Geometric local invariants and pure three-qubit states

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.

9 pages, 3 figures

References in corpus (1)

• Constructing N-qubit entanglement monotones from anti-linear operators

Cited by in corpus (14)

• Geometric phases in quantum information
• Entanglement between two subsystems, the Wigner semicircle and extreme value statistics
• Entanglement classes of permutation-symmetric qudit states: symmetric operations suffice
• Lectures on Quantum Tensor Networks
• Classification of Arbitrary Multipartite Entangled States under Local Unitary Equivalence
• Minimal Renyi-Ingarden-Urbanik entropy of multipartite quantum states
• Tensor Network Methods for Invariant Theory
• A Complete Set of Invariants for LU-Equivalence of Density Operators
• Using Partial Transpose and Realignment to generate Local Unitary Invariants
• Vector Properties of Entanglement in a Three-Qubit System
• Simple permutation-based measure of quantum correlations and maximally-3-tangled states
• Global asymmetry of many-qubit correlations: A lattice gauge theory approach
• SU(2) Invariants of Symmetric Qubit States
• Lorentz invariants of pure three-qubit states