Quantum Information Theory

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 analogues. Thee 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 and the Kraus Representation Theorem. Entanglement and some applications elucidating it susefulness as a resource in QIT will be discussed. This will be followed by a study of the von Neumann entropy, its properties and its interpretation as the data compression limit of a quantum information source. Schumacher’s theorem will be discussed in detail. The definition of ensemble average fidelity and entanglement fidelity will be introduced in this context. 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.

Desirable Previous Knowledge
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 Group Theory will be useful.

Introductory Reading

I would strongly advise you to read the following notes on some fundamentals of Quantum Mechanics:
http://cam.qubit.org/sites/default/files/prerequisites_12.pdf

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

1. M.A.Nielsen and I.L.Chuang, Quantum Computation and Quantum Information;
Cambridge University Press, 2000.

2. 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: Sergii Strelchuk

Example sheets distributed in class.

First Example Class : 3pm, Tuesday 7th of February, MR4
Second Example Class : 3pm, Tuesday, 6th of March, MR4
Third Example Class : 3.30pm, Thursday, 15th of March, MR4
Fourt Example Class : in the Easter term

Lecture Notes