Quantum Information Theory

Lent Term: M. W. F. 10am, MR5

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