Entanglement and Closest Product States of Graph States with 9 to 11 Qubits


The numbers of local complimentary inequivalent graph states for 9, 10 and 11 qubit systems are 440, 3132, 40457, respectively. We calculate the entanglement, the lower and upper bounds of the entanglement and obtain the closest product states for all these graph states. New patterns of closest product states are analyzed.

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Wang, C. , Jiang, L. and Wang, L. (2013) Entanglement and Closest Product States of Graph States with 9 to 11 Qubits. Journal of Applied Mathematics and Physics, 1, 51-55. doi: 10.4236/jamp.2013.14010.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] R. Raussendorf, D. E. Browne and H. J. Briegel, “Measurement-Based Quantum Computation on Cluster States,” Physical Review A, Vol. 68, No. 2, 2003, Article ID: 022312. http://dx.doi.org/10.1103/PhysRevA.68.022312
[2] D.-M. Schlingemann, Quant. Inf. Comp., Vol. 2, 2002, p. 307.
[3] D.-M. Schlingemann, Quant. Inf. Comp., Vol. 4, 2002, p. 287.
[4] M. Hein, J. Eisert and H. J. Briegel, “Multiparty Entanglement in Graph States,” Physical Review A, Vol. 69, No. 6, 2004, Article ID: 062311. http://dx.doi.org/10.1103/PhysRevA.69.062311
[5] M. Hein, W. Dur, J. Eisert, R. Raussendorf, M. Van den Nest and H. J. Briegel, In G. Casati, D. L. Shepelyansky, P. Zoller and G. Benenti, Eds., Quantum Computers, Algorithms and Chaos, IOS Press, Amsterdam, 2006.
[6] M. Grassl, A. Klappenecker and M. Rotteler, “Graphs, Quadratic Forms, and Quantum Codes,” Proceedings of 2002 IEEE International Symposium on Information Theory, Lausanne, Switzerland, p. 45. http://dx.doi.org/10.1109/ISIT.2002.1023317
[7] R. Raussendorf and H. J. Briegel, “A One-Way Quantum Computer,” Physical Review Letters, Vol. 86, No. 22, 2001, pp. 5188-5191. http://dx.doi.org/10.1103/PhysRevLett.86.5188
[8] G. Vidal and R. Tarrach, “Robustness of Entanglement,” Physical Review A, Vol. 59, No. 1, 1999, pp. 141-150. http://dx.doi.org/10.1103/PhysRevA.59.141
[9] V. Vedral, M. B. Plenio, M. A. Rippin and P. L. Knight, “Quantifying Entanglement,” Physical Review Letters, Vol. 78, No. 12, 1997, pp. 2275-2279. http://dx.doi.org/10.1103/PhysRevLett.78.2275
[10] V. Vedral and M. B. Plenio, “Entanglement Measures and Purification Procedures,” Physical Review A, Vol. 57, No. 3, 1998, pp. 1619-1633. http://dx.doi.org/10.1103/PhysRevA.57.1619
[11] T.-C. Wei and P. M. Goldbart, “Geometric Measure of Entanglement and Applications to Bipartite and Multipartite Quantum States,” Physical Review A, Vol. 68, No. 4, 2003, Article ID: 042307. http://dx.doi.org/10.1103/PhysRevA.68.042307
[12] M. Hayashi, D. Markham, M. Murao, M. Owari and S. Virmani, “Entanglement of Multiparty-Stabilizer, Symmetric, and Anti-symmetric States,” Physical Review A, Vol. 77, No. 1, 2008, Article ID: 012104. http://dx.doi.org/10.1103/PhysRevA.77.012104
[13] X. Y. Chen, “Entanglement of Graph States up to Eight Qubits,” Journal of Physics B, Vol. 43, No. 8, 2010, Article ID: 085507. http://dx.doi.org/10.1088/0953-4075/43/8/085507
[14] A. Cabello, A. J. Lopez-Tarrida, P. Moreno and J. R. Portillo, “Entanglement in Eight-Qubit Graph States,” Physics Letters A, Vol. 373, No. 26, 2009, pp. 2219-2225. http://dx.doi.org/10.1016/j.physleta.2009.04.055
[15] M. Hayashi, D. Markham, M. Murao, M. Owari and S. Virmani, “Bounds on Multipartite Entangled Orthogonal State Discrimination Using Local Operations and Classical Communication,” Physical Review Letters, Vol. 96, No. 4, 2006, Article ID: 040501. http://dx.doi.org/10.1103/PhysRevLett.96.040501
[16] D. Markham, A. Miyake and S. Virmani, “Entanglement and Local Information Access for Graph States,” New Journal of Physics, Vol. 9, 2007, pp. 194. http://dx.doi.org/10.1088/1367-2630/9/6/194
[17] L. E. Danielsen, “Database of Self-Dual Quantum Codes.” http://www.ii.uib.no/larsed/vncorbits/

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