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Unitary error bases have a great number of applications across quantuminformation and quantum computation, and are fundamentally linked to quantumteleportation, dense coding and quantum error correction. Werner'scombinatorial construction builds a unitary error basis from a family ofHadamard matrices and a Latin square. In this dissertation, I give a newcategorical axiomatisation of Latin squares, and use this to give a fullygraphical presentation and proof of the correctness of Werner's construction.The categorical approach makes clear that some of the Latin square axioms areunnecessary for the construction to go through, and I propose a generalisedconstruction scheme with the potential to create new classes of unitary errorbases.

Exploring quantum teleportation through unitary error bases