Robustness of entanglement for W and Greenberger–Horne–Zeilinger mixed states

Quantum entanglement is one of the most crucial resources in quantum information. Its robustness, in a certain sense, quantifies the tolerance of entanglement against noise and interference. By studying the robustness of entanglement for mixed states composed of entangled pure states, it’s helpful to understand the robustness of entanglement in more general states. In this work, we present an analytical method for evaluating the robustness of entanglement for W and GHZ mixed states involving three and four qubits. By employing the definition of the robustness of entanglement and analyzing the entanglement witness, we establish tight upper and lower bounds for the robustness of these mixed states. The calculated results demonstrate the accuracy of our approach, offering insights for studying the robustness of entanglement for general quantum states.