arXiv:1108.5339 · Journal of Mathematical Physics

Density conditions for quantum propositions

Hans Havlicek, Karl Svozil

Published Nov 1, 1996• 10 citations

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 {u,v,w}, successive application of the binary logical operation (x,y)⟼(x∨y)⊥ 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 {u,v,w} is orthogonal to the other ones.

DOI: 10.1063/1.531738

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