Density conditions for quantum propositions

Abstract

As has already been pointed out by Birkhoff and von Neumann, quantum logic can be formulated in terms of projective geometry. In three-dimensional Hilbert space, elementary logical propositions are associated with one-dimensional subspaces, corresponding to points of the projective plane. It is shown that, starting with three such propositions corresponding to some basis $\{\vec{u},\vec{v},\vec{w}\}$, successive application of the binary logical operation $(x,y)\mapsto (x\vee y{)}^{\perp }$ generates a set of elementary propositions which is countable infinite and dense in the projective plane if and only if no vector of the basis $\{\vec{u},\vec{v},\vec{w}\}$ is orthogonal to the other ones.

References in corpus (1)

• Hidden Variables and the Two Theorems of John Bell

Cited by in corpus (3)

• Embedding Quantum Universes into Classical Ones
• Contexts in quantum, classical and partition logic
• Satisfiability of cross product terms is complete for real nondeterministic polytime Blum-Shub-Smale machines