Fall Semester 2020, Quantum Information Theory
(Math 9100)


Instructor: Dietmar Bisch
Lecture: TuTh, 9:35am-10:50am, online on Zoom. Meeting ID: 969 6447 5678. Please email Dietmar Bisch for the passcode.
Office: SC 1405, (615) 322-1999
Office hours: Tu 11:00-11:45am on Zoom, meeting ID: 979 0316 0502, and Th 1:00-1:45pm on Zoom, meeting ID: 939 6879 7682. Waiting room feature enabled. Please email Dietmar Bisch for the passcode (same passcode for both office hours). Further office hours by appointment.
Mailbox: SC 1326

Prerequisites: Linear algebra, 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. There is a lot available online.

Lectures (8/25/20 to 12/3/20)
Click on the lecture for the pdf. Note: Access to lectures has been turned off, as the semester has ended. If you would like to see the lectures, please email the instructor.

  • Lecture 1, 8/25/20.
  • Lecture 2, 8/27/20. Here are links to some viewing/reading related to the material presented in lecture 2:
  • Lecture 3, 9/1/20.
  • Lecture 4, 9/3/20. Here are links to some viewing/reading related to the material presented in lecture 4:
  • Lecture 5, 9/8/20.
  • Lecture 6, 9/10/20. Here is some further reading menioned during the lecture:
  • Lecture 7, 9/15/20. Further viewing/reading:
  • Lecture 8, 9/17/20.
  • Lecture 9, 9/22/20. Additional reading on slice maps and purification of states in the context of operator algebras:
  • Lecture 10, 9/24/20. References and additional reading:
  • Lecture 11, 9/29/20. References and additional reading:
  • Lecture 12, 10/1/20. References and additional reading:
  • Lecture 13, 10/6/20. References and additional reading:
  • Lecture 14, 10/8/20. References and additional reading:
  • Lecture 15, 10/13/20. References and additional reading:
  • Lecture 16, 10/15/20. References and additional reading:
  • Lecture 17, 10/20/20. References and additional reading:
  • Lecture 18, 10/22/20. References and additional reading:
  • Lecture 19, 10/27/20. References and additional reading:
  • Lecture 20, 10/29/20. References and additional reading:
  • Lecture 21, 11/3/20. References and additional reading:
  • Lecture 22, 11/5/20.
  • Lecture 23, 11/10/20. References and additional reading:
  • Lecture 24, 11/12/20. References and additional reading:
  • Lecture 25, 11/17/20. References and additional reading:
  • Lecture 26, 11/19/20. References and additional reading:
  • Lecture 27, 12/1/20. Student project presentation by Michael Montgomery and Kai Toyasawa (project 8 below on biunitary connections, planar algebras and quantum error correction).
  • Lecture 28,12/3/20. Student project presentation by Changying Ding and Dumindu da Silva (project 5 below on Tsirelson's problem and Kirchberg's problem, and CEP).
  • Lecture 29,12/3/20. Student project presentation by Julio Caceres and Yuze Ruan (project 7, introduction to topological quantum computing).
    Additional reading: Qiskit: Open-source framework for quantum computing:
  • Qiskit, Wikipedia page.

    Syllabus: This course will be somewhat unusual due to the general situation with COVID-19. I plan to give live lectures on Zoom during the scheduled class time (see above). The pdfs of the lectures, or, alternatively, the pdfs of my notes for the lecture will be available to students enrolled in the class (at least that's my intention). Recording of classes has all kinds of legal ramifications, so I won't allow it, unless you convince me otherwise (and it is legal). Any student who records a class without my permission and/or publishes recordings outside the Vanderbilt ecosystem is guilty of an Honor Code violation. Please note that Vanderbilt's Honor Code applies to this course.

    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 (observables, states, measurements, density matrices 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, then they can be used to transmit quantum information on a classical channel (so-called quantum teleportation). 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.

    In the second part of the course I will present the basics of quantum computation and quantum algorithms. I plan to describe Peter Shor's famous factoring algorithm, which can be used to factor integers on a (hypothetical) quantum computer in polynomial time. Shor's result was quite a surprise since the best known 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 hope to be able to cover some of the basic ideas. If there is time, I will spend the the latter part of the course on topological quantum computing (TQC). The plan is to explain what the advantages of a topological quantum computer are, and what the mathematical approach to TQC entails (modular tensor categories, in particular).

    This course will concentrate on the more theoretical aspects of quantum computing and quantum information theory. For instance, I will most likely not discuss the numerous ideas currently on the market for actually building a quantum computer. Due to Google's recent breakthrough in this area, I will probably spend some time on it though.

    Grading: There will be no exams. 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 on Zoom during class time. I will provide guidance on the project, and also on how to use Zoom.

    Team Projects: Here are a few suggestions:
    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) Tsirelon's problem and the Connes embedding problem. References:
  • N. Ozawa, Tsirelson's problem and asymptotically commuting unitary matrices, arXiv:1211.2712.
  • W. Slofstra, The set of quantum correlations is not closed, arXiv:1703.08618.
  • W. Slofstra, Tsirelson's problem and an embedding theorem for groups arising from non-local games, arXiv:1606.03140.
  • N. Ozawa, About the Connes Embedding Conjecture---Algebraic approaches---, arXiv:1212.1700.
  • T. Fritz, Tsirelson's problem and Kirchberg's conjecture, arXiv:1008.1168.
  • V. B. Scholz, R. F. Werner, Tsirelson's Problem, arXiv:0812.4305.
  • Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, Henry Yuen, MIP*=RE, arXiv:2001.04383.
  • Gilles Pisier, Tensor Products of C*-Algebras and Operator Spaces: The Connes-Kirchberg Problem, Cambridge University Press, 2020.
    6) A topic pertaining to the study of entanglement measures (to be determined).
    7) 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.
    8) 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.
    9) A topic from the Mathematical Picture Language Project (Jaffe, Liu at Harvard & Tsinghua University)
    10) You can suggest a topic yourself, e.g. some of the further reading suggested during the lectures.