  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.
