(in Polish) Grafy kwantowe 1000-1M23GK

1. Introduction: finite dimensional C*-algebra, tensor products, completely positive maps.
2. Basic notions of quantum information theory: quantum channels, Kraus decomposition, Stinespring's theorem, channel capacity.
3. First approach to quantum graphs: operator systems, quantum Lovász function.
4. Classical graphs as quantum graphs: quantum invariants.
5. Quantum adjacency matrix: Schur product, Choi matrix of a completely positive map, the degree matrix of a quantum graph.
6. Random quantum graphs: construction of the random model, properties of the quantum adjacency matrix.
7. Symmetries of quantum graphs: symmetries of operator systems and of the quantum adjacency matrix, triviality of the automorphism group of a typical quantum graph.

Main fields of studies for MISMaP

mathematics
physics

Type of course

elective monographs

Mode

Classroom

Prerequisites

Functional Analysis

Prerequisites (description)

The course will be devoted to matricial versions of graphs, so the only formal prerequisite is a solid knowledge of linear algebra. Nonetheless, operator theory on Hilbert spaces can be useful, even though we will only deal with the finite dimensional case.

Course coordinators

Mateusz Wasilewski

Learning outcomes

The student:
1. Understands basic notions from quantum information theory.
2. Knows different definitions of a quantum graph and understands the relationship between them.
3. Sees the need for using various approaches to the theory of quantum graphs.
4. Has the knowledge of the field sufficient for carrying out their own research.

Assessment criteria

The course will end with a written final exam, which will determine their preliminary grade. Students interested in improving their grade will be asked to participate in the oral exam.

Bibliography

- Duan, Runyao; Severini, Simone; Winter, Andreas Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans. Inform. Theory 59 (2013), no. 2, 1164–1174.
- Weaver, Nik Quantum relations. Mem. Amer. Math. Soc. 215 (2012), no. 1010, v–vi, 81–140.
- Musto, Benjamin; Reutter, David; Verdon, Dominic A compositional approach to quantum functions. J. Math. Phys. 59 (2018), no. 8, 081706, 42 pp.
- Ortiz, Carlos M.; Paulsen, Vern I. Lovász theta type norms and operator systems. Linear Algebra Appl. 477 (2015), 128–147.
- Chirvasitu, Alexandru; Wasilewski, Mateusz Random quantum graphs. Trans. Amer. Math. Soc. 375 (2022), no. 5, 3061–3087.

Additional information

Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:

Master's degree, second cycle programme, Mathematics

Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system:

Description of 1000-1M23GK in USOSweb