Entanglement Classification of extended Greenberger-Horne-Zeilinger-Symmetric States

In this paper we analyze entanglement classification of the extended Greenberger-Horne-Zeilinger-symmetric states $ρ^ES$, which is parametrized by four real parameters $x$, $y_1$, $y_2$ and $y_3$. The condition for separable states of $ρ^ES$ is analytically derived. The higher classes such as bi-separable, W, and Greenberger-Horne-Zeilinger classes are roughly classified by making use of the class-specific optimal witnesses and map from extended Greenberger-Horne-Zeilinger symmetry to Greenberger-Horne-Zeilinger symmetry. From this analysis we guess that the entanglement classes of $ρ^ES$ are not dependent on $y_j (j=1,2,3)$ individually, but dependent on $y_1 + y_2 + y_3$ collectively. The difficulty arising in the extension of analysis with Greenberger-Horne-Zeilinger symmetry to the higher-qubit system is discussed.