## Quantum Information Theory

** Lent Term: M. W. F. 10am, MR9 **

**Nilanjana Datta**

**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 counterparts. 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 with a short introduction to some of the basic concepts

and tools of Classical Information Theory, which will prove useful in the study of QIT. Topics

in this part of the course will include a brief discussion of data compression, transmission of

data through noisy channels, Shannon’s theorems, entropy and channel capacity.

The quantum part of the course will commence with a study of open systems and a discussion

of how they necessitate a generalization of the basic postulates of quantum mechanics.

Topics will include quantum states, quantum operations, generalized measurements, POVMs,

the Kraus Representation Theorem and the Choi-Jamilkowski isomorphism. Along the way we will

prove results establishing 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. We will look at the strange phenomenon of entanglement,

and some applications elucidating its usefulness as a resource in QIT. In particular, we will

discuss the protocols of “superdense coding” and “teleportation” which make use of entanglement

to accomplish tasks that would otherwise be impossible.

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

notions of entropy and information, quantum data compression limit, channel capacities, as well

as random coding arguments. Various examples of quantum channels will be given and the different

capacities of a quantum channel will be discussed. The Holevo bound on the accessible information

and the Holevo-Schumacher-Westmoreland (HSW) Theorem will also be covered.

**Prerequisites**

Knowledge of basic quantum mechanics will be assumed.

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. T. M. Cover and J. A. Thomas, "Elements of Information Theory," Wiley Series in Telecommunications and Signal Processing.

2. 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: Eric Hanson

First Example Class : 2pm - 4pm, Thursday 8th of February, MR9.

*Example Sheet 1 *

Second Example Class: 3pm - 5pm, Tuesday 27th of February, MR20.

*Example Sheet 2 *

Third Example Class: 2pm - 4pm, Friday 16th of March, MR21.

*Example Sheet 3 *

Fourth Example Class:

*Example Sheet 4 *

**Discussions & Answers to Questions **

* Short response on source coding *

* Further details on source coding *

**Figures**

Notes 1:

* Plot of the binary entropy *

Notes 3:

* Venn Diagram illustration for entropies *

**Background and prerequisites **

** Mathematical Preliminaries ** (updated February 9, 2018).

*Quantum Mechanical Preliminaries *