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Department of Mathematical Sciences

# MATH3391 Quantum Information III

Quantum mechanics differs in a fundamental way from classical mechanics. Quantum systems can be in a superposition of states, and measurements probabilistically select one state. This can be used in quantum computing where in some sense processing a superposition corresponds to parallel computing, but the catch is that we cannot measure all results. Systems can also be entangled even when they are separated, and this has interesting applications to communication, although it does not allow faster than light communication.

Quantum Information Theory describes how information can be stored and processed in quantum systems. It uses the properties of quantum systems such as superposition and mutually incompatible measurements in a fundamental way. As will be explained in this module, this gives the potential to allow completely secure communication between two parties (and this has been implemented), and a quantum computer is potentially for some tasks much more powerful than a classical computer (but there remain significant practical difficulties in constructing a useful device.) We will also explore how concepts such as entanglement can be quantified, and how it can be clearly determined that a quantum system is not simply a complicated classical system with randomness due to lack of knowledge.

## Outline of Course

Aim: To provide an introduction to the application of quantum systems to processing information, specifically in terms of communication and computing. We will study the concept of quantum entanglement and demonstrate that quantum systems have properties that are fundamentally different from those of classical systems. We will see ways to quantify entanglement and how it can be used to achieve perfectly secure communication between two parties. We will also study some aspects of quantum information processing and investigate some examples of quantum algorithms.

### Term 1

• Quantum Mechanics Introduction (4 lectures): Review of wave mechanics, introduction of Dirac notation and the density matrix.
• Quantum Information (4 lectures): The qubit, Bloch sphere, bipartite systems and concept of pure and mixed states.
• Quantum properties and applications (6 lectures): Superdense coding, teleportation, quantum key distribution, EPR paradox, Hidden variable theories and Bell inequalities.
• Information, entropy and entanglement (6 lectures): Brief introduction to classical information theory including Shannon information and entanglement. Quantum entropy measures, von Neumann entropy, relative entropy and conditional entropy.

### Term 2

• Classical computing (4 lectures): Universal gates/circuit models, very brief discussion of computational complexity.
• Quantum computing (8 lectures): Quantum circuit model and universal gates, example algorithms (e.g. Grover's and Shor's), brief discussion of quantum computational complexity and comparison to classical examples (e.g. Shor's algorithm in context of RSA cryptography).
• Quantum error correction (6 lectures): Contrast to classical use of redundancy, examples of single qubit errors, use of entanglement to correct errors, example of Shor code. Discussion of error correction in quantum computing, including fault tolerant gates.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.