Qubit Logic, Algebra and Geometry

Abstract

A qubit is a two-state quantum system, in which one bit of binary information can be stored and recovered. A qubit differs from an ordinary bit in that it can exist in a complex linear combination of its two basis states, where combinations differing by a factor are identified. This projective line, in turn, can be regarded as an entity within a Clifford or geometric algebra, which endows it with both an algebraic structure and an interpretation as a Euclidean unit 2-sphere. Testing a qubit to see if it is in a basis state generally yields a random result, and attempts to explain this in terms of random variables parametrized by the points of the spheres of the individual qubits lead to contradictions. Geometric reasoning forces one to the conclusion that the parameter space is a tensor product of projective lines, and it is shown how this structure is contained in the tensor product of their geometric algebras.

Acknowledgements

The author thanks David Cory (MIT) and Chris Doran (Cambridge) for useful discussions on the topics covered herein. This work was supported by the U. S. Army Research Office under grant DAAG 55-97-1-0342 from DARPA/MTO.