We propose a method of constructing the separability criteria for multipartite quantum states on the basis of entanglement witnesses. The entanglement witnesses are obtained by finding the maximal expectation values of Hermitian operators and then optimizing over all possible Hermitian operators. We derive a set of tripartite separability criteria for the four-qubit Greenberger–Horne–Zeilinger (GHZ) diagonal states. The derived criterion set contains four criteria that are necessary and sufficient for the tripartite separability of the highly symmetric four-qubit GHZ diagonal states; the criteria completely account for the numerically obtained boundaries of the tripartite separable state set. One of the criteria is just the tripartite separability criterion of the four-qubit generalized Werner states.
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
Labc2⊂La4⊂La2b2⊂Gabcd,
La203⊕1ˉ⊂Gabcd, and
L03⊕1ˉ03⊕1ˉ⊂Gabcd 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.