Research

Fusion categories have played an important role in understanding structures arising from quantum physics, and lie at the heart of quantum algebra and quantum topology. Some fusion categories can be extended to ribbon fusion categories : since these are endowed with the topological properties of ribbon graphs, they naturally lend themselves to investigation from a skein-theoretic perspective. 

The main narrative of this paper entails extracting information from fusion rules using skein-theoretic methods and a rotation operator. Our focus is on quantum invariants, braid eigenvalues and 6j-symbols associated to these fusion rules.

So far, experimental evidence for the detection of nonabelian anyons in condensed matter systems has proved to be divisive. What if we could demonstrably create and manipulate a topological qubit? This would build a compelling case for the presence of anyons!

With this in mind, we construct simple procedures using 'Ising anyons' for (a) teleportation, and (b) superdense coding. In particular, these procedures do not rely on braiding (and hence, should be easier to realise). Both protocols are presented in the more general (d-ary) setting of 'Tambara-Yamagami' anyons, of which Ising anyons are a special case. We consequently find a braid-free realisation of the Pauli gates using Ising anyons.

As expected from the work of Abramsky & Coecke (2004), teleportation manifests as a zizag-like flow of quantum information in spacetime. In the anyonic setting, this flow is facilitated by the pivotal structure of the underlying fusion category.

See also : https://www.youtube.com/watch?v=9rksGBJBezc , https://www.youtube.com/watch?v=ljIF56U-ogI 

Exchange symmetry is a cornerstone of quantum mechanics, first appearing when we study systems of many identical particles. It tells us, for instance, that all fundamental particles are either bosons or fermions, and has many far-reaching consequences. When we suppress one spatial dimension, exchange symmetry indicates that effectively 2D quantum systems should support quasiparticles called anyons with exotic statistical behaviour.

Algebraically, anyonic systems are modelled using ribbon fusion categories. Our goal here is to patch the gap between the fundamental principle of exchange symmetry and the categorical framework.

By assuming that 2D quasiparticles are spatially localised, we show that the superselection sectors (i.e. the possible topological charges) of anyonic systems form a fusion algebra. Moreover, we prove that the superselection sectors of such systems are uniquely specified by the action of a special braid, which obeys some internal symmetries that characterise the behaviour of anyons.

[Annals of Physics, Vol. 429 (2021)]