Does three-tangle properly quantify the three-party entanglement for Greenberger-Horne-Zeilinger-type states?

Abstract

Some mixed states composed of only GHZ states can be expressed in terms of only W-states. This fact implies that such states have vanishing three-tangle. One of such rank-3 states, ${\Pi }_{GHZ}$, is explicitly presented in this paper. These results are used to compute analytically the three-tangle of a rank-4 mixed state $\sigma$ composed of four GHZ states. This analysis with considering Bloch sphere $S^{16}$ of $d=4$ qudit system allows us to derive the hyper-polyhedron. It is shown that the states in this hyper-polyhedron have vanishing three-tangle. Computing the one-tangles for ${\Pi }_{GHZ}$ and $\sigma$, we prove the monogamy inequality explicitly. Making use of the fact that the three-tangle of ${\Pi }_{GHZ}$ is zero, we try to explain why the W-class in the whole mixed states is not of measure zero contrary to the case of pure states.

10 pages, no figure V2: new calculational results are included. 11 pages: V3 accepted in the Rapid Communication of PRA, 4 pages (two column)

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• Tripartite thermal correlations in an inhomogeneous spin-star system
• Genuine Tripartite Entanglement in a Spin-Star Network at Thermal Equilibrium
• Efficient numerical method to calculate three-tangle of mixed states
• A measure of tripartite entanglement in bosonic and fermionic systems
• Three-Tangle in Non-inertial Frame
• Practical method to obtain a lower bound to the three-tangle
• Tripartite Entanglement Dynamics in the presence of Non-Markovian Environment
• Three-tangle for high-rank mixed states
• Efficient Test to Demonstrate Genuine Three Particle Nonlocality
• Monogamy relations and upper bounds for the generalized $W$-class states using Rényi-$\alpha$ entropy
• Tripartite Entanglement-Dependence of Tripartite Non-locality in Non-inertial Frame
• Geometric approach to entanglement quantification with polynomial measures
• Production of entangled x rays through nonlinear double Compton scattering
• Testing the Monogamy Relations via Rank-2 Mixtures
• Entanglement Classification of Restricted Greenberger-Horne-Zeilinger Symmetric States in Four-Qubit System
• Competition of Direct and Indirect Sources of Thermal Entanglement in a spin star network
• Linear response of tripartite entanglement to infinitesimal noise