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

TL;DR

This paper classifies entanglement types of extended GHZ-symmetric states, deriving conditions for separability and analyzing how entanglement classes depend on state parameters, with implications for higher-qubit systems.

Contribution

It provides an analytical framework for entanglement classification of extended GHZ-symmetric states using witnesses and symmetry mappings.

Findings

Separable states characterized analytically.

Entanglement classes depend on the sum of certain parameters.

Discussion on extending analysis to higher-qubit systems.

Abstract

In this paper we analyze entanglement classification of extended Greenberger-Horne-Zeilinger-symmetric states $\rho^{ES}$, which is parametrized by four real parameters $x$, $y_1$, $y_2$ and $y_3$. The condition for separable states of $\rho^{ES}$ 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 or map from the extended Greenberger-Horne-Zeilinger symmetry to the Greenberger-Horne-Zeilinger symmetry. From this analysis we guess that the entanglement classes of $\rho^{ES}$ are not dependent on $y_j \hspace{.2cm} (j=1,2,3)$ individually, but dependent on $y_1 + y_2 + y_3$ collectively. The difficulty arising in extension of analysis with Greenberger-Horne-Zeilinger symmetry to the higher-qubit system is discussed.