Graph structure of quantum mechanics

TL;DR

This paper develops a graph-theoretic framework to characterize quantum contextuality, revealing that finite and infinite dimensional quantum systems are structured by graphs and atoms, offering a new mathematical perspective on quantum mechanics.

Contribution

It introduces a novel approach using partial Boolean algebra and graph theory to analyze quantum contextuality, extending to infinite dimensions.

Findings

Quantum systems are determined by atoms in graph structures.

Finite dimensional quantum systems are characterized by two graph theorems.

Quantum mechanics can be viewed as a graph-structured combination of hidden-variable theories.

Abstract

The quantum mechanics is proved to admit no hidden-variable in 1960s, which means the quantum systems are contextual. Revealing the mathematical structure of quantum mechanics is a significant task. We develop the approach of partial Boolean algebra to characterize the contextuality theory with local consistency and exclusivity, and then prove that the finite dimensional quantum systems are determined by atoms using two graph structure theorems. We also generalize our work to infinite dimensional cases. Our conclusions indicate that the quantum mechanics is a graph-structured combination of multiple hidden-variable theories, and provide a precise mathematical framework for quantum contextuality.