Mathematical Methods for Quantum Information Processing

Overview

1. Operatorial quantum mechanics: review of linear algebra in Dirac notation; basics of quantum mechanics for pure states.
2. Density matrix and mixed states; Bloch sphere; generalised measurements.
3. Maps and operations: complete positive maps; Kraus operators.
4. Quantum Communication protocols: quantum cryptography; cloning; teleportation; dense coding.
5. Quantum computing: review of classical circuits and logic gates; quantum circuits and algorithms; implementation of quantum circuits on small prototypes of quantum computers (IBM Quantum Experience); examples of physical Hamiltonians implementing quantum gates.
6. Theory of entanglement: basic notions and pure-state entanglement manipulation; detection of entanglement; measures of entanglement; entanglement and non-locality, Bell's inequality; multipartite entanglement.

Learning Objectives

On completion of the module, it is intended that students will be able to:
1. Express linear operators in terms of the Dirac notation; derive both the matrix and outer-product representation of linear operators in Dirac notation; recognise Hermitian, normal, positive and unitary operators, and put in use their respective basic properties; construct Kronecker products and functions of operators.
2. Comprehend and express the postulates of quantum mechanics in Dirac notation; define projective measurements and calculate their outcome probabilities and output states; give examples of destructive and non-destructive projective measurements; prove the uncertainty principle for arbitrary linear operators; define positive-operator-valued measurement and use their properties to discriminate between non-orthogonal states; prove the no-cloning theorem for generic pure states.
3. Explain the necessity of using the mixed-state description of quantum systems; define the density operator associated with an ensemble of pure states; express the postulate of quantum mechanics with the density operator formalism; distinguish pure and mixed states; describe two-level system in the Bloch sphere; geometrically describe generic n-level systems; calculate the partial trace and the reduced density operator of a tensor-product system.
4. Describe the dynamics of a non-isolated quantum system with the formalism of completely-positive and trace preserving (CPTP) dynamical maps; derive the operator-sum representation of a CPTP map; give examples of CPTP maps.
5. Demonstrate the most relevant communication protocols for pure states using the Dirac notation for states and linear operators: super-dense coding, quantum teleportation and quantum key distribution.
6. Describe the basic properties of classical circuit for classical computing in terms of elementary logic gates; define the main model of quantum computation in terms of quantum circuits and gates; comprehend and construct basic quantum algorithms: Grover, Deutsch-Josza and Shor algorithms; construct, implement, and test small quantum circuits on prototypes of quantum computers (IBM Quantum Experience).
7. Define quantum entanglement for pure and mixed states; identify entangled states; manipulate pure entangled state via LOCC operations; calculate the amount of entanglement in simple quantum systems; define Bell inequalities and calculate their violation; define the entanglement in multiple composite systems.

Skills

• Mathematical modelling of quantum systems, including problem solving aspects in the context of quantum technologies.
• Assimilating abstract ideas.
• Using abstract ideas to formulate specific problems.
• Applying a range of mathematical methods to solving specific problems.

Assessment

None