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 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 its usefulness 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 Linear Algebra will be useful.

Introductory Reading

I would strongly advise you to read the following notes on some fundamentals of Quantum Mechanics:

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

Example Classes
Course Instructor: Felix Leditzky

Example sheets distributed in class.

Lecture Notes

Lecture 1 -- Introduction and Shannon Entropy 84.57 KB
Lecture 2 -- Shannon's Source Coding Theorem75.39 KB
Lecture 3 -- Entropy of a pair of random variables55.69 KB
Lectures 4 & 5 -- Shannon's Noisy Channel Coding Theorem76.32 KB
Lectures 6 & 7 -- Open Systems: States97.49 KB
Lecture 8 -- Schmidt decomposition, Purification & the No-Cloning Theorem67.9 KB
Lectures 9 & 10 -- Time evolution in open quantum systems83.78 KB
Lectures 11 & 12 -- Distance between states79.2 KB
Lectures 13 & 14 -- Generalized Measurements76.23 KB
Lectures 15 & 16 -- Entanglement and its applications80.33 KB
Lectures 17 & 18 -- Quantum Entropy74.65 KB
Lectures 19 & 20 -- Quantum Data Compression75.68 KB
Lecture 21 -- Quantum Channels58.84 KB
Lecture 22 -- Accessible information and the Holevo bound55.05 KB
Lecture 23 -- Properties of the Holevo quantity & the HSW theorem86.32 KB
Lecture 24 -- Coherent Information and the quantum data-processing inequality53.06 KB