## Overview

Welcome to my course "Math 595: Quantum channels", which focuses on the fundamental notion of a quantum channel, a mathematical model for the noisy evolution of a quantum system.

The first part of the course is an introduction to the theory of quantum channels in the finite-dimensional setting of quantum information theory. We discuss the various mathematically equivalent representations of quantum channels, focus on some important subclasses of channels, and make various connections to information-theoretic aspects of quantum information theory.

In the second part of the course we turn to the theory of quantum Markov chains. We first discuss the quantum relative entropy and its fundamental property, the data-processing inequality, and give a proof of this inequality that naturally leads to equality conditions and the concept of recovery channels. Specializing this analysis to the partial trace, we obtain the strong subadditivity property of the von Neumann entropy, as well as a natural definition of quantum Markov chains. We then review a structure theorem for quantum Markov chains, the fundamental differences to classical Markov chains, and—time permitting—discuss the notion of approximate quantum Markov chains.

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

- Required:
- Math 416 Abstract Linear Algebra (or equivalent)
- Useful, but not required:
- Intro to Quantum Mechanics/Information (such as ECE 404, Phys 486/487, Phys 513)

### 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

- Basics of finite-dimensional quantum information theory:

Quantum systems and quantum states, Measurements, Composite systems and entanglement, Distance measures - Representations of quantum channels:

Isometric picture, Unitary evolution, Kraus representation, Linear operator representation - Classes of quantum channels:

Mixed-unitary and unital channels, Entanglement-breaking channels, Symmetric and antidegradable channels, PPT channels - Unital channels and majorization:

Majorization for real vectors, Majorization for Hermitian operators, Schur-Horn Theorem - Covariant quantum channels:

Definition, properties, examples, Holevo information and minimum output entropy - Quantum relative entropy:

Definition and operational interpretation, Joint convexity and data-processing inequality - Equality conditions for data-processing:

Petz's proof of the equality condition, Formulation in terms of recovery channels - Quantum Markov chains: Strong subadditivity of von Neumann entropy, Quantum conditional mutual information and its operational interpretations, Structure theorem for exact quantum Markov chains
- Approximate quantum Markov chains:

Classical vs. quantum setting, Approximate recovery channels

## Notes & exercises

### Lecture notes

We will use the following lecture notes, which have been typed up by Jacob Beckey and Louis Schatzki. These notes are being updated throughout the course so please always make sure you have the newest version. They are based on a previous lecture, for which handwritten notes can be found here and here.

We will start the course with a recap of the basics of finite-dimensional quantum information theory, for which handwritten notes are available.

### Exercise sheets

- Sheet 1 (Last update: Jan 24)

### Further reading in quantum information theory

Throughout the lecture I am trying to give some quantum information-theoretic context while discussing quantum channels and their properties. Unfortunately, it is beyond the scope of this lecture to discuss these information-theoretic aspects in more detail. Along with Mark M. Wilde's book on Quantum Information Theory (see the course literature below), the lecture Theory of Quantum Communication (Fall 2020) taught by Debbie Leung at the University of Waterloo is an excellent resource to learn more about this topic.

## Literature

- Stephane Attal: Quantum channels, Lecture notes.
- Ben Ibinson et al.: Robustness of quantum Markov chains, Communications in Mathematical Physics 277.2 (2008), pp. 289–304.
- Michael A. Nielsen and Denes Petz: A Simple Proof of the Strong Subadditivity Inequality, Quantum Information & Computation 5.6 (2005), pp. 507–513.
- Denes Petz: Quantum Information Theory and Quantum Statistics. Springer, 2008.
- Denes Petz: Monotonicity of quantum relative entropy revisited, Reviews in Mathematical Physics 15.01 (2003), pp. 79–91.
- Mary Beth Ruskai: Inequalities for quantum entropy: A review with conditions for equality, Journal of Mathematical Physics 43.9 (2002), pp. 4358–4375.
- David Sutter: Approximate quantum Markov chains, Vol. 28. SpringerBriefs in Mathematical Physics, 2018.
- John Watrous: Theory of Quantum Information, Cambridge University Press, 2018.
- Mark M. Wilde: Quantum Information Theory, Cambridge University Press, 2017.
- Michael M. Wolf: Quantum Channels & Operations: Guided Tour, Lecture notes, 2012.

## Contact

Email: <mylastname>@illinois.edu

Homepage: felixleditzky.info

Postal address:

Office 39, Computing Applications Building

605 E Springfield Ave

Champaign, IL 61820

USA