## 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:

http://cam.qubit.org/sites/default/files/prerequisites_13.pdf

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

http://www.theory.caltech.edu/~preskill/ph229/#lecture

**Example Classes**

Course Instructor: Felix Leditzky

Example sheets distributed in class.

**Lecture Notes**