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

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.

9 pages, 10 figures, revtex4

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Cited by in corpus (12)

• Quantifying entanglement resources
• Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information
• Freudenthal triple classification of three-qubit entanglement
• Entanglement of three-qubit random pure states
• Geometric local invariants and pure three-qubit states
• Invariant-based entanglement monotones as expectation values and their experimental detection
• A brief introduction to multipartite entanglement
• Many-body entanglement: Permutations and equivalence classes
• Global asymmetry of many-qubit correlations: A lattice gauge theory approach
• SU(2) Invariants of Symmetric Qubit States
• Degrees of entanglement for multipartite systems
• Lorentz invariants of pure three-qubit states