My work is focused on categorical and algebraic structures and their applications in quantum information.
- With Jamie Vicary I am investigating quantum protocols in the situation of reference frame misalignment:
- 'Perfect tight quantum teleportation without a shared reference frame.' (https://arxiv.org/abs/1710.01060). This paper shows that for finite reference frames perfect teleportation may often be achieved with no communication of reference frame information, using algebraic structures called 'equivariant unitary error bases'. We investigate these structures and completely classfiy them for qubit systems. (A very early version of this work appeared in the proceedings of QPL 2016 as 'Tight reference frame independent quantum teleportation'.)
- A paper on application of this approach to infinite reference frames is in preparation.
- In 'Coherence for braided and symmetric pseudomonoids' (I extend the coherence theorems for braided and symmetric monoidal categories to arbitrary braided and symmetric pseudomonoids using diagrammatic 'movie' calculus. This work is a categorification of the theory of PROs, PROBs and PROPs for monoids and commutative monoids. I recently gave two talks about this work at HDRA2017 and PSSL 101; the slides are available here.
- Benjamin Musto, David Reutter and I have proposed a theory of quantum sets and quantum functions, which brings together ideas in noncommutative topology and quantum information. The first paper in this project is available here. In a forthcoming paper we apply this framework to the investigation of quantum graph isomorphisms. We are particularly interested in quantum graph isomorphisms between nonisomorphic graphs, which give instances of quantum pseudo-telepathy.