My work is focused on categorical and algebraic structures and their applications in quantum mechanics. Some key words are: topological quantum field theory, quantum information, categorical quantum mechanics, tensor categories, bicategories, etc. Some particular areas of research are indicated below.
- I have investigated categorical quantum mechanics in the category of finite dimensional unitary representations of a compact group. Several applications have already been developed:
- If one takes the group action to instantiate reference frame transformations, one can perform perfect reference frame independent quantum teleportation for finite group actions using certain algebraic structures within this category analogous to those used in normal quantum teleportation protocols. A very early version of this work appeared in the proceedings of QPL 2016. http://arxiv.org/pdf/1603.08866.pdf. A much newer version with a complete classification of such protocols for a qubit will appear soon. This is joint work with Jamie Vicary.
- This analysis is extended to increase purity of teleportation for infinite group actions, where perfect teleportation is impossible. This is also joint work with Jamie Vicary.
- I am working on applications of this approach to device-dependent RFI QKD.
- In higher algebra, I have extended the coherence theorems for braided and symmetric monoidal categories to arbitrary braided and symmetric pseudomonoids using a higher diagrammatic rewriting approach. This work is a categorification of the theory of PROs, PROBs and PROPs for monoids and commutative monoids.
- With Benjamin Musto I have been investigating constructions of quantum Latin squares and their relation to their classical equivalents, particuarly with regard to computational complexity of their completion.