Graph Approach to Quantum Systems

TL;DR

This paper corrects previous lattice descriptions of 3-dimensional quantum systems using a graph approach, introduces new hypergraph-based lattices, and explores their properties to advance quantum information research.

Contribution

It provides corrected lattice models for 3D quantum systems, introduces a bipartite graph generation method, and links hypergraph structures to quantum information applications.

Findings

Corrected 3D Kochen-Specker lattice descriptions

Generated up to 41 hypergraph lattices with bipartite graphs

Discovered new properties of hypergraph lattices such as superposition and orthoraguesian equations

Abstract

Using a graph approach to quantum systems, we prove that descriptions of 3-dim Kochen-Specker (KS) setups as well as descriptions of 3-dim spin systems by means of Greechie lattices that we find in the literature are wrong. Correct lattices generated by McKay-Megill-Pavicic (MMP) hypergraphs and Hilbert subspace equations are given. To enable exhaustive generations of 3-dim KS setups by means of recently found "stripping technique," bipartite graph generation is used to provide us with lattices with equal numbers of elements and blocks (orthogonal triples of elements) - up to 41 of them. We obtain several new results on such lattices and hypergraphs, in particular on properties such as superposition and orthoraguesian equations. Since a bipartite graph approach has recently been applied to CSS (Calderbank-Shor-Steane) and graph states on the one hand, and span programs, quantum walks, and quantum search on the other, our results also enable the study of these quantum information fields by means of hypergraphs and lattices.