Linear orthogonality spaces as a new approach to quantum logic

TL;DR

This paper explores a minimalistic approach to quantum logic using orthogonality spaces, focusing on their structure and relation to Hermitian spaces, especially in finite cases of rank up to 3.

Contribution

It introduces a new perspective on quantum logic through orthogonality spaces and characterizes their connection to Hermitian spaces in finite cases.

Findings

Orthogonality spaces of rank at most 3 are analyzed.

Conditions under which orthogonality spaces relate to Hermitian spaces are identified.

The approach simplifies understanding of quantum logic structures.

Abstract

The notion of an orthogonality space was recently rediscovered as an effective means to characterise the essential properties of quantum logic. The approach can be considered as minimalistic; solely the aspect of mutual exclusiveness is taken into account. In fact, an orthogonality space is simply a set endowed with a symmetric and irreflexive binary relation. If the rank is at least $4$ and if a certain combinatorial condition holds, these relational structures can be shown to give rise in a unique way to Hermitian spaces. In this paper, we focus on the finite case. In particular, we investigate orthogonality spaces of rank at most $3$.