We investigate the relation between the entanglement and the robustness of a multipartite system to a depolarization noise. We find that the robustness of a two-qubit system in an arbitrary pure state depends completely on its entanglement. However, this is not always true in a three-qubit system. There is a residual effect on the robustness of a three-qubit system in an arbitrary superposition of a Greenberger-Horne-Zeilinger state and a
W state. Its entanglement determines the trend of its robustness. However, there is a splitting on its robustness under the same entanglement. Its robustness not only has the same periodicity as its three-tangle but also alters with its three-tangle synchronously. There is also a splitting on the robustness of an
n-qubit (
n>3) system although it is more complicated.
In practice, the interaction between quantum system and environment will lead to the decoherence of entanglement. By quantifying the entanglement through the correlation tensor, we systematically investigate the entanglement robustness of two-qubit states under the four types of noisy channels and corresponding correlated channels. We use an analytical method to find all of the most robust and fragile states under the noisy channels. The puzzling phenomenon that the most robust and fragile states have the same entanglement under noisy channels has been investigated. Moreover, we compare the entanglement robustness of a three-qubit Greenberger-Horne-Zeilinger state with a
W state. Our method shows the behavior of entanglement under the noisy channels and characterizes the entanglement robustness of the multiqubit system analytically.
Abstract We study the robustness of entanglement for W and GHZ mixed states of three and four qubits. By using the definition of the robustness of entanglement and constructing multi-partite fully separable states, we obtain the analytical upper bound of the robustness of entanglement. By analyzing the entanglement witness , we obtain the lower bound of the robustness of entanglement. Then we find that the upper bound and the lower bound are fully coincident, which shows that our results are accurate and valuable.