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Greenberger-Horne-Zeilinger Symmetry in Four Qubit System

Like a three-qubit Greenberger-Horne-Zeilinger(GHZ) symmetry we explore a corresponding symmetry in the four-qubit system, which we call GHZ$_4$ symmetry. While whole GHZ-symmetric states can be represented by two real parameters, the whole set of the GHZ$_4$-symmetric states is represented by three real parameters. In the parameter space all GHZ$_4$-symmetric states reside inside a tetrahedron. We also explore a question where the given SLOCC class of the GHZ$_4$-symmetric states resides in the tetrahedron. Among nine SLOCC classes we have examined five SLOCC classes, which results in three linear hierarchies $L_{abc_2} \subset L_{a_4} \subset L_{a_2b_2} \subset G_{abcd}$, $L_{a_20_{3\oplus\bar{1}}} \subset G_{abcd}$, and $L_{0_{3\oplus\bar{1}}0_{3\oplus\bar{1}}} \subset G_{abcd}$ which hold, at least, in the whole set of the GHZ$_4$-symmetric states. Difficulties arising in the analysis of the remaining SLOCC classes are briefly discussed.