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, ${\Pi }_{GHZ}$, is explicitly presented in this Rapid Communication. These results are used to compute analytically the three-tangle of a rank-4 mixed state $\sigma$ 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 ${\Pi }_{GHZ}$ 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.

• Received 17 January 2009

DOI:https://doi.org/10.1103/PhysRevA.80.010301