Three-Party Entanglement in Tripartite Teleportation Scheme through Noisy Channels

We have tried to interpret the physical role of the three-tangle and $\pi$-tangle in the real physical information process. For the model calculation we adopt the three-party teleportation scheme through the various noisy channels. The three parties consist of sender, accomplice and receiver. It is shown that the $\pi$-tangles for the X- and Z-noisy channels vanish at $\kappa t \to \infty$ limit, where $\kappa t$ is a parameter introduced in the master equation of Lindblad form. In this limit the receiver's maximum fidelity reduces to the classical limit 2/3. However, this nice feature is not maintained at the Y- and isotropy-noise channels. For Y-noise channel the $\pi$-tangle vanishes at $0.61 \leq \kappa t$. At $\kappa t = 0.61$ the receiver's maximum fidelity becomes 0.57, which is much less than the classical limit. Similar phenomenon occurs at the isotropic noise channel. We also computed the three-tangles analytically for the X- and Z-noise channels. The remarkable fact is that the three-tangle for Z-noise channel is exactly same with the corresponding $\pi$-tangle. In the X-noise channel the three-tangle vanishes at $0.10 \leq \kappa t$. At $\kappa t = 0.10$ the receiver's fidelity can be reduced to the classical limit provided that the accomplice performs the measurement appropriately. However, the receiver's maximum fidelity becomes 8/9, which is much larger than the classical limit. Since the Y- and isotropy-noise channels are rank-8 mixed states, their three-tangles are not computed explicitly. Instead, we have derived their upper bounds with use of the analytical three-tangles for other noisy channels. Our analysis strongly suggests that we need different three-party entanglement measure whose value is between three-tangle and $\pi$-tangle.