Remarks on orthogonality spaces

Abstract

We provide two results. The first gives a finite graph constructed from consideration of mutually unbiased bases that occurs as a subgraph of the orthogonality space of ${\mathbb{C}}^3$ but not of that of ${\mathbb{R}}^3$. The second is a companion result to the result of Tau and Tserunyan \cite{Tau} that every countable graph occurs as an induced subgraph of the orthogonality space of a Hilbert space. We show that every finite graph occurs as an induced subgraph of the orthogonality space of a finite orthomodular lattice and that every graph occurs as an induced subgraph of the orthogonality space of some atomic orthomodular lattice.