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	Serge Fehr	Cryptology Group

Quantum Information Theory (Fall 2021)

• Lecturer: Serge Fehr (serge.fehr@cwi.nl)
• Teaching Assistant: Jelle Don (jelle.don@cwi.nl)

Time and Place

• The course takes place on Thursdays, from 16.15 to 18.00 in Huygens 204.

Course Description

We introduce and study the mathematics of quantum information science, which brings together quantum mechanics and theoretical computer science, with a focus on information theoretic (instead of computational) aspects in this course. In the first few lectures, we discuss the so-called density-operator formalism of quantum mechanics and study the relevant mathematical concepts: density operators, partial trace, purification, CPTP maps, and the Schatten norms. After that, the focus will be on measures of quantum information, i.e., ways to quantify the information content of quantum states. This is done by means of studying a quantum-version of the classical Renyi entropies. We analyze the crucial properties of these information measures and discuss (on some examples) their operational significance. From a technical perspective and to some extent, the course can be seen as a non-commutative extension of classical probability- and information-theory. Along the way, we will touch upon different mathematical concepts, e.g., covering some elements of matrix analysis.

List of topics that are covered: density operators, partial trace, purification, CPTP maps, Choi-Jamiolkowski isomorphism, Kraus and Stinespring representations, Schatten norm, Hoelder inequality, trace distance, (conditional) min-entropy, Renyi divergence, (conditional) Renyi entropy, operator monotonicity and convexity, Lieb-Ando Theorem, data processing inequality, duality of Renyi entropy, monotonicity for classical systems, entropic uncertainty relation, privacy amplification.

Prerequisites: Prior knowledge on quantum information science (e.g. a course on quantum computing) is helpful by not necessary; familiarity with complex numbers and linear algebra is sufficient.

Lecture Notes

• Title page
• Chapter 0: Preliminaries
• Chapter 6: The Density-Operator Formalism
• Chapter 7: Norms and Distance Measures
• Chapter 8: Measures of Quantum Information (Part i)
• Chapter 8: Measures of Quantum Information (Part ii)
• Chapter 9: Applications
• Appendix A: Probability and Information Theory
• Appendix B: (Operator) Monotonicity & Convexity

Exercise Sets and Self Study

• Homework: Exercise 1. Due: September 23. Solutions.
• Homework: Exercise 2. Due: October 14. Solutions.
• Homework: Exercise 3. Due: November 11. Solutions.
• Homework: Exercise 4. Due: December 9. Solutions.

Homework Exercises

There will be 4 to 5 sets of homework exercises; the exact schedule for these exercises will be announced during class. The exercises will not be graded, but the rule is that sufficiently many of the exercises need to be solved sufficiently well in order to be admitted to the final exam. Working in small groups is allowed, but every student has to write - and not copy-paste! - his/her own individual answer sheet. The answer sheet should be handed in by e-mail to the TA (see below) in pdf format, either typeset (e.g. using LaTeX) or neatly hand-written and scanned. Model solutions to the exercises will be handed out.

Examination and Grade

• The final grade for the course will be determined by means of an oral exam. In order to be admitted to the exam, the exercises need to be done sufficiently well (see above). The oral exams take place on January 5 in room 408 and on January 6 in room 412. The final exam schedule has been announced by email. If you believe that you are eligible for the exam but have not been contacted by email about the schedule, please contact the lecturer immediately.

Consultation Hour

• During the semester, the lecturer is available for questions on Thursday mornings (11:00 h to 12:00 h, room 230) on a walk-in basis, and of course during the break and right after the lecture. Alternatively, you can always send an email. The lecturer is also available on Monday January 3, 2022 (11:00 h to 17:00 h, room 230).
• For technical questions regarding the exercises, please email the teaching assistant.

Additional Reading Material

The material treated in the course (and what you need to know in the exam) is covered by (the union of) the lecture notes (above), the exercises and their solutions (above), and the physical lectures. If you would like to read beyond what is treated in the course, I can recommend Mark Wilde's book From Classical to Quantum Shannon Theory, or John Watrous' lecture notes. If you are interested in the quantum computing part of my lecture notes, then you can get them here. NB: It is not expected from you that you look into these additional sources; these are not part of the course.

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