Remarks on orthogonality spaces

TL;DR

We address how graphs embed into orthogonality spaces arising from projection lattices and more general orthomodular lattices. The paper provides a finite-example separation showing a graph occurring in the orthogonality space of $\mathcal{P}(\mathbb{C}^3)$ but not in $\mathcal{P}(\mathbb{R}^3)$, and develops constructive embedding methods that yield every finite graph as a fully embedded orthogonality graph of a finite orthomodular lattice and, in particular, of an atomic oml. The methods combine Greechie-diagram encodings, Kalmbach coatom extensions, and pasting techniques, together with a compactness argument to realize the result for atomic omls. Collectively, the results illuminate the expressive power of orthogonality spaces beyond standard Hilbert-space projections and show how quantum-logical structures can encode arbitrary graphs.

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.

Figures (2)

• Figure 1: A partial configuration
• Figure 2: Diagram for $\langle u\rangle$ to be a center of the block of $\langle a_1\rangle$, $\langle a_2\rangle$, $\langle a_3\rangle$.

Theorems & Definitions (23)

• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4
• Theorem 2.5
• proof
• Theorem 2.6
• proof