Section information

General description of the course

This is an introduction to the mathematical foundations of quantum theory aimed at advanced undergraduate/beginning graduate students in Mathematics and Engineering, although students from other disciplines are equally welcome to attend.

Topics include: the mathematical postulates of quantum theory from a quantum probability perspective, multilinear algebra on finite dimensional Hilbert spaces, rudiments of Lie groups and Lie algebras, spectral theory of self-adjoint operators, simple physical systems in finite and infinite dimensions, quantum probability and quantum information.

What is different about this course

It intends to be a bridge between the more traditional courses in quantum physics mostly taken by physics majors and courses on quantum computing and quantum information that are now becoming common in the curriculum of computer science and engineering majors.

Quantum physics, as taught at the physics department, assumes familiarity with a number of ideas and examples that physics majors are exposed to in their mechanics, electromagnetism, statistical physics, and lab courses. Their goal is to efficiently cover as much physics content as possible at a sufficient level of mathematical sophistication. Courses on quantum computing, on the other hand, are typically limited to finite dimensional theory, with a strong emphasis on algorithms and quantum circuits, and limited contact with the underlying physics.

In Math 444, we will spell out the fundamental concepts around which quantum theory is organized (say, from an operational quantum physics perspective), providing a solid basis for further studies in quantum information theory at the graduate level, and will also explore textbook physics systems, not as thoroughly as a physics major would see, but likely in greater mathematical depth.

For the math major in particular, Math 444 emphasizes the connections between what they learn (in algebra, analysis, topology/geometry, probability theory and statistics) and a topic of great scientific interest.

Learning objectives

This course has two main objectives: one is to explore the fundamental mathematical concepts that arise in quantum physics; the second is to familiarize the student with the way quantum theory itself is structured, providing a solid basis for further study. The course has in mind to serve, on one hand, students from engineering and computer science interested in quantum information and quantum computing and, on the other, math majors who wish to become acquainted with the mathematical structure of quantum theory and explore the connections between this theory and what they learn in standard math courses. (I hope that physics majors can also benefit from seeing their subject from a different perspective.)

On the math side, we will learn about

  1. Linear and multilinear algebra
  1. Group theory
  1. Hilbert spaces
  1. Some probability theory and stochastic processes relevant to quantum theory

See below the more detailed list of topics we expect (and in some cases hope, given sufficient time) to cover.

Prerequisites

It is expected that you have a good grounding in linear algebra (roughly similar to Math 429) and multivariate calculus, as well as familiarity with probability theory, although I will make every effort to make the course as self-contained as possible.

Coursework

Coursework will consist of weekly homework and reading assignments. The final grade will be calculated on the basis of the weekly assignments and final exam, with special emphasis on the former.

Assignments will be handled through Gradescope. The plan is to set the due date for homework submission to Fridays 11:59PM. (The minus one minute from midnight is simply to avoid confusion due to clock discontinuity.)

Text

There is, unfortunately, no single text that approaches the subject matter of this course quite at the level and from the perspective we plan to cover it. My intention is to make available to you a variety of sources throughout the semester. They will consist of typed course notes, occasional chapters from various texts and expository papers. I should have a fair amount of this material collated earlier on in the semester and available in Canvas.

Material needed for homework assignments will be clearly identified on the problem sets themselves.

For approximately half the course (mostly about finite dimensional quantum theory) we will rely on course notes which will be available in Canvas. For the second half I will take material from a number of texts. I expect the following to be useful (you don’t need to buy any of them for this course; I will make whichever parts we need available to you in some form):

-Groups and Symmetries: From Finite Groups to Lie Groups, 2nd ed. by Yvette Kosmann-Schwarzbach, Universitext, Springer, 2022. (Pdf can be downloaded from the Olin Library.)

-Quantum Theory for Mathematicians by Brian C. Hall, Graduate Texts in Mathematics, Springer 2013. (Pdf can be downloaded from the Olin Library.)

-From Classical to Quantum Mechanics by G. Esposito, G. Marmo, G. Sudarsham, Cambridge University Press, 2004.

Topics we hope to cover.

The course will be divided into two parts: Part I covers the foundations of Quantum Theory in a finite dimensional setting from a perspective that emphasizes non-commutative probability theory. It should be accessible to students with a strong background in linear algebra (although I will review the necessary facts). Part II is dedicated to systems governed by differential (Schrödinger) operators.

The following is a very tentative list of topics. Some will be developed in detail while others may be surveyed more superficially.

  1. Mathematical postulates of quantum probability spaces
    • Hilbert spaces, operators, the spectral theorem;
    • Gleason’s theorem and Born’s rule, the uncertainty relation;
    • Dynamics in finite dimensional QT;
    • Quantum trajectories, strong and weak measurements;
    • The unitary group;
    • The qubit and the Bloch sphere.
  2. Composite systems and entanglement
    • Tensor products of Hilbert spaces;
    • Symmetric and antisymmetric tensor products (indistinguishability, bosons and fermions);
    • Fock spaces;
    • Tensor diagrams and quantum circuits;
    • von Neumann entropy, Schmidt decomposition and rank; entanglement;
    • EPR states, Bell’s inequalities, superdense coding, quantum teleportation.
  3. Infinite dimensional systems
    • Self-adjointness, Schrödinger operators;
    • Wave packets and the Fourier transform;
    • Examples of quantum systems (mostly in dimension 1):
      • Particle in a box;
      • Harmonic oscillator;
      • Quantum graphs;
  4. Open systems and basic concepts in quantum stochatic processes

Coursework and grades

Coursework will consist mainly of weekly homework assignments (I plan around 10 or 11 assignments), and a final exam. Final grades are determined as follows: homework 90%, final exam 10%. Your lowest homework grade will be dropped. Assignments will be handled through Gradescope and grades will be posted on Canvas.

Grades may be curved, but will not be less than the following scale:

A+ A A- B+ B B- C+ C C- D F
TBA 90+ [85,90) [80,85) [75,80) [70,75) [65,70) [60,65) [55,60) [50,55) [0,50)

If you are taking this class Pass/Fail, you need a C- to pass. There is no requirement for auditing other than a good attendance.

Recording

I plan to record the in-person lectures using Zoom and make them available in Canvas. I expect, and strongly urge, that students will attend classes in person, but the option of attending online is available if you cannot be there for health related reasons.

Academic integrity

I will follow the University’s academic integrity policy, which you can read here. Work submitted under your name is expected to be your own. You are permitted, and encouraged, to collaborate on assignments. You may research broadly over the internet and in books when working on assignments.

If you have any concerns or questions about this policy or academic integrity in class, please contact me.

Email

Please include Math 444 in the subject line of any email message that pertains to this course.

Homework assignments

Weekly homework assignments and solutions will be posted in Canvas.