script Quantum Information Theory by Renato Renner...

script Quantum Information Theory by Renato Renner

1 Introduction

The very process of doing physics is to acquire information about the world around us.

At the same time, the storage and processing of information is necessarily a physical

process. It is thus not surprising that physics and the theory of information are inherently

connected.1 Quantum information theory is an interdisciplinary research area whose goal

is to explore this connection.

As the name indicates, the information carriers in quantum information theory are

quantum-mechanical systems (e.g., the spin of a single electron). This is in contrast to

classical information theory where information is assumed to be represented by systems

that are accurately characterized by the laws of classical mechanics and electrodynamics

(e.g., a classical computer, or simply a piece of paper). Because any such classical system

can in principle be described in the language of quantum mechanics, classical information

theory is a (practically significant) special case of quantum information theory.

The course starts with a quick introduction to classical probability and information theory.

Many of the relevant concepts, e.g., the notion of entropy as a measure of uncertainty,

can already be defined in the purely classical case. I thus consider this classical part as a

good preparation as well as a source of intuition for the more general quantum-mechanical

treatment.

We will then move on to the quantum setting, where we will spend a considerable

amount of time to introduce a convenient framework for representing and manipulating

quantum states and quantum operations. This framework will be the prerequisite for

formulating and studying typical information-theoretic problems such as information storage

and transmission (with possibly noisy devices). Furthermore, we will learn in what

sense information represented by quantum systems is different from information that is

represented classically. Finally, we will have a look at applications such as quantum key

distribution.

I would like to emphasize that it is not an intention of this course to give a complete

treatment of quantum information theory. Instead, the goal is to focus on certain key

concepts and to study them in more detail. For further reading, I recommend the standard

textbook by Nielsen and Chuang [6]. Also, I would like to mention the course on

quantum computation [11] by Stefan Wolf (Computer Science Department). Wolf’s course

is somewhat complementary in the sense that it focuses on quantum computation, while

this course is on quantum information.

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