Genuine tripartite entanglement and quantum phase transition

A new formulation called as entanglement measure for simplification, is presented to characterize genuine tripartite entanglement of $(2\times 2\times n)-$dimensional quantum pure states. The formulation shows that the genuine tripartite entanglement can be described only on the basis of the local $(2\times 2)-$dimensional reduced density matrix. In particular, the two exactly solvable models of spin system studied by Yang (Phys. Rev. A \textbf{71}, 030302(R) (2005)) is reconsidered by employing the entanglement measure. The results show that a discontinuity in the first derivative of the entanglement measure or in the entanglement measure itself of the ground state just corresponds to the existence of quantum phase transition, which is obviously prior to concurrence. Hence, the given entanglement measure may become a new alternate candidate to help study the connection between quantum entanglement and quantum phase transitions.