Pure spinors, impure spinors and quantum mechanics

The geometry of spinors in higher dimensional spaces is used to elucidate a potential ambiguity in the concept of a pure quantum state, and a `toroidal entropy is introduced to provide a measure of the geometrical `impurity' of spinors. The geometry of the sub-manifold of geometrically pure spinors is described. The relationship of toroidal entropy with the preparation of a pure quantum state is discussed. It is shown that the toroidal entropy is trivial in 3 dimensions or for a single qubit system, but may be relevant to the physics of general quantum computation. A generalization of these concepts to general Lie group representations is also presented.