Restoration of three-qubit entanglements and protection of tripartite quantum state sharing over noisy channels via environment-assisted measurement and reversal weak measurement

The restoration of three-qubit entanglement is investigated under the amplitude damping (AD) decoherence with environment-assisted measurement (EAM) and reversal weak measurement (RWM). The results show that there exists a critical strength of RWM dependent of the initial three-qubit entangled state under a given damping rate of the AD channel, i.e., if the selected RWM strength is higher than the critical strength, the entanglement will be reduced compared to one without RWM. Some three-qubit entangled states cannot be restored. We calculated the restorable condition of the initial entanglement and illustrated the valid area for three-qubit GHZ state and W state. Fortunately, an optimal strength of RWM corresponding to a certain damping rate of AD channels can be found within the valid area for a restorable initial state, by which a noise-infected entanglement can be restored to its maximum value. Particularly, when three qubits of W state are subjected to their respective AD channels, due to the symmetry of three qubits, the W state cannot be decohered provided the EAM is successful, and no RWM is required. This is beneficial to quantum communication over the noisy channel. Applying this protection regime to tripartite QSS and taking appropriate initial entangled state as the quantum channel, the fidelity of the shared state can be improved to the maximum 1 probabilistically. Thus, the decoherence effect of the noisy channels can be significantly suppressed or even avoided.