Abstract
We explain the use of category theory in describing certain sorts of anyons . Yoneda’s lemma leads to a simplification of that description. For the particular case of Fibonacci anyons , we also exhibit some calculations that seem to be known to the experts but not explicit in the literature.
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Notes
- 1.
Other answers explained the physics, in terms of excitations, but these matters are not the subject of this paper, which is specifically about mathematics except for the introductory material summarized in Sect. 8.2.
- 2.
For more on the notion of fusion, see Remark 1 at the end of this introduction.
- 3.
Here we use the so-called Schrödinger picture of quantum mechanics . A physically equivalent alternative view, the Heisenberg picture, has the states remaining constant in time, while the operators modeling properties of the state evolve by conjugation with a one-parameter group of unitary operators.
- 4.
A few discrete symmetries can be modeled by anti-unitary transformations.
- 5.
In the infinite-dimensional case, the description is similar but one must take into account the possibility of a continuous spectrum of the operator, in addition to or instead of discrete eigenvalues.
- 6.
- 7.
There are set-theoretic issues if \(\mathscr {C}\) is a proper class rather than a set, but these issues need not concern us here. The finiteness conditions imposed on our anyon category \(\mathscr {A}\) ensure that it is equivalent to a small, i.e., set-sized, category.
- 8.
We have chosen to regard \(V_1\) and \(V_\tau \) as each being a single space, independent of the parenthesization. The different parenthesizations give (possibly) different bases for these spaces. An alternative view is that each parenthesization gives its own \(V_1\) and \(V_\tau \), isomorphic to \(\mathbb {C}\) and \(\mathbb {C}^2\) respectively, with their standard bases, while \(\alpha \) gives an isomorphism between the two \(V_1\)’s and an isomorphism between the two \(V_\tau \)’s. The two viewpoints are easily intertranslatable and the computations that follow would be the same in either picture.
References
Bagchi, B., & Misra, G. (2000). A note on the multipliers and projective representations of semi-simple Lie groups. Sankhyā: The Indian Journal of Statistics, 62, 425–432.
Bonderson, P. H. (2007). Non-abelian anyons and interferometry. Ph.D. thesis, California Institute of Technology.
Freyd, P. (1964). Abelian categories: An introduction to the theory of functors. Harper & Row.
Giulini, D. (2007). Superselection rules. http://arxiv.org/pdf/0710.1516.pdf.
Kauffman, L., & Lomonaco, S. (2010). Topological quantum information theory. In S. Lomonaco (Ed.), Quantum information science and its contribution to mathematics. Proceedings of symposia in pure mathematics (Vol. 68, pp. 103–177). American Mathematical Society.
Kitaev, A. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics, 303, 2–30. http://arxiv.org/abs/quant-ph/9707021.
Mac Lane, S. (1971). Categories for the working mathematician. Graduate Texts in Mathematics 5. Springer.
Nayak, C., Simon, S., Stern, A., Freedman, M., & Das Sarma, S. (2008). Non-abelian anyons and topological quantum computation. Reviews of Modern Physics, 80, 1083–1159.
Panangaden, P., & Paquette, É. O. (2011). A categorical presentation of quantum computation with anyons, Chapter 15. In B. Coecke (Ed.), New structures for physics. Lecture Notes in Physics (Vol. 813, pp. 983–1025). Springer.
Raghunathan, M. S. (1994). Universal central extensions. Reviews of Modern Physics, 6, 207–225.
Wang, Z. (2010). Topological quantum computation. CBMS Regional Conference Series in Mathematics (Vol. 112). American Mathematical Society.
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Blass, A., Gurevich, Y. (2016). On Quantum Computation, Anyons, and Categories. In: Omodeo, E., Policriti, A. (eds) Martin Davis on Computability, Computational Logic, and Mathematical Foundations. Outstanding Contributions to Logic, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-41842-1_8
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