Bi-directional quantum teleportation of GHZ-like states

In this paper we propose a method through which $n$-qubit states can simultaneously be bi-directionally transmitted between two users. We assume that Alice and Bob, the legitimate users, each have a $n$-qubit GHZ-like state and want to teleport it to the other party. Also, a four-qubit cluster state plays the role of the quantum channel of this bi-directional quantum teleportation. The protocol is based on the method that at first, each user, through a series of $\mbox{CNOT}$ gates, converts the $n$-qubit state into a single qubit and some $0$ qubits. Then, by means of the Bell state measurement and proper operation, the single qubit state is transferred over the channel between the two sides. By re-applying the $\mbox{CNOT}$ gates on the transmitted qubits and auxiliary $0$ states, each user reconstructs the initial GHZ-like state. Finally, we investige the effects of some kind of noises on the density marix of the channel due to its interaction with the environment and present a method to protect the channel against the bit-flip error.