Spring Semester 2026, Quantum Information Theory
(Math 9101)


Instructor: Dietmar Bisch
Lecture: WF, 9:20am-10:35am, SC 1312
Office: SC 1405, (615) 322-1999
Office hours: W 10:40-11:30am, F 1:00-1:50pm and by appointment. These times are tentative.
Mailbox: SC 1326

Prerequisites: Linear algebra, graduate real analysis and basic functional analysis. Some knowledge of quantum mechanics would be helpful, but I will spend a few lectures reviewing what we need.

Recommended Literature: There will be no textbook. The following books contain part of what I plan to cover in the course:

1) Michael Nielsen, Isaac Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2011 (first edition: 2000).
2) A. Kitaev, A.H. Shen, M.N. Vyali, Classical and quantum computation , American Mathematical Society 2002.
3) Zhenghan Wang, Topological Quantum Computation, American Mathematical Society 2010.
4) Mark Wilde, Quantum Information Theory, Cambridge University Press; 2nd edition (February 6, 2017).
5) Klaas Landsman, Foundations of Quantum Theory: From Classical Concepts to Operator Algebras, Springer 2017.
6) David J. Griffith, Introduction to Quantum Mechanics, Prentice Hall 1995.
7) Vern Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2003.

Further references to various (online) LN and research articles will be given throughout the course. I plan to post them on this website as well.

Syllabus:
I will discuss in this course some of the fundamental aspects of quantum computing and quantum information theory, that is the theory of information processing using a quantum physical system.

After a short introduction to quantum mechanics (Stern-Gerlach experiment, observables, pure states, mixed states, density matrices, quantum measurements, Heisenberg uncertainty relation, and all that), I plan to discuss the notion of entanglement of quantum systems. Entanglement is a feature of quantum mechanics, which does not exist in classical physics. Einstein called it the ``spooky action at a distance''. It expresses a correlation of subsystems of a quantum physical system which appears naturally as soon as the commutative algebras of functions in classical physics are replaced by non-commutative algebras of operators (matrices) in quantum physics. If correlated quantum systems contain ``enough'' entanglement, they can be used to transmit quantum information on a classical channel, despite the no-cloning theorem. This surprising feature leads to quantum teleportation, which has been realized experimentally.

Mathematically, noncommutative structures described by operator theory, the theory of operator algebras and the theory of completely positive maps are at the heart of these phenomena. I will try to emphasize the connections to operator algebras, in particular to the theory of von Neumann algebras and subfactor theory, throughout the course, if applicable.

The course will cover the basics of quantum computation and quantum algorithms. I plan to describe the Deutsch-Josza algorithm and Shor's famous factoring algorithm, which can be used to factor integers on a (hypothetical) quantum computer in polynomial time. Shor's discovery was a huge surprise, since the best known factoring algorithms on a classical computer to date are exponential in the number of digits of the integer to be factored. Quantum error correction is another important chapter, and I plan to present the Shor code and the Knill-Laflamme model for error correction. If there is time, I plan to discuss the use of bi-unitary matrices from subfactor theory to construct error correcting codes.

This course will concentrate on the theoretical aspects of quantum computing and quantum information theory. For instance, I will most likely not have time to discuss the numerous ideas currently on the market for actually building a quantum computer. If I have time, I will cover some ideas of topological quantum computing (anyons, modular tensor categories, quantum spin chains etc.).

Grading: There will be no exams. I will assign exercises during the lectures, but they will be optional. The course grade will be based on attendance and team projects. Details for the projects will be provided as soon as I have some idea what interests students have. Each team will work out a presentation of a particular topic related to the subject of the course and present the project in class (one lecture -- 75 minutes). I will provide guidance on the projects.

Possible Team Projects:
1) A presentation of the Bell inequalities. References:

  • Nielsen/Chuang, chapter 2.6: "EPR and the Bell inequality".
  • Bell inequalities and entanglement, Reinhard Werner, Michael Wolf.
  • Bell's Theorem, Bachelor's Thesis of Sven Etienne (supervisor: Klaas Landsman).

    2) Universality of quantum gates. Single qubit gates and CNOT are universal. Hadamard, phase, \pi/8 and CNOT gates are universal. Reference: Nielsen/Chuang, chapter 4.5 (up to 4.5.5).

    3) The Solovay-Kitaev theorem. Reference: Nielsen/Chuang, Appendix 3.

    4) Presentation of a quantum error correcting code (exact reference to be determined).

    5) A topic pertaining to the study of entanglement measures (to be determined).

    6) Topics from topological quantum computing: anyons, Jones braid group representation, modular tensor categories, topological phases (see Z. Wang's book above). Here some other references:
  • V.T. Lahtinen, J.K. Pachos, A Short Introduction to Topological Quantum Computatione
  • Eric C. Rowell, Zhenghan Wang, Mathematics of Topological Quantum Computing (arxiv 2017) .
  • B. Field, T. Simula, Introduction to topological quantum computation with non-Abelian anyons
  • J. K. Pachos, Introduction totopological quantum computation
  • George Toh, TQC (talk 2017).
  • Sankar Das Sarma, Michael Freedman, and Chetan Nayak, Topological quantum computation
  • Graham Collins, Computing with Quantum Knots, Scientific American 2006.
  • Natalie Wolchover, Physicists Aim to Classify All Possible Phases of Matter, Quanta Magazine 2018.
  • Colleen Delaney, A categorical perspective on symmetry, topological order, and quantum information, PhD thesis UCSB 2019.
  • John Preskill's notes for Quantum Computation course at Caltech.

    7) Planar algebras and quantum information theory (Reutter's articles can be found on his web page https://www.davidreutter.com/):
  • Vijay Kodiyalam, Sruthymurali, V. S. Sunder, Planar algebras, quantum information theory and subfactors,
  • David Reutter, Jamie, Vicary, Biunitary constructions in quantum information. High. Struct. 3 (2019), no. 1, 109–154.
  • David Reutter, Jamie Vicary, Shaded tangles for the design and v erification of quantum circuits. Proc. A. 475 (2019), no. 2224, 20180338, 26 pp.

    8) You can suggest a topic yourself, e.g. some of the further reading suggested during the lectures.