About the Course

Time and Venue: Michaelmas : Mondays, Wednesdays and Fridays at 12 noon in MR12.
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. The concept of decoherence and error correction will be introduced and various examples of quantum-error correcting codes 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. (See prerequisites). However, an additional lecture could 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.

Lecturer: Nilanjana Datta
Lecture Notes

• Basics of Classical Information Theory
• Shannon Entropy
• Information Transmission Through Classical Channels:
• Classical Data Compression
• Open Quantum Systems - States
• Schmidt Decomposition, Purification, No-Cloning Theorem
• Open Quantum Systems - Time Evolution
• Kraus Representation Theorem
• Open Quantum Systems - Generalized Measurements
• Entanglement
• Quantum Entropy
• Quantum Data Compression
• Quantum Errors
• Properties of Quantum Error-Correcting Codes
• Examples of QECC
• Quantum Channels
• Accessible Information and the Holevo Bound
• HSW Theorem
• Coherent Information
• Distance between States

Class Instructor: Sergeii Strelchuk
Example Classes: (example sheets distributed in class)

• Class 1: Wednesday 21st of Oct. in MR12 from 2pm.
• Class 2: Wednesday, 4th of November, 2pm, MR12
• Class 3: Monday, 23rd of November, 2pm, MR4
• Class 4:

Textbooks:

• Quantum Computation and Quantum Information by M.A. Nielsen and I.L. Chuang (CUP)
• Quantum Computing by Jozef Gruska (McGraw Hill)
• Classical and quantum computation by A.Yu. Kitaev, A.H. Shen and N.N. Vyalyi (AMS)
• Elements of Information Theory by T.M.Cover and J.M.Thomas (Wiley & Sons)
• John Preskill's lecture notes on Quantum Computation