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.
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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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