Quantum Information Theory
Michaelmas Term: days/times tba
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.
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.
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
Course Instructor: Felix Leditzky
Example sheets distributed in class.