Quantum Information Processing

, Volume 14, Issue 12, pp 4691–4713 | Cite as

Quantum state representation based on combinatorial Laplacian matrix of star-relevant graph



In this paper the density matrices derived from combinatorial Laplacian matrix of graphs is considered. More specifically, the paper places emphasis on the star-relevant graph, which means adding certain edges on peripheral vertices of star graph. Initially, we provide the spectrum of the density matrices corresponding to star-like graph (i.e., adding an edge on star graph) and present that the Von Neumann entropy increases under the graph operation (adding an edge on star graph) and the graph operation cannot be simulated by local operation and classical communication (LOCC). Subsequently, we illustrate the spectrum of density matrices corresponding to star-alike graph (i.e., adding one edge on star-like graph) and exhibit that the Von Neumann entropy increases under the graph operation (adding an edge on star-like graph) and the graph operation cannot be simulated by LOCC. Finally, the spectrum of density matrices corresponding to star-mlike graph (i.e., adding m nonadjacent edges on the peripheral vertices of star graph) is demonstrated and the relation between the graph operation and Von Neumann entropy, LOCC is revealed in this paper.


Quantum state representation Combinatorial Laplacian matrix Star-relevant graph Von Neumann entropy 



This project was supported by NSFC (Grant Nos. 61272514, 61170272, 61121061, 61411146001,61202082), NCET (Grant No. NCET-13-0681), the National Development Foundation for Cryptological Research (Grant No. MMJJ201401012) and the Fok Ying Tung Education Foundation (Grant No. 131067).


  1. 1.
    Von Neumann, J.: Mathematical Foundations of Quantum Mechanics, vol. 2, p. 5. Princeton University Press, Princeton (1955)Google Scholar
  2. 2.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information, . pp. 101573–101578. Cambridge University Press, Cambridge (2010)Google Scholar
  3. 3.
    Balachandran, A.P., et al.: Algebraic approach to entanglement and entropy. Phys. Rev. A 88(2), 022301 (2013)CrossRefADSGoogle Scholar
  4. 4.
    Adhikari, B., Adhikari, S., Banerjee, S.: Graph representation of quantum states. arXiv preprint arXiv:1205.2747 (2012)
  5. 5.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bennett, C.H., et al.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70(13), 1895 (1993)MathSciNetCrossRefADSMATHGoogle Scholar
  7. 7.
    Bouwmeester, D., et al.: Experimental quantum teleportation. Nature 390(6660), 575–579 (1997)CrossRefADSGoogle Scholar
  8. 8.
    Bennett, C.H., Wiesner, S.J.: Communication via one-and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69(20), 2881 (1992)MathSciNetCrossRefADSMATHGoogle Scholar
  9. 9.
    Liu, X.S., et al.: General scheme for superdense coding between multiparties. Phys. Rev. A 65(2), 022304 (2002)CrossRefADSGoogle Scholar
  10. 10.
    Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59(3), 1829 (1999)MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Chen, X.-B., et al.: Controlled quantum secure direct communication with quantum encryption. Int. J. Quantum Inf. 6(03), 543–551 (2008)CrossRefMATHGoogle Scholar
  12. 12.
    Bennett, C.H., Brassard, G.: Quantum cryptography: public key distribution and coin tossing. Theor. Comput. Sci. 560, 7–11 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Liu, B., Lai, H.-J.: Matrices in Combinatorics and Graph Theory, vol. 3. Springer Science & Business Media, New York (2013)Google Scholar
  14. 14.
    Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Springer Science & Business Media, New York (2011)MATHGoogle Scholar
  15. 15.
    Newman, M.: Networks: An Introduction. Oxford University Press, Oxford (2010)CrossRefMATHGoogle Scholar
  16. 16.
    Kocay, W., Kreher, D.L.: Graphs, Algorithms, and Optimization. CRC Press, Boca Raton (2004)MATHGoogle Scholar
  17. 17.
    Hein, M., et al.: Entanglement in graph states and its applications. arXiv preprint quant-ph/0602096 (2006)
  18. 18.
    Braunstein, S.L., Ghosh, S., Severini, S.: The Laplacian of a graph as a density matrix: a basic combinatorial approach to separability of mixed states. Ann. Comb. 10(3), 291–317 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hildebrand, R., Mancini, S., Severini, S.: Combinatorial laplacians and positivity under partial transpose. Math. Struct. Comput. Sci. 18(01), 205–219 (2008)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Hassan, A., Saif, M., Joag, P.S.: A combinatorial approach to multipartite quantum systems: basic formulation. J. Phys. A Math. Theor. 40(33), 10251 (2007)CrossRefADSMathSciNetMATHGoogle Scholar
  21. 21.
    Hassan, A., Saif, M., Joag, P.S.: On the degree conjecture for separability of multipartite quantum states. J. Math. Phys. 49(1), 012105 (2008)MathSciNetCrossRefADSMATHGoogle Scholar
  22. 22.
    Wu, C.W.: Conditions for separability in generalized Laplacian matrices and diagonally dominant matrices as density matrices. Phys. Lett. A 351(1), 18–22 (2006)MathSciNetCrossRefADSMATHGoogle Scholar
  23. 23.
    Wu, C.W.: On graphs whose Laplacian matrixs multipartite separability is invariant under graph isomorphism. Discret. Math. 310(21), 2811–2814 (2010)CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Hassan, A., Saif M.: Detection and Quantification of Entanglement in Multipartite Quantum Systems Using Weighted Graph and Bloch Representation of States. arXiv preprint arXiv:0905.0312 (2009)
  25. 25.
    Passerini, F., Severini, S.: The von Neumann entropy of networks. Available at SSRN 1382662 (2008)Google Scholar
  26. 26.
    Watrous, J.: Theory of Quantum Information, p. 128. University of Waterloo Fall, Waterloo (2011)Google Scholar
  27. 27.
    Horn, R.A., Johnson, C.R.: Matrix Analysis, pp. 387. Cambridge University Press, Cambridge (2012)Google Scholar
  28. 28.
    Chitambar, E., et al.: Everything you always wanted to know about LOCC (but were afraid to ask). Commun. Math. Phys. 328(1), 303–326 (2014)MathSciNetCrossRefADSMATHGoogle Scholar
  29. 29.
    Nielsen, M.A., Vidal, G.: Majorization and the interconversion of bipartite states. Quantum Inf. Comput. 1(1), 76–93 (2001)MathSciNetMATHGoogle Scholar
  30. 30.
    Marshall, A.W., Olkin, I., Arnold, B.: Inequalities: Theory of Majorization and Its Applications. Springer Science & Business Media, New York (2010)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Information Security Center, State Key Laboratory of Networking and Switching TechnologyBeijing University of Posts and TelecommunicationsBeijingChina

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