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Similar to the three-qubit Greenberger-Horne-Zeilinger (GHZ) symmetry weexplore the four-qubit GHZ symmetry group and its subgroup called restrictedGHZ symmetry group. While the set of symmetric states under the whole grouptransformation is represented by three real parameters, the set of symmetricstates under the subgroup transformation is represented by two real parameters.After comparing the symmetric states for whole and subgroup, the entanglementis examined for the latter set. It is shown that the set has only two SLOCCclasses, $L_{abc_2}$ and $G_{abcd}$. Extension to the multi-qubit system isbriefly discussed.

Entanglement Classification of Restricted Greenberger-Horne-Zeilinger Symmetric States in Four-Qubit System