Bipartite and multipartite entanglement in entangled graphs

Document Type : Full length research Paper

Authors

1 Department of Physics, Payame Noor University, Tehran, Iran

2 Department of Physics, Payame noor University, Tehran, Iran

Abstract

In this study, we have obtained a parametric relationship for the entanglement measurement between each pair of qubits for graphs with more than four qubits. We have also calculated the value of entanglement in five-qubit entangled graphs. Analysis of our results shows that the total of 1024 five-qubit entangled graphs based on the maximum entanglement between each pair of qubits, categorizes into 31 groups and if we consider the number of graph edges and degrees of vertices then these states are classified in 40 classes. Also based on the numerical results obtained from multipartite entanglement measures such as generalized concurrence, global measurements, and Meyer-Wallach measurements, we show that 1024 graphs of the five-qubit system are in 24, 32 and 23 categories, respectively. As well as we conclude that the maximum amount of entanglement belongs to the cycle graph and the minimum value belongs to the one-edge graph. While the maximum amount of entanglement between each pair of qubits is in the one-edge graph and the minimum value is in the complete graph.

Keywords

Main Subjects


[1] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement, Reviews of Modern Physics 81 (2009) 865-942.
 
[2] M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University Press, (2000).
 
[3] M. Plesch and V. Bu┼żek, Entagled graph: Bipartite entanglement in multiqubit systems, Physical Review A 67 (2003) 012322, pp. 1-6.
 

[4] W. Dür, Multipartite entanglement that is robust against disposal of particles, Physical Review A 63 (2001) 020303(R) pp. 1-4.

 
[5] C. Sabín and G. García-Alcaine, A classification of entanglement in three-qubit systems, The European Physical Journal  D 48 (2008) 435-442.
 
[6] P. Jakubczyk, Y. Kravets, and D. Jakubczyk, Entanglement of one-magnon Schur-Weyl states, The European Physical Journal  D 61 (2011) 507-512.
 

[7] M. Gharahi Ghahi and S.J. Akhtarshenas, Entangled graphs: a classification of four-qubit entanglement, The European Physical Journal  D 70 (2016) 54-59.

 
[8] L. Assadi and M. Jafarpour, Classification of 4-qubit entagled graph states according to bipartite entanglement, multipartite entanglement and non-local properties,International Journal of Theoretical Physics 55 (2016) 4809-4821.
 
[9] R. Diestel, Graph theory, Springer, Heidelberg, (2010).
 
[10] M. Hein, J. Eisert and H.J. Briegel, Multiparty entanglement in graph states, Physical Review A69 (2004) 062311, pp. 1-20.
 
[11] M. Hein, W. Dur, J. Eisert, R. Raussendorf, M. Van den Nest and H.J. Briegel, Entanglement in graph states and its application, Proc. Int. School Phys. Enrico Fermi. Quantum Computers, Algorithms and Chaos. 162 (2006), pp. 1-115.
 

[12] H. Ma, F. Li, N. Mao, et al., Network-based arbitrated quantum signature scheme with graph State, International Journal of Theoretical Physics 56 (2017) 2551-2561.

 

[13] L. Jian-Wu, L. Xiao-Shu, S. Jin-Jing, et al., Multiparty quantum blind signature scheme based on graph states, International Journal of Theoretical Physics 57 (2018) 2404-2414.

 

[14] A. Akhound, S. Haddadi and M.A.Chaman Motlagh, Calculation of entanglement in graph states up to five-qubit based on generalized concurrence, arXiv:1610.02560 (2016), 1-5.

 
[15] S. Hill and W.K. Wootters, Entanglement of a pair of quantum bits, Physical Review Letters 78 (1997) 5022-5025.
 
[16] W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Physical Review Letters 80 (1998) 2245-2248.
 
[17] A.R.R. Carvalho et al., Decoherence and multipartite entanglement, Physical Review Letters 93 (2004) 230501, 1-4.
 
[18] X.N. Zhu and Sh.M. Fei, Lower bound of concurrence for qubit systems,Quantum Information Processing 13 (2014) 815-823.
 
[19] X.N. Zhu, M. Li and Sh.M. Fei, A lower bound of concurrence for multipartite quantum systems, Quantum Information Processing 17 (2018) 30-39.
 
[20] D.A. Meyer and N.R. Wallach, Global entanglement in multiparticle systems, Journal of Mathematical Physics 43 (2002) 4273-4278.
 
[21] P.J. Love, et al., A characterization of global entanglement, Quantum Information Processing 6 (2007) 187-195.
 
[22] A.J. Scott, Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions,Physical Review A 69 (2004) 052330, 1-10.