Greenberger-Horne-Zeilinger Symmetry in Four Qubit System

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 $L_{ab{c}_2}\subset L_{a_4}\subset L_{a_2{b}_2}\subset G_{abcd}$, $L_{a_2{0}_{3\oplus \bar{1}}}\subset G_{abcd}$, 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.

26 pages, 9 figures. arXiv admin note: text overlap with arXiv:1305.2012

References in corpus (4)

• Entangled three-qubit states without concurrence and three-tangle
• Inductive Entanglement Classification of Four Qubits under SLOCC
• Four-qubit entanglement from string theory
• Entanglement of three-qubit Greenberger-Horne-Zeilinger-symmetric states