Exhaustive Generation of Orthomodular Lattices with Exactly One Non-Quantum State

TL;DR

This paper introduces a method for generating orthomodular lattices with a single state that do not correspond to Hilbert space properties, revealing smaller examples than previously known.

Contribution

It presents a reverse Kochen-Specker theorem and uses MMP algorithms to generate the smallest such orthomodular lattices, expanding understanding of their structure.

Findings

Generated smallest known OMLs with 35-38 atoms and blocks.

Most generated OMLs admit exactly one state.

Discovered properties of these OMLs, contrasting with larger known examples.

Abstract

We propose a kind of reverse Kochen-Specker theorem that amounts to generating orthomodular lattices (OMLs) with exactly one state that do not admit properties of the Hilbert space. We apply MMP algorithms to obtain smallest OMLs with 35 atoms and 35 blocks (35-35) and all other ones up to 38-38. We find out that all but one of them admit exactly one state and discover several other properties of theirs. Previously known such OMLs have 44 atoms and 44 blocks or more. We present some of them in our notation.