Remarks on orthogonality spaces

TL;DR

This paper explores the structure of orthogonality spaces, demonstrating how various graphs can be embedded within them, with implications for quantum logic and the mathematical foundations of quantum theory.

Contribution

It introduces new results on the embedding of finite and countable graphs into orthogonality spaces, including those derived from orthomodular lattices.

Findings

A finite graph from mutually unbiased bases exists in $ ext{C}^3$ but not in $ ext{R}^3$.

Every finite graph can be embedded in the orthogonality space of a finite orthomodular lattice.

All graphs can be embedded in the orthogonality space of some atomic orthomodular lattice.

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.