### Quantum Information Theory - M1 - 8EC

Course description

With the birth of Quantum Mechanics a century ago, our understanding of the physical world has profoundly expanded, and so has our understanding of information. While a classical bit assumes only discrete values, represented by the binary values zero and one, a quantum-mechanical bit or “qubit” can assume a continuum of intermediate states. Quantum Information Theory studies the remarkable properties of this new type of information, ways of processing it, as well as its advantages and limitations.

This course offers a mathematical introduction to Quantum Information Theory. We will start with the fundamentals (such as quantum states, measurements, and entropy) and then discuss some more advanced topics (entanglement theory and quantum communication) and techniques (semidefinite programming and representation theory).

This course complements Ronald de Wolf’s course on Quantum Computing. Neither course requires the other, but students interested in writing a thesis in quantum information/computing are encouraged to follow both courses.

Prerequisites

Familiarity with basic linear algebra and probability theory. Concretely, you should be familiar with the majority of the material in Sections 1.1 and 1.2.2 of this textbook. We are happy to remind you of the more difficult bits in class (but please let us know before the term starts). In addition, some mathematical maturity is required. Concretely, you should have some experience writing down correct and complete mathematical proofs. Some of the homework problems will require programming. You can use the programming language of your choice; examples and solutions will be given in Python.

Prior exposure to the formalism of quantum mechanics or information theory can be helpful, but is not necessary.

Tentative Syllabus

• Lecture 1: Introduction, formalism of quantum information theory
• Lecture 2: Reduced states, purifications, fidelity
• Lecture 3: Quantum channels
• Lecture 4: Measurements
• Lecture 5: Shannon entropy and data compression
• Lecture 6: From classical to quantum data compression
• Lecture 7: Entropy and subsystems
• Lecture 8: Holevo bound and relative entropy
• Lecture 9: Entanglement
• Lecture 10: Separable maps and LOCC
• Lecture 11: Majorization and Nielsen’s theorem
• Lecture 12: Distillable entanglement and entanglement cost
• Lecture 13: Monogamy of entanglement
• Lecture 14: Quantum state merging
• Lecture 15: Semidefinite programming
• Lecture 16: Completely bounded trace norm

Lecturer

Michael Walter (UvA), Maris Ozols (UvA/ILLC)