Exploring quantum teleportation through unitary error bases

TL;DR

This paper presents a new categorical framework for understanding Latin squares and their role in constructing unitary error bases, which are crucial for quantum information tasks like teleportation.

Contribution

It introduces a categorical axiomatisation of Latin squares, provides a graphical proof of Werner's construction, and proposes a generalized scheme for creating new unitary error bases.

Findings

Categorical axioms simplify Latin square requirements

Graphical proof of Werner's construction correctness

Proposed generalized construction for new error bases

Abstract

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.