Mathematical Foundations of Quantum Information

Mathematical Foundations of Quantum Information (WiSe 26/27)

Quantum Information Theory is a rapidly growing field of research at the intersection of mathematics, computer science and physics. While its origins are closely connected to quantum mechanics, many of its central concepts can be studied from a rigorous mathematical perspective. This makes the field accessible from different backgrounds and opens the door to a wide range of applications, from quantum computing and quantum communication to quantum-inspired methods that can already be implemented on classical computers.

This lecture provides a mathematical introduction to Quantum Information Theory. The aim is to develop the fundamental language and structures needed to understand quantum information, with an emphasis on clear definitions, precise formulations and basic examples. Starting from the mathematical framework underlying quantum systems, we will gradually build towards central notions such as quantum states, transformations, correlations and information-theoretic quantities. Towards the end of the course, we will also indicate how these concepts enter quantum algorithms and related topics in quantum information. 
No prior knowledge of physics is required. This lecture is part of the curriculum in the new Master's programme “Quanteninformationstheorie”.

More specifically, the lecture will cover:

• Repetition of Complex numbers, Hilbert spaces, linear operators on Hilbert spaces, tensor products
• Basics of Quantum Computing
• Non-cloning theorem
• Quantum teleportation
• Partial traces and measurements
• Quantum channels
• Basics of Quantum algorithms
• Optional: Concepts of entropy

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Lecture and Exercise Sessions

Lecturer	Prof. Dr. Moritz Weber (Room 310 in E2 4)
Assistant	Jonas Metzinger (Room 430 in E2 4)
Lecture time	Tuesdays 10-12 and Thursdays 10-12 in TBA
Exercise sessions	To be announced, weekly

The course will be taught in English unless all participants speak German.

For any questions regarding the course, please contact Jonas Metzinger.

Prerequisites

No prior knowledge of physics is required. The lecture is primarily designed for Bachelor’s and Master’s students in mathematics or computer science. Students from physics and related fields are equally welcome. Participants are expected to be familiar with one- and multidimensional analysis and linear algebra as covered in Mathematics for Computer Scientists 1–3 or equivalent the courses Analysis 1&2 and Linear Algebra 1.
 

Exam and admission requirements

The course is worth 9 CP. The examination will be either written or oral; the format will be announced during the semester.
To be admitted to the examination, students must obtain at least 50% of the total points available on the exercise sheets.

Literature

• Nielsen, Michael A. and Chuang, Isaac. Quantum Computation and Quantum Information, 2010.
• Rieffel, Eleanor and Polak, Wolfgang. Quantum Computing: A Gentle Introduction, 2014.
• Watrous, John. The Theory of Quantum Information, 2018.
• Watrous, John. Understanding Quantum Information and Computation, with lectures on Youtube, 2025
• de Wolf, Ronald. Quantum Computing: Lecture Notes, 2023 arXiV
• Wilde, Mark. Quantum Information Theory, 2017.
• Lenze, Burkhard. Mathematik und Quantum Computing, 2018

See also in the Semesterapparat.