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Classification of 4-qubit Entangled Graph States According to Bipartite Entanglement, Multipartite Entanglement and Non-local Properties

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Abstract

We use concurrence to study bipartite entanglement, Meyer-Wallach measure and its generalizations to study multi-partite entanglement and MABK and SASA inequalities to study the non-local properties of the 4-qubit entangled graph states, quantitatively. Then, we present 3 classifications, each one in accordance with one of the aforementioned properties. We also observe that the classification according to multipartite entanglement does exactly coincide with that according to nonlocal properties, but does not match with that according to bipartite entanglement. This observation signifies the fact that non-locality and multipartite entanglement enjoy the same basic underlying principles, while bipartite entanglement may not reveal the non-locality issue in its entirety.

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Authors and Affiliations

  1. Physics Department, Shahid Chamran University of Ahvaz, Ahvaz, Iran

    Leila Assadi & Mojtaba Jafarpour

Authors
  1. Leila Assadi
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  2. Mojtaba Jafarpour
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Correspondence to Mojtaba Jafarpour.

Appendices

Appendix: 1

Defining

$$\left( {\alpha^{2}+\frac{\gamma^{2}}{2}} \right)^{2}=A, \quad \left( {\beta^{2}+\frac{\gamma^{2}}{2}} \right)^{2}=B, \quad \left( {\alpha^{2}+\frac{\gamma^{2}}{6}} \right)^{2}=C, $$
$$\left( {\beta^{2}+\frac{\gamma^{2}}{6}} \right)^{2}=D, \quad \left( {\alpha^{2}+\frac{3\gamma^{2}}{5}} \right)^{2}=F, \quad \left( {\beta^{2}+\frac{2\gamma^{2}}{5}} \right)^{2}=G, $$
$$\left( {\alpha^{2}+\frac{2\gamma^{2}}{5}} \right)^{2}=H, \quad \left( {\beta^{2}+\frac{3\gamma^{2}}{5}} \right)^{2}=I, \quad \left( {\alpha^{2}+\frac{\gamma^{2}}{5}} \right)^{2}=J, $$
$$\left( {\beta^{2}+\frac{\gamma^{2}}{5}} \right)^{2}=K, \quad \left( {\alpha^{2}+\frac{3\gamma^{2}}{4}} \right)^{2}=L, \quad \left( {\beta^{2}+\frac{\gamma^{2}}{4}} \right)^{2}=M, $$
$$\left( {\alpha^{2}+\frac{\gamma^{2}}{4}} \right)^{2}=N, \quad \left( {\beta^{2}+\frac{3\gamma^{2}}{4}} \right)^{2}=O, \quad \alpha^{4}+\beta^{4}+\gamma^{4}=Z, $$
$$\left( {\alpha^{2}+\frac{\gamma^{2}}{3}} \right)^{2}=P, \quad \left( {\beta^{2}+\frac{2\gamma^{2}}{3}} \right)^{2}=Q, \quad \left( {\alpha^{2}+\frac{2\gamma^{2}}{3}} \right)^{2}=R, $$
$$\gamma^{2}\left( {\alpha +\beta } \right)^{2}=E, \quad \left( {\beta^{2}+\gamma^{2}} \right)^{2}=T, \quad \left( {\alpha^{2}+\gamma^{2}} \right)^{2}=Y, \quad \left( {\beta^{2}+\frac{\gamma^{2}}{3}} \right)^{2}=S, $$

We obtain

Table 5
Full size table

Appendix: 2

Table 5 MABK and SASA inequalities for the 4-qubit entangled graphs
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Assadi, L., Jafarpour, M. Classification of 4-qubit Entangled Graph States According to Bipartite Entanglement, Multipartite Entanglement and Non-local Properties. Int J Theor Phys 55, 4809–4821 (2016). https://doi.org/10.1007/s10773-016-3104-x

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