Greenberger-Horne-Zeilinger Symmetry in Four Qubit System

TL;DR

This paper investigates a four-qubit GHZ symmetry, characterizing symmetric states with three parameters, and explores their SLOCC class hierarchies within a geometric tetrahedral representation.

Contribution

It introduces the concept of GHZ$_4$ symmetry for four-qubit systems and analyzes the SLOCC class hierarchies within the symmetric state set.

Findings

GHZ$_4$-symmetric states form a tetrahedral parameter space.

Five SLOCC classes exhibit three linear hierarchy relations.

Certain SLOCC classes' analysis faces difficulties.

Abstract

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.