
Mathematics 2000
Geometric Algebra in Quantum Information ProcessingAbstract: This paper develops a geometric model for coupled twostate quantum systems (qubits), which is formulated using geometric (aka Clifford) algebra. It begins by showing how Euclidean spinors can be interpreted as entities in the geometric algebra of a Euclidean vector space. This algebra is then lifted to Minkowski spacetime and its associated geometric algebra, and the insights this provides into how density operators and entanglement behave under Lorentz transformations are discussed. The direct sum of multiple copies of spacetime induces a tensor product structure on the associated algebra, in which a suitable quotient is isomorphic to the matrix algebra conventionally used in multiqubit quantum mechanics. Finally, the utility of geometric algebra in understanding both unitary and nonunitary quantum operations is demonstrated on several examples of interest in quantum information processing.
