Quantum mechanics has existed for about 100 years, but especially during the last 10 years subjects such as quantum information theory, quantum computing and quantum cryptology have had a lot of attention. This has made the statistical aspects of quantum theory more relevant than ever before. The purpose of this course is to give a careful introduction to the mathematical/statistical foundation of quantum theory in such a way that some of the more remarkable new results can be described in a mathematically correct way. To avoid technical complications many of the results will only be proved in their finite dimensional versions.

• State
• Representation of a measurement
• Group representations
• Estimations of parameters and Cramer-Rao inequalities
• Tensor product
• Bayesian networks, causality and Bell's inequalities
• Quantum computers
• Shor's algorithm
• Bits, qubits and data compression
• Error correction
• Completely positive maps
• Capasity(ies) of quantum channels
• Teleportation
• Entanglement
• Quantum cryptology
• Quantum theory and thermodynamics

The course is intended for graduate students in mathematics, physics, statistics and computer science. Elementary knowledge about probability and Hilbert spaces is necessary corresponding to the courses 2SS and 2AN. No knowledge about physics is assumed.

Lectures Tuesdays and Thursdays 13.15-15.00 in Aud. 9 at H. C. Ørstedsinstituttet. First lecture 4/2. Last lecture 22/5.

Notes are produced during the course. Latest version can be found here as PostScript.

Week 6-9. Simple quantum systems

    4/2 Introduction and definition of a state. Convexity.

    6/2 Convexity. Separation Theorem and Caratheodory's Theorem.

    11/2 Group representations on convex sets. Spin.

    13/2 Matrix algebras and spectral theory.

    18/2 Matrix algebras and spectral theory. The group algebra.

    20/2 States and measurement in matrix algebras.

    25/2 Group representations on Hilbert spaces. Stone's Theorem.

    27/2 Covariant measurements, estimation and Cramér-Rao inequality.

Week 10-14. Composed quantum systems

Week 15-21. Quantum computing, information and cryptology

To get point one should do the exercises. The format is PostScript. 11/2

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