Papers and supplementary material on Ariel
· T. Heinosaari e M. Ziman, The Mathematical Language of Quantum Theory, Cambridge, 2012
· Papers and supplementary material on Ariel
Assessment methods and Criteria
The exam consists of an oral interview (lasting from 45 to 75 minutes) in which both the knowledge acquired during the lectures and the ability to deal with problems related to the topics discussed in class will be assessed.
FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 3
FIS/03 - PHYSICS OF MATTER - University credits: 3
FIS/03 - PHYSICS OF MATTER - University credits: 3
Lessons: 42 hours
Professor:
Smirne Andrea
Professor(s)
Reception:
On appointment (also remotely on Zoom, if needed)
5th floor, building LITA room A/5/C4