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Quantum Information and Computation

Entanglement classification of relaxed Greenberger-Horne-Zeilinger-symmetric states

Eylee Jung, DaeKil Park

Published Sep 1, 2014• 0 citations
Publisher

Abstract

In this paper we analyze entanglement classification of relaxed 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 relaxed 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.

DOI
10.26421/qic14.11-12-3

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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.
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.
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Recently, a new type of symmetry for three-qubit quantum states was introduced, the so-called Greenberger-Horne-Zeilinger (GHZ) symmetry. It includes the operations which leave the three-qubit standard GHZ state unchanged. This symmetry is powerful as it yields families of mixed states that are, on the one hand, complex enough from the physics point of view and, on the other hand, simple enough mathematically so that their properties can be characterized analytically. We show that by using the properties of GHZ-symmetric states it is straightforward to derive optimal witnesses for detecting class-specific entanglement in arbitrary three-qubit states.
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.
Recently, a new type of symmetry for three-qubit quantum states was introduced, the so-called Greenberger-Horne-Zeilinger (GHZ) symmetry. It includes the operations which leave the three-qubit standard GHZ state unchanged. This symmetry is powerful as it yields families of mixed states that are, on the one hand, complex enough from the physics point of view and, on the other hand, simple enough mathematically so that their properties can be characterized analytically. We show that by using the properties of GHZ-symmetric states it is straightforward to derive optimal witnesses for three-qubit entanglement.
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.