## Quantum Information Theory

** Michaelmas Term: days/times tba **

Will Matthews

**Course Description**

Quantum Information Theory (QIT) is an exciting, young field which lies at the intersection of

Mathematics, Physics and Computer Science. It was born out of Classical Information Theory,

which is the mathematical theory of acquisition, storage, transmission and processing of information.

QIT is the study of how these tasks can be accomplished, using quantum-mechanical

systems. The underlying quantum mechanics leads to some distinctively new features which

have no classical analogues. These new features can be exploited, not only to improve the

performance of certain information-processing tasks, but also to accomplish tasks which are

impossible or intractable in the classical realm.

This is an introductory course on QIT, which should serve to pave the way for more advanced

topics in this field.

The course will start by introducing a mathematical framework, based on the postulates of

quantum mechanics and widely used in the study of quantum information theory, in which we

can describe the time evolution of open systems (quantum operations) and very general forms of

measurement (instruments and POVMs). Along the way we will prove results establishing the

non-locality of quantum mechanics (Bell’s theorem), the fact that quantum information cannot

be perfectly copied (the “no-cloning” theorem), and fundamental limits on how well different

states of a quantum system can be distinguished by measurements.

Building on this we will develop some of the major results of classical and quantum information

theory, which concern data compression and the reliable transmission of information over noisy

communication channels. Key mathematical ideas introduced in the process will be the classical

and quantum notions of entropy and information, channel capacities, as well as random coding

arguments. We will also look at the remarkable “dense coding” and “teleportation” protocols,

which make use of the strange phenomenon of entanglement to accomplish tasks that would

otherwise be impossible, and look at various ways of classifying and quantifying entangled states.

**Pre-requisites**

Knowledge of basic quantum mechanics will be assumed. However, an additional lecture can

be arranged for students who do not have the necessary background in quantum mechanics.

Elementary knowledge of Probability Theory, Vector Spaces and Linear Algebra will be useful.

**Background reading**

The following books and lecture notes provide some interesting and relevant reading material.

On classical information theory:

1. D. J. C. MacKay, "Information Theory, Inference, and Learning Algorithms", CUP 2003, available online: http://www.inference.phy.cam.ac.uk/mackay/itila/book.html

On quantum information theory:

1. M. A. Nielsen and I. L. Chuang, "Quantum Computation and Quantum Information";

Cambridge University Press, 2000.

2. M. M. Wilde, "From Classical to Quantum Shannon Theory", CUP; http://arxiv.org/abs/1106.1445.

3. J. Preskill, Chapter 5 of his lecture notes: Lecture notes on Quantum Information Theory

http://www.theory.caltech.edu/~preskill/ph229/#lecture

**Example Classes**

Course Instructor: Felix Leditzky

Example sheets distributed in class.