Math 490 Mathematical aspects of Quantum Information Theory

**Time:** MWF 12:00-12:50pm

https://illinois.zoom.us/j/83227321296?pwd=NS9lOEdSWjNxN2FXUm5EQzFvT0h6Zz09

**Location: ** Virtual

Recordings:

**Instructor:** Marius
Junge

Office hours: Thursday 5.00-6.00,

Grading: HW 50%, Final 50% (or presentation, depending on #students). For presentation: Please find a topic by looking an exercise problem on quantum channels or Bell inequalities or a sunbject of your choice. Notify me, what you want to do, and then you will prepare a text-that will pass as your final exam.

Course description :

Quantum information science may become a new driver for a new discovery in science. Be this as it may, mathematical aspects of quantum information theory provide an insteresting blend of mathematics while still combining this with physical and operational aspects. Best example: What is a quantum algorithm.

This course will focus the mathematical foundations of quantum information theory. For better or worse, these foundations are spread out in different areas of math: Linear algebra and probability, and some beginnig of information theory. And there is the physics side....

We will gradually develop the basic tools

O: Introduction

I) Classical Gates: What is gate classical gate, and how many functions can we compute with limited resources?

II) Classical Information Theory: How can information be send with a noisy device. A first glance at Shannon's model of noisy channels, and the background we need from probability. In particular, we will discuss joint distributions and independence.

III) Normed spaces and linear maps: Basic definitions and examples, norm of linear maps, convex sets and extreme points.

IV) Hilbert spaces, unitaries and densities: We discuss differen norms on the space of matrices and positive definite matrices.

V) Quantum gates, measurements, entanglement, purification and fidelity.

VI) Entropy, classical and quantum, mutual information, equiparititon theorem.

VII) Quantum channels, Stinespring theorem, complementary channel, Data processing inequality.

VIII) Capacity (with entanglement)

IX) Qauntum devices beating classical ones.

For Notes: See Box

Books: (We will mostly follow Wilde's book)

Michael A. Nielsen & Isaac L. Chuang: Quantum Computation and Quantum Information, Cambridge University Press 2010.

Mark Wilde: From Classical to Quantum Shannon Theory, Cambridge University Press, 2013.

John Preskill: Quantum Computation , lectures notes from CalTech

John Watrous: The Theory of Quantum Information , Cambridge University press 2018.

G. J.A. Jameson: Summing and nuclear norms in Banach spaces, Cambridge University press, 2009

G. Pisier: Factorization of Linear Operator and geometry of Banach Spaces, AMS series no 60

Lecture notes from previous course (the oder is slightly different) Video Recordings (last year)Lectures for week 1: see 1-24, 1-27, 1-29 (ignore number one) (This is about fundation from probability)

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