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arXiv:1305.2012 · Physical Review A

Entanglement classification of restricted Greenberger-Horne-Zeilinger–symmetric states in a four-qubit system

DaeKil Park

Published May 27, 2014• 2 citations
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Abstract

Similar to the three-qubit Greenberger-Horne-Zeilinger (GHZ) symmetry we explore the four-qubit GHZ symmetry group and its subgroup, the 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 group and subgroup, the entanglement is examined for the latter set. It is shown that the set has only two stochastic local operations and classical communication classes, ${L}_{ab{c}_{2}}$ and ${G}_{abcd}$. Extension to the multiqubit system is briefly discussed.

DOI
10.1103/physreva.89.052326

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Like a three-qubit Greenberger-Horne-Zeilinger(GHZ) symmetry we explore a corresponding symmetry in the four-qubit system, which we call GHZ4_44​ symmetry. While whole GHZ-symmetric states can be represented by two real parameters, the whole set of the GHZ4_44​-symmetric states is represented by three real parameters. In the parameter space all GHZ4_44​-symmetric states reside inside a tetrahedron. We also explore a question where the given SLOCC class of the GHZ4_44​-symmetric states resides in the tetrahedron. Among nine SLOCC classes we have examined five SLOCC classes, which results in three linear hierarchies Labc2⊂La4⊂La2b2⊂GabcdL_{abc_2} \subset L_{a_4} \subset L_{a_2b_2} \subset G_{abcd}Labc2​​⊂La4​​⊂La2​b2​​⊂Gabcd​, La203⊕1ˉ⊂GabcdL_{a_20_{3\oplus\bar{1}}} \subset G_{abcd}La2​03⊕1ˉ​​⊂Gabcd​, and L03⊕1ˉ03⊕1ˉ⊂GabcdL_{0_{3\oplus\bar{1}}0_{3\oplus\bar{1}}} \subset G_{abcd}L03⊕1ˉ​03⊕1ˉ​​⊂Gabcd​ which hold, at least, in the whole set of the GHZ4_44​-symmetric states. Difficulties arising in the analysis of the remaining SLOCC classes are briefly discussed.
The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamental problems in quantum physics. We study this topic in three-qubit systems considering the entanglement classification of stochastic local operations and classical communication (SLOCC). The equivalence under SLOCC divides three-qubit states into separable, biseparable, WWW, and Greenberger-Horne-Zeilinger (GHZ) classes. The WWW and GHZ are two subclasses of genuine tripartite entanglement. We adopt the family of GHZ-symmetric states as a research subject, which share the symmetries of the GHZ state and have a complete characterization of SLOCC classes. In the biseparable region (with bipartite entanglement), there exist GHZ-symmetric states that are found to be fully local. In addition, there are bilocal states in both the WWW and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.
In this paper we analyze entanglement classification of extended Greenberger-Horne-Zeilinger-symmetric states ρESρ^{ES}ρES, which is parametrized by four real parameters xxx, y1y_1y1​, y2y_2y2​ and y3y_3y3​. The condition for separable states of ρESρ^{ES}ρES 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 ρESρ^{ES}ρES are not dependent on yj(j=1,2,3)y_j \hspace{.2cm} (j=1,2,3)yj​(j=1,2,3) individually, but dependent on y1+y2+y3y_1 + y_2 + y_3y1​+y2​+y3​ collectively. The difficulty arising in extension of analysis with Greenberger-Horne-Zeilinger symmetry to the higher-qubit system is discussed.
The first characterization of mixed-state entanglement was achieved for two-qubit states in Werner's seminal work [Phys. Rev. A 40, 4277 (1989)]. A physically important extension concerns mixtures of a pure entangled state [such as the Greenberger-Horne-Zeilinger (GHZ) state] and the unpolarized state. These mixed states serve as benchmark for the robustness of multipartite entanglement. They share the symmetries of the GHZ state. We call such states GHZ symmetric. Here we give a complete description of the entanglement in the family of three-qubit GHZ-symmetric states and, in particular, of the three-qubit generalized Werner states. Our method relies on the appropriate parametrization of the states and on the invariance of entanglement properties under general local operations. An application is the definition of a symmetrization witness for the entanglement class of arbitrary three-qubit states.
In this paper we analyze entanglement classification of relaxed Greenberger-Horne-Zeilinger-symmetric states ρES\rho^{ES}ρES, which is parametrized by four real parameters xxx, y1y_1y1​, y2y_2y2​ and y3y_3y3​. The condition for separable states of ρES\rho^{ES}ρES 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 relaxed Greenberger-Horne-Zeilinger symmetry to the Greenberger-Horne-Zeilinger symmetry. From this analysis we guess that the entanglement classes of ρES\rho^{ES}ρES are not dependent on yj(j=1,2,3)y_j \hspace{.2cm} (j=1,2,3)yj​(j=1,2,3) individually, but dependent on y1+y2+y3y_1 + y_2 + y_3y1​+y2​+y3​ collectively. The difficulty arising in extension of analysis with Greenberger-Horne-Zeilinger symmetry to the higher-qubit system is discussed.
Mixed states appear naturally in experiment over pure states. So for studying different notions of nonlocality and their relation with entanglement in realistic scenarios, one needs to consider mixed states. In a recent paper [Phys. Rev. Lett. 108, 020502 (2012)], a complete characterization of entanglement of an entire class of mixed three-qubit states with the same symmetry as the Greenberger-Horne-Zeilinger state, known as GHZ-symmetric states, has been achieved. In this paper we investigate different notions of nonlocality of the same class of states. By finding the analytical expressions of maximum violation value of most efficient Bell inequalities we obtain the conditions of standard nonlocality and genuine nonlocality of this class of states. Also the relation between entanglement and nonlocality is discussed for this class of states. Interestingly, genuine entanglement of GHZ-symmetric states is necessary to reveal standard nonlocality. However, it is not sufficient to exploit the same.