Quantum Information

Quantum Information deals with the study of quantum mechanics from the point of view of information theory, as well as with the use of quantum mechanical systems for the purpose of information processing and computation. On the one hand, this includes quantum information theory, with topics such as quantum teleportation, the transmission of information through quantum channels, quantum cryptography, and the quantification of quantum entanglement as a resource for the aforementioned tasks. On the other hand, it involves quantum computation, i.e., computation based on the laws of quantum mechanics, covering topics such as quantum algorithms, quantum error correction, and the physical realization of quantum computers.

The module provides a comprehensive introduction to the field of Quantum Information.  It covers:

• States, evolution, and measurement
• Quantum entanglement
• Quantum channels
• Quantum cryptography
• Quantum computation and quantum algorithms
• Quantum error correction
• Implementations of quantum information processing

After successful completion of the module the students:

1. are able to explain the pure and mixed state formalism of quantum mechanics
2. understand the Bloch sphere picture
3. understand Schmidt decomposition, its relevance for understanding entanglement, and purifications of mixed states
4. can explain completely positive maps and their relevance, the Kraus representation, and the Stinespring dilation
5. know about POVM measurements and their relation to projective measurements
6. understand pure state entanglement, in particular the relevance of the entropy of entanglement
7. know about basics of mixed state entanglement, in particular the PPT criterion, entanglement witnesses, and negativity
8. understand the importance of Bell inequalities
9. have learned the basics of quantum computing, in particular quantum circuits, the role of reversibility, and basic alsorithms, in particular the Deutsch algorithm, Grover's algorithm, and Shor's algorithm
10. know basics of quantum error correction, in particular the 9-qubit Shor code

Very good knowledge of Linear Algebra (e.g. MA9201) is essential for this module. Knowledge of quantum mechanics (e.g. PH0007) is certainly useful, but not strictly necessary. (However, please let the lecturer know in advance should you have no prior knowledge of quantum mechanics.)

The module consists of a lecture and exercise classes. The lecture will be taught on the blackboard.  Lecture notes will be published on the lecture website http://www.mpq.mpg.de/~nys/QI1819.  The lecture will be complemented by homework sheets and exercises, which serve to deepen the understanding of the topics covered in the lecture through specific examples and complementary perspectives.  Study of the homework sheets and attendance of the exercises is therefore strongly encouraged.

blackboard presentation, lecture notes, homework sheets

• J. Preskill, Quantum Computation lecture notes.
• M. Nielsen and I. Chuang, Quantum Information and Computation. (Cambridge University Press, 2010)

There will be an oral exam of 25 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using comprehension questions and examples.

For example an assignment in the exam might be:

• How does the Deutsch algorithm work?
• What is a completely positive map?
• What is the Bloch sphere?
• What is the idea behind quantum error correction?
• What is the PPT criterion?
• What is the Schmidt decomposition, and what is its relevance to entangement?
• What is a purification of a mixed state?
• What happens to continuous erros in quantum error correction?
• How can we find a Schmidt decomposition?
• What is the Choi-Jamiolkowski isomorphism?

Participation in the exercise class is strongly recommended since the exercises prepare for the problems of the exam and rehearse the specific competencies.