Asymmetric Controlled Bidirectional Remote Preparation of Single- and Three-Qubit Equatorial State in Noisy Environment.

We present an asymmetric controlled bidirectional remote preparation scheme under the control of Charlie, where Alice transmits an arbitrary single-qubit equatorial state to Bob and Bob transmits an arbitrary three-qubit equatorial state to Alice. First, the quantum channel is constructed by using Hadamard (H) and CNOT operations. Second, we consider our scheme in the noiseless environment and construct the appropriate measurement bases. In addition, the participants can recover the prepared states through implementing the corresponding recovery operations determinately. Third, we discuss our scheme in five types of noisy environments (amplitude-damping, phase-damping, bit-flip, phase-flip, and depolarizing noisy environment). The fidelities of the output states are calculated, which depend on the coefficients of the prepared states and the decoherence rates. Finally, some discussions and conclusions are given. Compared with previous schemes, our scheme does not need additional operations or auxiliary qubits. So it is slightly more efficient.