(in Polish) Grafy kwantowe

mathematics
physics

elective monographs

Functional Analysis 1000-135AF

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.

Classroom

The course is dedicated to students interested in functional analysis in a broad sense, mathematical formalism of quantum mechanics or modern, yet elementary, mathematics.

Quantum graphs arise in quantum information and are natural counterparts of graphs. We will start with a quick introduction to quantum information theory, necessary to motivate the notion of a quantum graph. Afterwards we will introduce three equivalent definitions of quantum graphs, we will explain how to translate between them and will illustrate how each of them is useful in its own way. We will show how to construct plenty of examples of quantum graphs. In the last part of the course we will study properties of random quantum graphs: we will prove that a typical quantum graph does not admit any nontrivial symmetries.

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.

- 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.

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.

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.