Three-tangle and Three-π for a class of tripartite mixed states

Abstract: We study the tripartite entanglement for a class of mixed states defined by the mixture of GHZ and W states, \rho=p|GHZ><GHZ|+(1-p)|W><W|. Based on the Caratheodory theorem and the periodicity assumption, the possible optimal decomposition of the states has been derived, which is not independent on the detailed measure of entanglement. We find that, according to p, there are two different decompositions containing 3 or 4 quantum states in the decomposition respectively. When the decomposition contains 3 quantum states, the tripartite entanglement of the mixed state is simply the entanglement of superposition states of GHZ and W. When the decomposition contains 4 quantum states, the tripartite entanglement of the mixed state is a liner function of p. We also study the relations between the three-tangle and three-\pi. It is shown that the three-tangle is smaller than the three-\pi. Moreover, the three-\pi\ has a minimal point in the interval 0 and 1, while the three-tangle is a non decreasing function of p.