Math 595: Representation-theoretic methods in quantum information theory

Instructor: Felix Leditzky (<mylastname>@illinois.edu)
Meeting times (Fall 2022): Tue, Thu 2-3.20pm, 147 Altgeld Hall.
Office hours: By appointment.

Course information

Welcome to my course "Math 595: Representation-theoretic methods in quantum information theory"! In this course we study symmetries in quantum information theory using tools from representation theory. Two fundamental symmetry groups in quantum information are the symmetric group, acting by permuting subsystems in a tensor product of identical Hilbert spaces, and the unitary group, acting diagonally on a tensor product space. Schur-Weyl duality establishes a close relationship between these two representations, giving rise to a useful decomposition of the representation space into irreducible representations. This structure result allows us to succinctly describe invariant objects and characterize optimal information-processing protocols in the presence of permutation and unitary symmetries.

The first half of the course starts with a brief review of the basics of quantum information theory and representation theory. We then discuss the representation theory of the symmetric and unitary groups and how they relate to each other via Schur-Weyl duality. These findings can be applied to characterize symmetric quantum states such as Werner and isotropic states. In the second half of the course we use these representation-theoretic methods to characterize quantum information-processing tasks such as data compression, spectrum estimation, quantum state tomography, and quantum state merging. Depending on available time and interest, we will also discuss useful results such as de Finetti theorems and the decoupling theorem.

Please see the Table of contents below for a detailed list of topics that we will cover.

Lectures

Lectures will be given in person at the above time and place.

Grading policy

There will be no mandatory homework assignments or written exams for this course. Grading will be based on class participation. I will provide exercises that we can discuss in office hours.

Prerequisites

Code of conduct

I am dedicated to providing an inclusive and safe classroom experience for everyone, regardless of gender, gender identity and expression, sexual orientation, disability, physical appearance, body size, race, age or religion. I will not tolerate harassment and discriminating or disrespectful behavior between any classroom participants (including myself) in any form, whether in person or online. Violations of this code of conduct will be reported appropriately. (This code of conduct is based on a template provided by the Geek Feminism Wiki.)

Table of contents

Lectures

Handwritten lecture notes will be made available here as we go along.

Literature

Contact

Email: <mylastname>@illinois.edu
Homepage: felixleditzky.info

Postal address:
Office 39, Computing Applications Building
605 E Springfield Ave
Champaign, IL 61820
USA