Course content

Operator Algebraic methods for Quantum Information explores the mathematical foundations of modern quantum technologies.The course introduces C*-algebras, von Neumann algebras, operator systems and completely positive maps as the core language for quantum communication, quantum computation, and entanglement theory. Students will learn how abstract functional analysis becomes a powerful tool for understanding and designing quantum information processes.

Learning outcome

1. Knowledge

The course provides a rigorous introduction to the theory of C*-algebras, von Neumann algebras, and operator spaces/systems with emphasis on completely positive maps, quantum channels, noncommutative probability, and mathematical foundations of quantum information theory. The concepts will be illustrated through examples from quantum communication, error correction, correlations, and entanglement theory.

2. Skills

Students will acquire strong analytical tools from operator algebra theory and be able to:

• work mathematically with quantum channels, completely positive maps, Stinespring dilations, and tensor product structures,
• apply operator algebra methods to central problems in quantum information, quantum communication, and quantum computation,
• use operator systems and operator spaces in the study of nonlocality, Bell inequalities, and entanglement structures.

3. Competence

Students will be prepared to:

• engage in research-level discussions in operator algebras and quantum information theory,
• contribute scientifically to problems at the interface between functional analysis, operator algebras, and quantum technologies,
• participate in international research communities focusing on mathematically grounded quantum information.

Learning methods and activities

The course will be taught as needed. If there are few PhD students, the course is only given as a guided self-study.

Recommended previous knowledge

TMA4230 Functional Analysis or equivalent.

Course materials

Information will be given at the first meeting.