Necessary and sufficient fully separable criterion and entanglement of three-qubit Greenberger-Horne-Zeilinger diagonal states

We analytically prove that the necessary criterion coincides with the sufficient criterion for the full separability of three-qubit Greenberger-Horne-Zeilinger (GHZ) diagonal states. Our result closes the discussion on the separability of three-qubit GHZ diagonal states. Based on the criterion, the corresponding entanglement is exactly calculable for some GHZ diagonal states and is tractable for the others in terms of the relative entropy of entanglement.