Orbit classification and analysis of qutrit graph states under local complementation and local scaling

TL;DR

This paper characterizes entanglement classes of 7-qutrit graph states using orbit analysis under local operations, revealing connections between graph properties and entanglement measures, and providing computational tools for qudit quantum information.

Contribution

It extends local complementation analysis from qubits to qutrits, offering a complete classification and new insights into the structure of entanglement in higher-dimensional systems.

Findings

Complete orbit classification for 7-qutrit graph states.

Identified correlations between graph connectivity and entanglement measures.

Provided computational tools for orbit extraction and analysis.

Abstract

Graph states and their entanglement properties are pivotal for the development of quantum computing and technologies. For qubits, local complementation, a graphical rule that connects all the equivalent states under Local Clifford (LC) operations, was used for the complete characterization of all the LC equivalence classes up until 12 particles, assisting applications in quantum error correction and state preparation protocols optimization. This concept has been extended for qudits. In this work, we provide a complete characterization of the entanglement classes up until 7 qutrits, mapping each class into an orbit. The graph-theoretic properties of the orbits are studied, illuminating the rich structure they have. Clear connections between the connectivity of the orbits and the entanglement properties are observed. The correlations between the graph-theoretic properties and the Schmidt measure provide useful insights regarding qudit state preparation and fault-tolerance. This work is accompanied by a repository consisting of the code to extract the orbits and perform the presented statistical analysis. The strong interplay between quantum theory and graph theory is well-known and extensively studied for qubits. Our work provides practical tools and results to assist this endeavor for the qudit case.