Three-tangle does not properly quantify tripartite entanglement for Greenberger-Horne-Zeilinger-type states

Abstract

Some mixed states composed of only Greenberger-Horne-Zeilinger (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, ${\ensuremath{\Pi}}_{GHZ}$, is explicitly presented in this Rapid Communication. These results are used to compute analytically the three-tangle of a rank-4 mixed state $\ensuremath{\sigma}$ composed of four GHZ states. This analysis with considering Bloch sphere ${S}^{16}$ of $d=4$ qudit system allows us to derive the hyperpolyhedron. It is shown that the states in this hyperpolyhedron have vanishing three-tangle. Computing the one-tangles for ${\ensuremath{\Pi}}_{GHZ}$ and $\ensuremath{\sigma}$, we prove the monogamy inequality explicitly. Making use of the fact that the three-tangle of ${\ensuremath{\Pi}}_{GHZ}$ 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.