QST Theory: Quantum Information

The PH1010 QST Theory: Quantum Information offers an introduction to the theoretical foundations of Quantum Science and Technology. The course starts with a brief motivation and an introduction to fundamental concepts and the basic formalism (pure/mixed states, evolution, completely positive maps, measurements Schmidt decomposition, tomography, quantum estimation, hypothesis testing). Then the concept of entanglement is discussed in detail, including the distinction between pure and mixed-state entanglement, entanglement entropy, quantification and conversion. Subsequently, some of the revolutionary promises of exploiting entanglement are presented, including dense coding, quantum teleportation and quantum cryptography. Next the Bell inequalities, characterizing the quantum weirdness of entanglement and non-locality, are introduced and discussed in detail. Subsequent chapters cover central applications of quantum information theory: quantum computation, quantum algorithms such as those of Deutsch, Shor and Grover, and quantum simulation. Final core topics are decoherence, Lindbladian descriptions thereof, and error correction schemes to counteract the consequences of decoherence and protect fragile quantum information. The module will typically also include one or more optional topics, such as many-body entanglement, topological quantum computation, quantum complexity, or tensor networks, which link quantum information theory to many-body physics.

After participation in the Module the student is able to:

1. Explain fundamental concepts such as the distinction between pure and mixed states, quantum evolution, completely positive maps, and quantum measurements.

2. Explain and quantify the notion of entanglement in various contexts (pure states, mixed states, purification, Bell inequalities).

3. Understand and explain central applications of quantum information theory, such as quantum cryptography, quantum computation, and quantum simulation.

4. Understand the central ideas underlying the quantum algorithms of Grover and Shor.

5. Understand the notion and the consequences of decoherence, model it using Lindbladians, and explain central elementary error correction strategies.

6. Competently perform quantum mechanical computations relevant for the above topics.

No prerequisites in addition to the requirements for the Master’s program in Quantum Science and Technology. Familiarity with quantum mechanics is assumed, at the level of an introductory course from a Bachelor degree in physics.

The module consists of a lecture series (4 SWS) and exercise classes (2 SWS), comprising two lecture sessions and one exercise session per week. 

The main teaching material is presented on the blackboard or by beamer. The central role of entanglement, providing cross-links between different topics, will be emphasized at each level.

Lectures are supplemented by weekly problem sets, deepening the understanding of core concepts through concrete calculations. Solutions to the problem sets are discussed within the exercise classes.

Participation in the exercise classes is strongly recommended, since the exercises offer thorough training in core conceptual and computational skills, thereby greatly aiding exam preparation.

Blackboard, presentations (slides), electronic board (with handouts).

Standard textbooks on Quantum Information, for example:

• Quantum Computation (Lecture Notes), John Preskill. http:/www.theory.caltech.edu/people/preskill/ph229
   A famous set of lecture notes, continually being refined throughout the last 20 years.

• Quantum Computation and Quantum Information, 10th Anniversary Edition, Michael A. Nielsen, Isaac L. Chuang, Cambridge University Press, 2010.
   One of the most cited books in physics of all time, providing a general, accessible and wide-ranging introduction to the topic.

• Quantum Computing: A Gentle Introduction (Scientific and Engineering Computation), Eleanor Rieffel, Wolfgang Polak, The MIT Press, 2011. A thorough exposition of quantum computing and the underlying concepts of quantum physics.

• Quantum Information Theory, Mark. M. Wilde, Cambridge University Press, 2013.
   Aims to introduce readers familiar with classical communication and information theory to the novel aspects of quantum communication and information theory.

• Quantum Computer Science: An Introduction, N. David Mermin, Cambridge University Press, 2007. Develops the subject without assuming any background in physics.

• An Introduction to Quantum Computing, Phillip Kaye, Raymond Laflamme, Michele Mosca, Oxford University Press, 2010.
   An introduction to quantum computing aimed at advanced undergraduate and beginning graduate students in physics, mathematics, computer science or engineering.

• Classical and Quantum Computation, A. Yu. Kitaev, A. H. Shen, M. N Vyalyi, Graduate Studies in Mathematics, Vol. 47, American Mathematical Society, 2002.                                                                    For students interested in the mathematical aspects of quantum information theory.

There will be a written exam of 180 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using conceptual questions and computational tasks.

For example an assignment in the exam might be:

• How does the Hadamard gate act on the Bloch sphere?
• What is an example of a map which is positive but not completely positive?
• What is the Schmidt decomposition, and what is its relevance to entanglement?
• How can a Schmidt decomposition be found?
• What is the PPT criterion?
• What is a purification of a mixed state?
• How can the state 1/3*|00>+2/3*|11> be converted to a maximally entangled state?
• Construct a basis of maximally entangled states for two qutrits.
• Characterize a measurement of an observable in a system of two qubits when restricted to an effective qutrit inside the two-qubit system.
• Find and characterize a protocol for teleporting an arbitrary two-qubit state.
• How does the Deutsch algorithm work?
• Why is the oracle in the Deutsch algorithm not acting as |x> -> |f(x)>?
• Find a quantum circuit performing a controlled-U two-qubit gate, where U is an arbitrary unitary.
• What is the idea behind quantum error correction?
• What happens to continuous errors in quantum error correction?
• What is the Choi-Jamiolkowski isomorphism?