Operator Algebraic Tools in Quantum Information Theory

**Time:** Tuesday 4-5.30 pm Wednesday 4-5.30 pm

**Location: ** 03.06.011/ 02.10.011

Course description :

The focus of this course lies in information theoretical foundations based on common tools from functional analysis. The major gools are on entropic inequatities including

SSA

Data processing

Recovery

Additivity of squashed entanglement

Decay for semigroups

Lectures

Quantum mechanics and simple norms

Entropy, weak typicality, and channels

Relative entropy, minimal output entropy for classical channel

Interpolation

Classical Data processing inequality

Conditional expectations

Impartant concepts in mathematics have been developped to understand solutions, and solutions spaces of certain PDE's. This allows to classify and treat certain classes of PDE's. Fourier analysis is one the most of important of these theories. In this sense this couse is also a first step towards mathematics on the graduate level.

Plan:

I: Basics on PDE's

What is a PDE? Linear PDE's, characteristic curves and lines,

well-posedness.

II Wave equation

How to find a solution in 1D? causality, solution formulae, reflection principle, inhomogenous problem (3.3-3.4), well-posedness.

Duhamel

III Fourier analysis

Hilbert spaces and inner products, abstract Fourier coefficients, concrete Fourier coefficients, different form of convergence, Gibbs phenomena, a little peek at continuous Fourier coefficients.

IV Heat equation

Solution formulae, reflection, source, well-posedness.

V Boundary problems in 1D

separation of variables.

VI Laplace equation

The equation, rectangles and cubes, Poisson formula

VII Boundary problems in plane and space

solid virbration, Legendre functions.

wave equation

Books:

Walter A. Strauss, Partial Differential Equations, An Introduction, Second editionAttendance: If you miss the class more than three times unexcused, you may be dropped! It is you task to drop the class in time, and make space who actually need the class. If I can not recognize a student, he will not be handed out an exam.

Link to lectures

Grading:

Homework (20%)Practice probems

Midterm1 January 31 (25%)

Midterm2 February 28 (25%)

Final (7:00-10:00 p.m., Wednesday, May 9) (30%)

Practice probems solutions

Exam1

Practice problems2

Practice problems2-solutions

Practice problems final

Practice problems final solutions

hw1

hw2

hw3

hw4

hw5

hw6

hw7

hw8

hw9

hw10

hw11

hw12