A Representation of Projection Lattices and Their States in Euclidean Space

Abstract

We propose a representationr:L∪Ω→Rν, where L is the collection of closed subspaces of ann-dimensional real, complex, or quaternionic Hilbert space H, or equivalently, the projection lattice of this Hilbert space, where Ω is the set of all states ω:L→[0,1]. The value that ω∈Ω takes ina∈L is given by the scalar product of the representative points (r(a) andr(ω)). The representationr(a∨b) of the join of two orthogonal elementsa,b∈L is equal tor(a)+r(b). The convex closure of the representation of Σ, the set of atoms of L, is equal to the representation of Ω.