Geometric local invariants and pure three-qubit states

TL;DR

This paper introduces a geometric method inspired by lattice gauge theory to generate local invariants for multi-qubit states, providing a new way to analyze quantum entanglement and correlations.

Contribution

It develops a novel geometric framework using Wilson loop-like constructs to generate local invariants for qubits, applicable to pure and mixed states of any dimension.

Findings

Complete set of invariants for three qubits obtained

Framework extends to mixed states and higher-dimensional systems

Operational interpretation of invariants in terms of observables

Abstract

We explore a geometric approach to generating local SU(2) and $SL(2,\mathbb{C})$ invariants for a collection of qubits inspired by lattice gauge theory. Each local invariant or 'gauge' invariant is associated to a distinct closed path (or plaquette) joining some or all of the qubits. In lattice gauge theory, the lattice points are the discrete space-time points, the transformations between the points of the lattice are defined by parallel transporters and the gauge invariant observable associated to a particular closed path is given by the Wilson loop. In our approach the points of the lattice are qubits, the link-transformations between the qubits are defined by the correlations between them and the gauge invariant observable, the local invariants associated to a particular closed path are also given by a Wilson loop-like construction. The link transformations share many of the properties of parallel transporters although they are not undone when one retraces one's steps through the lattice. This feature is used to generate many of the invariants. We consider a pure three qubit state as a test case and find we can generate a complete set of algebraically independent local invariants in this way, however the framework given here is applicable to mixed states composed of any number of $d$ level quantum systems. We give an operational interpretation of these invariants in terms of observables.