Three-party entanglement in tripartite teleportation scheme through noisy channels

Abstract

In this paper we have tried to interpret the physical role of three-tangle and $\pi$-tangle in real physical information processes. For the model calculation we adopt the tripartite teleportation scheme through 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 the limit $\kappa t \rightarrow \infty$, where $\kappa t$ is a decoherence parameter introduced in the master equation in the Lindblad form. At this limit the maximum fidelity of the receiver's state reduces to the classical limit $2/3$. However, this nice feature is not maintained for the Y- and isotropy-noise channels. For the Y-noise channel the $\pi$-tangle vanishes when $0.61 \leq \kappa t$. At $\kappa t = 0.61$ the maximum fidelity becomes $0.57$, which is much less than the classical limit. Similar phenomenon occurs for the isotropic noise channel. We also compute analytically the three-tangles for the X- and Z-noise channels. The remarkable fact is that the three-tangle for the Z-noise channel coincides exactly with the corresponding $\pi$-tangle. In the X-noise channel the three-tangle vanishes when $0.10 \leq \kappa t$. At $\kappa t = 0.10$ the fidelity of the receiver's state can reduce to the classical limit provided that the accomplice performs the measurement appropriately. However, the 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 in this paper. Instead, their upper bounds are derived by making use of the analytic formulas of the three-tangle for other noisy channels. Our analysis strongly suggests that different tripartite entanglement measure is needed whose value is between three-tangle and $\pi$-tangle.