We investigate two-party quantum teleportation through noisy channels for multi-qubit Greenberger-Horne-Zeilinger (GHZ) states and find which state loses less quantum information in the process. The dynamics of states is described by the master equation with the noisy channels that lead to the quantum channels to be mixed states. We analytically solve the Lindblad equation for $n$-qubit GHZ states $nin{4,5,6}$ where Lindblad operators correspond to the Pauli matrices and describe the decoherence of states. Using the average fidelity we show that 3GHZ state is more robust than $n$GHZ state under most noisy channels. However, $n$GHZ state preserves same quantum information with respect to EPR and 3GHZ states where the noise is in $x$ direction in which the fidelity remains unchanged. We explicitly show that Jung ${it et, al.}$ conjecture [Phys. Rev. A ${bf 78}$, 012312 (2008)], namely, average fidelity with same-axis noisy channels are in general larger than average fidelity with different-axis noisy channels is not valid for 3GHZ and 4GHZ states.
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