Three-tangle for rank-three mixed states: Mixture of Greenberger-Horne-Zeilinger, W, and flipped-W states

Three-tangle for a rank-3 mixture composed of Greenberger-Horne-Zeilinger, W, and flipped-W states is analytically calculated. The optimal decompositions in the full range of parameter space are constructed by making use of the convex-roof extension. We also provide an analytical technique, which determines whether or not an arbitrary rank-3 state has vanishing three-tangle. This technique is developed by making use of the Bloch sphere S-8 of the qutrit system. The Coffman-Kundu-Wootters inequality is discussed by computing one-tangle and concurrences. It is shown that the one-tangle is always larger than the sum of squared concurrences and three-tangle. The physical implication of three-tangle is briefly discussed.