  With a two step optimization method of entanglement witness, we analytically
propose a set of necessary and sufficient entanglement criteria for four qubit
symmetric Greenberger-Horne-Zeilinger (GHZ) diagonal states. The criterion set
contains four criteria. Two of them are linear with density matrix elements.
The other two criteria are nonlinear with density matrix elements. The
criterion set has a nest structure. A proper subset of the criteria is
necessary and sufficient for the entanglement of a proper subset of the states.
We illustrate the nest structure of criterion set with the general Werner state
set and its superset the highly symmetric GHZ diagonal state set, they are
subsets of the symmetric GHZ diagonal state set.
