  Unitary error bases have a great number of applications across quantum
information and quantum computation, and are fundamentally linked to quantum
teleportation, dense coding and quantum error correction. Werner's
combinatorial construction builds a unitary error basis from a family of
Hadamard matrices and a Latin square. In this dissertation, I give a new
categorical axiomatisation of Latin squares, and use this to give a fully
graphical presentation and proof of the correctness of Werner's construction.
The categorical approach makes clear that some of the Latin square axioms are
unnecessary for the construction to go through, and I propose a generalised
construction scheme with the potential to create new classes of unitary error
bases.
