Exploring Quantum Contextuality with the Quantum Moebius-Escher-Penrose hypergraph

TL;DR

This paper introduces a novel hypergraph model inspired by paradoxical objects to explore quantum contextuality, demonstrating how quantum and classical representations differ and highlighting violations of classical probability conditions.

Contribution

The paper constructs the quantum Moebius-Escher-Penrose hypergraph using Hilbert space representations and analyzes its contextuality through classical and quantum probability frameworks.

Findings

Demonstrates violations of exclusivity and completeness in the hypergraph.

Shows the hypergraph's embedding within Boolean algebra and quantum frameworks.

Highlights the inherent contextuality via violations of Boole's conditions.

Abstract

This paper presents the quantum Moebius-Escher-Penrose hypergraph, drawing inspiration from paradoxical constructs such as the Moebius strip and Penrose's `impossible objects'. The hypergraph is constructed using faithful orthogonal representations in Hilbert space, thereby embedding the graph within a quantum framework. Additionally, a quasi-classical realization is achieved through two-valued states and partition logic, leading to an embedding within a Boolean algebra. This dual representation delineates the distinctions between classical and quantum embeddings, with a particular focus on contextuality, highlighted by violations of exclusivity and completeness, quantified through classical and quantum probabilities. The study also examines violations of Boole's conditions of possible experience using correlation polytopes, underscoring the inherent contextuality of the hypergraph. These results offer deeper insights into quantum contextuality and its intricate relationship with classical logic structures.