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

TL;DR

This paper classifies entanglement in four-qubit states with restricted GHZ symmetry, identifying only two SLOCC classes and discussing potential extensions to larger systems.

Contribution

It introduces a classification of four-qubit symmetric states under a restricted GHZ symmetry group, revealing only two SLOCC classes and extending the analysis to multi-qubit systems.

Findings

The set of symmetric states under the subgroup has two SLOCC classes.

Comparison with the whole group shows differences in entanglement classification.

Extension to multi-qubit systems is briefly discussed.

Abstract

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