  A perfect teleportation protocol requires pure maximally shared entangled
states. While in reality the shared entanglement is severely degraded due to
the inevitable interaction with the noisy environment, which leads to mixed
entangled state and extremely deteriorates the performance of teleportation.
Here, we propose a teleportation protocol to teleport an unknown qubit through
the amplitude damping channels with a fidelity up to one with a single copy of
the entangled state. Our proposed teleportation protocol, while illustrated
using the Bell and W entangled states as examples, can be utilized with any
type of entangled states. In our protocol we employ environment-assisted
measurement during the entanglement distribution, and further modify the
original teleportation protocol by applying weak measurement in the last step
of teleportation. We find a balance between teleportation fidelity and success
probability by varying the strength of the weak measurement. Furthermore, we
investigate the protection of controlled teleportation protocols, where all the
qubits of the entangled state pass through the amplitude damping channel. In
particular, for the controlled teleportation with the W state, the decoherence
of the shared entanglement can be totally suppressed by using EAM, hence no
weak measurement is required to achieve an average teleportation fidelity of
unity. The numerical simulation results reveal that our proposed teleportation
protocol outperforms both the weak measurement based probabilistic
teleportation protocol and the original teleportation protocol without
protection.
