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Speed of disentanglement in multi-qubit systems under depolarizing channel

We investigate the speed of disentanglement in the multiqubit systems under the local depolarizing channel, in which each qubit is independently coupled to the environment. We focus on the bipartition entanglement between one qubit and the remaining qubits constituting the system, which is measured by the negativity. For the two-qubit system, the speed for the pure state completely depends on its entanglement. The upper and lower bounds of the speed for arbitrary two-qubit states, and the necessary conditions for a state achieving them, are obtained. For the three-qubit system, we study the speed for pure states, whose entanglement properties can be completely described by five local-unitary-transformation invariants. An analytical expression of the relation between the speed and the invariants is derived. The speed is enhanced by the the three-tangle which is the entanglement among the three qubits, but reduced by the the two-qubit correlations outside of the concurrence. The decay of the negativity can be restrained by the other two negativity with the coequal sense. The unbalance between two qubits can reduce speed of disentanglement of the remaining qubit in the system, even can retrieve the entanglement partially. For the k-qubit systems in an arbitrary superposition of GHZ state and W state, the speed depends almost entirely on the amount of the negativity when k increases to five or six. An alternative quantitative definition for the robustness of entanglement is presented based on the speed of disentanglement, with comparison to the widely studied robustness measured by the critical amount of noise parameter where the entanglement vanishes. In the limit of large number of particles, the alternative robustness of the GHZ-type states is inversely proportional to k, and the one of the W states approaches 1/\sqrt{k}.