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Entropic measure and hypergraph states

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Abstract

We investigate some properties of the entanglement of hypergraph states in purely hypergraph theoretical terms. We first introduce an approach for computing local entropic measure on qubit \(t\) of a hypergraph state by using the Hamming weight of the so-called \(t\)-adjacent subhypergraph. Then, we quantify and characterize the entanglement of hypergraph states in terms of local entropic measures obtained by using the above approach. Our results show that full-rank hypergraph states of more than two qubits can not be converted into any graph state under local unitary transformations.

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Notes

  1. In fact, the local entropic measure \(E_2^t \left( {\left| \phi \right\rangle } \right) \) is usually given by the smallest eigenvalue of the reduced density matrix \(\rho _t \).

  2. Local maximally entangled states shown in this paper are not the same as locally maximally entangled states in [22]. In fact, Ref. [23] shows that any hypergraph state is locally maximally entangled.

References

  1. Horodecki, R., et al.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  3. Hein, M., Eisert, J., Briegel, H.J.: Multiparty entanglement in graph states. Phys. Rev. A 69, 062311 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Aschauer, H., Dur, W., Briegel, H.J.: Multiparticle entanglement purification for two-colorable graph states. Phys. Rev. A 71, 012319 (2005)

    Article  ADS  Google Scholar 

  5. Raussendorf, R., Browne, D.E., Briegel, H.J.: Measurement-based quantum computation on cluster states. Phys. Rev. A 68, 022312 (2003)

    Article  ADS  Google Scholar 

  6. Raussendorf, R., Briegel, H.J.: A one-way quantum computer. Phys. Rev. Lett. 86, 5188 (2001)

    Article  ADS  MATH  Google Scholar 

  7. Briegel, H.J., Raussendorf, R.: Persistent entanglement in arrays of interacting particles. Phys. Rev. Lett. 86, 910 (2001)

    Article  ADS  Google Scholar 

  8. Gottesman, D.: Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A 54, 1862 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  9. Gottesman, D.: Theory of fault-tolerant quantum computation. Phys. Rev. A 57, 127 (1998)

    Article  ADS  Google Scholar 

  10. Ionicioiu, R., Spiller, T.P.: Encoding graphs into quantum states: an axiomatic approach. Phys. Rev. A 85, 062313 (2012)

    Article  ADS  Google Scholar 

  11. Looi, S.Y., et al.: Quantum-error-correcting codes using qudit graph states. Phys. Rev. A 78, 042303 (2008)

    Article  ADS  Google Scholar 

  12. Menicucci, N.C., Flammia, S.T., van Loock, P.: Graphical calculus for Gaussian pure states. Phys. Rev. A 83, 042335 (2011)

    Article  ADS  Google Scholar 

  13. Verstraete, F., et al.: Criticality, the area law, and the computational power of projected entangled pair states. Phys. Rev. Lett. 96, 220601 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Pérez-García, D., et al.: Quantum Inf. Comput. 8, 650 (2008); N. Schuch et al. Phys. Rev. Lett. 98, 140506 (2007)

    Article  MathSciNet  Google Scholar 

  15. Perseguers, S., et al.: Quantum random networks. Nat. Phys. 6, 539 (2010)

    Article  Google Scholar 

  16. Qu, R., et al.: Encoding hypergraphs into quantum states. Phys. Rev. A 87, 022311 (2013)

    Article  ADS  Google Scholar 

  17. Rossi et al.: arXiv:1211.5554[quant-ph]

  18. Qu, R., et al.: Multipartite entanglement and hypergraph states of three qubits. Phys. Rev. A 87, 032329 (2013)

    Article  ADS  Google Scholar 

  19. Vidal, G.: Entanglement monotones. J. Mod. Opt. 47, 355 (2000)

  20. Eisert, J., Briegel, H.J.: Schmidt measure as a tool for quantifying multiparticle entanglement. Phys. Rev. A 64, 022306 (2001)

    Article  ADS  Google Scholar 

  21. Bruß, D., Macchiavello, C.: Multipartite entanglement in quantum algorithms. Phys. Rev. A 83, 052313 (2011)

    Google Scholar 

  22. Kruszynska, C., Kraus, B.: Local entanglability and multipartite entanglement. Phys. Rev. A 79, 052304 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  23. Qu, R., et al.: Relationship among locally maximally entangleable states, W states, and hypergraph states under local unitary transformations. Phys. Rev. A 87, 052331 (2013)

    Article  ADS  Google Scholar 

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Acknowledgments

This work is supported by the Chinese National Program on Key Basic Research Project (973 Program, Grant Nos. 2014CB744605 and 2013CB329304), the Natural Science Foundation of China (Grants Nos. 61170178, 61272254 and 61272265), and the European Union Framework 7 Marie-Curie International Research Staff Exchange Programme (Grant No. 247590). This work is completed during our academic visit at the Department of Computing, the Open University, United Kingdom.

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Authors and Affiliations

  1. School of Computer Science and Technology, Tianjin University, Tianjin, 300072, China

    Ri Qu, Yi-ping Ma, Yan-ru Bao, Juan Wang & Zong-shang Li

  2. Tianjin Key Laboratory of Cognitive Computing and Application, Tianjin, 300072, China

    Ri Qu, Yi-ping Ma, Yan-ru Bao, Juan Wang & Zong-shang Li

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  1. Ri Qu
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  2. Yi-ping Ma
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  3. Yan-ru Bao
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  4. Juan Wang
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  5. Zong-shang Li
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Correspondence to Yan-ru Bao.

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Qu, R., Ma, Yp., Bao, Yr. et al. Entropic measure and hypergraph states. Quantum Inf Process 13, 249–258 (2014). https://doi.org/10.1007/s11128-013-0646-1

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