Classification of qubit entanglement: SL(2,C) versus SU(2) invariance

TL;DR

This paper investigates the limitations of SU(2) invariants for classifying three-qubit entanglement and argues that SL(2,C) invariance is necessary for a complete characterization, highlighting the insufficiency of SU(2) invariants alone.

Contribution

It demonstrates that SU(2) invariants like I_5 are insufficient for entanglement classification, advocating for SL(2,C) invariance as a more complete approach.

Findings

I_5 invariant is independent only with W-type entanglement and threetangle

Constant I_5 can occur across a range of entanglement measures

SL(2,C) invariance is necessary for proper entanglement classification

Abstract

The role of SU(2) invariants for the classification of multiparty entanglement is discussed and exemplified for the Kempe invariant I_5 of pure three-qubit states. It is found to being an independent invariant only in presence of both W-type entanglement and threetangle. In this case, constant I_5 admits for a wide range of both threetangle and concurrences. Furthermore, the present analysis indicates that an SL^3 orbit of states with equal tangles but continuously varying I_5 must exist. This means that I_5 provides no information on the entanglement in the system in addition to that contained in the tangles (concurrences and threetangle) themselves. Together with the numerical evidence that I_5 is an entanglement monotone this implies that SU(2) invariance or the monotone property are too weak requirements for the characterization and quantification of entanglement for systems of three qubits, and that SL(2,C) invariance is required. This conclusion can be extended to general multipartite systems (including higher local dimension) because the entanglement classes of three-qubit systems appear as subclasses.