Lectures on General Probabilistic Theories: From Introduction to Research Participation

(The deadline of the registration is on Sep 30.)

100 years have passed since quantum mechanics was born. The mathematical model has been describing the physical world remarkably well. However, the foundations of this model still remain unclear. A comprehensive understanding of quantum theory, including its foundations, is becoming even more important in an era where the demands of realizing quantum information technologies pose significant theoretical and experimental challenges.

The framework of General Probabilistic Theories (GPTs) is a modern approach to the foundations of quantum theory. It deals with mathematical generalizations of both classical and quantum theories and has attracted increasing attention in recent years. Roughly speaking, research on GPTs has three major objectives: characterizing the models of classical and quantum theories, investigating the fundamental limits of physical and information-theoretic properties arising from operational requirements, and deepening our understanding of the mathematical structures underlying classical and quantum theories. The studies of GPTs have provided many new perspectives on these topics. However, at the same time, there remain many important open problems in the field. For this reason, more researchers are encouraged to enter and contribute to research on GPTs.

This intensive three-day lecture series is designed to provide researchers and graduate students with the essential knowledge necessary for research on GPTs, starting from an introduction to the subject. The lectures will cover the mathematical foundations, physical and information-theoretic concepts, and both the established results and future directions of GPT research. The 1st day will present the necessary mathematical structures, including convex geometry, positive cones, and the operational formulation of probabilistic models. The 2nd day will explore composite systems, information-theoretic quantities, symmetries, and Euclidean Jordan algebras. The 3rd day will survey key results on discrimination and communication tasks, the characterization of classical and quantum theories, and open problems that connect GPTs to quantum information science and beyond.

Note: The content of each lecture may extend into the next slot or be covered earlier, depending on the pace of discussion and participant questions.

The 1st day (6th Oct.): Mathematical Introduction to GPTs
Venue: Large Meeting Room, 2F, Wako Welfare & Conference Building
10:30-12:00 Lecture 1 (Introduction and Mathematics on Positive Cones)
12:00-13:30 Lunch time
13:30-15:00 Lecture 2 (Mathematics on Positive Cones)
15:00-15:30 Coffee break
15:30-17:00 Lecture 3 (Introduction to General Models and Relation between Operational Probability Theories)

The 2nd day (7th Oct.): Physical and Information Theoretical Concepts in GPTs
Venue: Large Meeting Room, 2F, Wako Welfare & Conference Building
10:30-12:00 Lecture 4 (Composite Systems in GPTs)
12:00-13:30 Lunch time
13:30-15:00 Lecture 5 (Information Quantities)
15:00-15:30 Coffee break
15:30-17:00 Lecture 6 (Dynamics, Symmetry, and Euclidean Jordan Algebras)

The 3rd day (8th Oct): Previous and Future Studies in GPTs
Venue: Meeting Room 435-437, 4F, Wako Main Research Building
10:30-12:00 Lecture 7 (Discrimination and Communication Tasks)
12:00-13:30 Lunch time
13:30-15:00 Lecture 8 (Characterization of Classical and Quantum Theories)
15:00-15:30 Coffee break
15:30-17:00 Lecture 9 (Other Topics, Open Problems, and Future Directions)
18:00- Dinner

The day of no lecture (9th Oct): Open Discussion and Q&A
Research discussions will take place between the lecturer and participants in areas such as the hallways on the 3rd and 4th floors of the Main Research Bldg, RIKEN Wako Campus.