Information entropy and the space of decoherence functions in generalized quantum theory

In standard quantum theory, the ideas of information entropy and of pure states are closely linked. States are represented by density matrices ρ on a Hilbert space and the information entropy - tr(ρ ln ρ) is minimized on pure states (pure states are the vertices of the boundary of the convex set of states). The space of decoherence functions in the consistent histories approach to generalized quantum theory is also a convex set. However, by showing that every decoherence function can be written as a convex combination of two other decoherence functions, we demonstrate that there are no ``pure'' decoherence functions. The main content of the paper is a notion of information entropy in generalized quantum mechanics which is applicable in contexts in which there is no a priori notion of time. Information entropy is defined first on consistent sets and then we show that it decreases upon refinement of the consistent set. This information entropy suggests an intrinsic way of giving a consistent set selection criterion.