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.