Density conditions for quantum propositions

TL;DR

This paper explores the density of elementary quantum propositions in the projective plane, showing that a basis without orthogonal vectors leads to a dense set through a specific logical operation.

Contribution

It demonstrates that starting from a basis with no orthogonal vectors, the generated propositions form a countably infinite dense set in the projective plane.

Findings

Generated propositions are dense if and only if the basis vectors are not orthogonal.

The set of propositions becomes countably infinite and dense under the specified operation.

Orthogonality of basis vectors prevents the propositions from being dense.

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.