Quantum Information Theory

Module Identifier: MAM5820
Module Title: Quantum Information Theory
Academic Year: 2025/2026
Semester: Semester 1
Co-ordinator: Dr Rolf Gohm
Reading List: View on Aspire
Pre-Requisite: MA26010 or MA21410
Exclusive (Any Acad Year): MA35810
Other Staff: Dr Rolf Gohm

Assessment

Assessment Type	Assessment details	Proportion
Semester Exam	Exam: 3 Hours	100%
Supplementary Exam	Exam: 3 Hours	100%

Learning Outcomes

On successful completion of this module students should be able to:

State various concepts of information and entropy and explain the relationships betweenthem.
Achieve efficient data compression by coding procedures, guided by theoretical limits
Explain the notion of a channel as a model of information transmission.
State Shannon’s main theorems about channel capacity and coding.
Reproduce the main assumptions and arguments leading to these theorems.
Apply the theoretical results to construct and to analyse a variety of important channels.
Set up an analogous setting of information processing in the context of quantum theory
Derive various quantum protocols and compare them with the classical situation.

Brief description

C. Shannon’s seminal paper ‘A mathematical theory of communication’ (1948) created information theory as a part of mathematics. It provides the tools for a rigorous understanding of information processing and communication. In this module we carefully develop main concepts like entropy, data compression and coding, channels and their capacity. We explain the main theorems and results and apply them to various classes of examples. In the quantum part we start with a short axiomatic introduction to quantum theory for mathematicians. For composite quantum systems we encounter the non-classical phenomenon of entanglement and we give some applications. We investigate a quantum variant of data compression known as Schumacher compression and compare with the classical situation.

Content

• Introduction, history, what is information, examples
• Entropy, joint and conditional entropy, relative entropy
and mutual information, rules and inequalities
MAF
• Asymptotic equipartition property, typical sets and source
coding
• Data compression, Kraft inequality and optimal codes,
Huffman codes
• Channels and channel capacity, examples
• Shannon’s channel coding theorem, zero error codes,
Hamming codes
• Source-channel coding theorem, binary case
• Information transmission guided by theory, detailed
discussion of examples
• Structure of quantum theory: states, operations, channels
• States of composite systems, entanglement
• Protocols: superdense coding, teleportation
• Von Neumann entropy, Schumacher compression

Module skills

Skills type	Skills details
Adaptability and resilience	Good understanding of the contentsrequires considerable intellectual effortover an extended period of time.
Co-ordinating with others	Discussing the theory and solving problemstogether during the module is encouraged.
Creative Problem Solving	Problem sessions based on problem sheetsto be solved independently. This is crucialto prepare for the problems in the exam.
Critical and analytical thinking	Theory is developed rigorously andcompared with real world situations.
Digital capability	Insights are provided into the mathematicalprinciples of digital information processing.
Professional communication	Discussing the theory and solving problemstogether during the module is encouraged.
Real world sense	Theory is compared with real worldapplications.
Reflection	Intuitive ideas need to be translated intomathematical reasoning.
Subject Specific Skills	Discussing the theory and solving problemstogether during the module is encouraged.

Notes

This module is at CQFW Level 7