# Quantum Information Theory (Spring 2020)

**Lecturers:** Michael Walter and Maris Ozols

**Teaching assistants:** TBD

**Schedule:** Mon 14:00-16:45, Spring 2020

**Further information:** MasterMath, DataNose, Studiegids

See here for last year’s course material.

## 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

## Rules about Homework and Exam

The final grade will be determined 60% by the final written exam (or the re-sit, if you take it) and 40% by the homework grade. In addition, your grade for the exam should be at least 5.0 in order to pass the course.

There will be one homework problem set per week, posted on the course homepage by Monday, and you should submit your completed assignment before the lecture the week after (either in class or by email). Assignments will be accepted late only if you have extenuating circumstances (such as sickness or family emergency) and provided you confirm with the lecturer before the deadline. Your lowest two scores on the problem sets will be ignored (this includes any problem set you did not submit).

You are allowed to bring one self-prepared “cheat sheet” to the exam (A4 paper, hand-written, you can use both sides).

## Literature

Lecture notes (hand-written or LaTeX’ed) and video recordings will be provided for each lecture.

Additional literature that may be useful:

- last year’s course material
- John Watrous, Theory of Quantum Information, lectures notes and book.
- Mark M. Wilde, Quantum Information Theory, Cambridge University Press (2013)
- John Preskill’s lecture notes, Chapters 1–5 and 10
- Michael A. Nielsen, Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2010)
- Fernando G.S.L. Brandao, Matthias Christandl, Aram W. Harrow, Michael Walter, The Mathematics of Entanglement
- Roger A. Horn, Charles R. Johnson, Matrix Analysis, Cambridge University Press (2012)
- Lectures notes of the UvA course Introduction to Information Theory (2019)
- Lectures notes of the UvA course Symmetry and Quantum Information (2018)

## FAQ

*Is this the first time this class is offered?*

This is the second time this class is offered at MasterMath. See here for last year’s course material. In 2018, Michael Walter offered a related course on Symmetry and Quantum Information.

*Can this course only be taken by students enrolled in the Master’s of Mathematics?*

No! You should be able to take the course as part of the “free choice” component of your Master’s program (in which case you can get credit) or as an additional course (which should show up on your Master’s diploma as well).

Moreover, auditors are very welcome, too, so feel free to drop by to watch a particular lecture if there’s a topic that interests you particularly.

*Where can I learn more advanced material?*

Attend the QuSoft seminar and write a Master’s thesis at QuSoft!