Mentor Initiated Controlled Bi-directional Remote State Preparation Scheme For \((2 \iff 4)\)-Qubit Entangled States in Noisy Channel

In this paper we present a bi-directional protocol for mutual remote preparation of a two and a four-qubit non-maximally entangled state where the parties intending to remotely prepare the respective states are not initially entangled. There is a controller of the protocol who oversees the performances of other parties and acts to signal for the execution of the final step in the protocol. There is a Mentor whose action creates entanglement between the rest of the parties and also determines one of the several possible courses of the communication scheme. After that the Mentor quits. The effect of three different noises, namely, Bit-flip, Phase-flip and Amplitude-damping noises are analyzed using the Kraus operator on the otherwise perfect protocol. The decreased fidelity in the presence of noise is numerically studied with respect to noise and other parameters. It is found that in all the three cases the fidelity tends to one as the noise parameter tends to zero.