Categorical Quantum Mechanics: 2011-2012
OverviewCategory theory gives a powerful mathematical framework for working with quantum theory, and provides a high-level computer science perspective with which to understand it. This course gives an overview of some of the recent research in this exciting field, much of which was carried out here at the Department of Computer Science. The focus is on reformulating quantum-mechanical concepts in category-theoretical terms, and applying this approach to quantum foundations and quantum information. The categorical formalism has a pictorial representation which makes deductions intuitive, and this will form a major part of the course.
While we concentrate on the applications to quantum theory, category theory forms an enormously important part of the modern mathematical landscape, and the tools we introduce in this course have close relationships to other area of mathematics, including representation theory, quantum field theory and knot theory. They also have relevance to more applied disciplines, such as programming language semantics and computational linguistics. Further topics may be investigated if time permits.
This course can currently only be taken by students enrolled on the DPhil or MSc in MFoCS programmes. However, everyone is welcome to sit in and follow the lectures.
Lecture notes are available for download via the "Course materials" link in the right-hand menu.
LecturesMonday 11am-12pm. Lecture Theatre B.
Wednesday 2pm-3pm. Room 380, except Week 4 (8 Feb), when we will be in Lecture Theatre B.
ClassesFriday 12pm-1pm, weeks 4, 5, 6 & 7, Room 051.
PracticalThe course will involve a practical session using the graphical reasoning package “Quantomatic”. This will be held on Tuesday of Week 8, 12pm-2pm, Room 379.
Ideal foundations for this course are given by the Michaelmas term course ''Categories, Proofs and Processes'', and the Hilary term course ''Quantum Computer Science''. Students who have not taken these courses will need to be familiar with basic topics from category theory and linear algebra, including categories, functors, natural transformations, vector spaces, Hilbert spaces and the tensor product.
Students wishing to do their dissertation with the Quantum Group are expected to sit this course, as well as the two mentioned above.
This syllabus gives a suggestion of the topics which might be covered, and may not be rigidly followed.
- Symmetric monoidal categories
- Graphical calculus
- Duals for morphisms
- Duals for objects
- Copying and deleting
- Frobenius algebras and classical structures
- Modelling quantum protocols
- Categories of completely positive maps
- Complementary observables
- Axiomatizing entangled states
- Advanced topics
Materials  to  cover topics relevant to the course in an introductory way; papers  to  are more advanced and provide good further reading.
- Samson Abramsky and Nikos Tzevelekos (2010) Introduction to categories and categorical logic. In: New Structures for Physics, B. Coecke (ed), pages 3-94. Lecture Notes in Physics 813. Springer-Verlag. arXiv:1102.1313 [math.CT] (these are the lecture notes of the Categories, Proofs and Processes course)
- John C. Baez and M. Stay (2010) Physics, topology, logic and computation: a Rosetta Stone. In: New Structures for Physics, B. Coecke (ed), pages 95-172. Lecture Notes in Physics 813. Springer-Verlag. arXiv:0903.0340 [quant-ph]
- Bob Coecke (2010) Quantum picturalism. Contemporary Physics 51, 59-83. arXiv:0908.1787 [quant-ph]
- Bob Coecke and Eric O. Paquette (2010) Categories for the practicing physicist. In: New Structures for Physics, B. Coecke (ed), pages 173-286. Lecture Notes in Physics 813, Springer-Verlag. arXiv:0905.3010 [quant-ph]
- Peter Selinger (2011) A survey of graphical languages for monoidal categories. In: New Structures for Physics, B. Coecke (ed.), pages 289-356. Lecture Notes in Physics 813, Springer-Verlag. arXiv:0908.3347 [math.CT]
- Samson Abramsky and Bob Coecke (2004) A categorical semantics of quantum protocols. In: Proceedings of 19th IEEE conference on Logic in Computer Science, pages 415–425. IEEE Press. arXiv:quant-ph/0402130. Revised version (2009): Categorical quantum mechanics. In: Handbook of Quantum Logic and Quantum Structures, K. Engesser, D. M. Gabbay and D. Lehmann (eds), pages 261–323. Elsevier. arXiv:0808.1023 [quant-ph]
- Samson Abramsky, No-Cloning in Categorical Quantum Mechanics. In Semantic Techniques in Quantum Computation, ed. S. Gay and I. Mackie, pages 1--28, Cambridge University Press 2010. arXiv:0910.2401
- Krysztof Bar, Lucas Dixon, Ross Duncan, Benjamin Frot, Alex Merry, Aleks Kissinger and Matvey Soloviev (2011), “Quantomatic” software. http://sites.google.com/site/quantomatic/
- Bob Coecke and Ross Duncan (2011) Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics 13, 043016. arXiv:0906.4725 [quant-ph]
- Peter Selinger (2007) Dagger compact closed categories and completely positive maps. Electronic Notes in Theoretical Computer Science 170, pages 139-163. http://www.mscs.dal.ca/~selinger/papers.html#dagger
- Jamie Vicary (2008) A categorical framework for the quantum harmonic oscillator. International Journal of Theoretical Physics 47, 3408-3447. arXiv:0706.0711 [quant-ph]
Related research at the Department of Computer Science