Laplacian matrices of weighted digraphs represented as quantum states

  • Bibhas Adhikari
  • Subhashish Banerjee
  • Satyabrata Adhikari
  • Atul Kumar
Article

Abstract

Representing graphs as quantum states is becoming an increasingly important approach to study entanglement of mixed states, alternate to the standard linear algebraic density matrix-based approach of study. In this paper, we propose a general weighted directed graph framework for investigating properties of a large class of quantum states which are defined by three types of Laplacian matrices associated with such graphs. We generalize the standard framework of defining density matrices from simple connected graphs to density matrices using both combinatorial and signless Laplacian matrices associated with weighted directed graphs with complex edge weights and with/without self-loops. We also introduce a new notion of Laplacian matrix, which we call signed Laplacian matrix associated with such graphs. We produce necessary and/or sufficient conditions for such graphs to correspond to pure and mixed quantum states. Using these criteria, we finally determine the graphs whose corresponding density matrices represent entangled pure states which are well known and important for quantum computation applications. We observe that all these entangled pure states share a common combinatorial structure.

Keywords

Combinatorial Laplacian Signless Laplacian Eigenvalues Pure and mixed states Density matrix Quantum entanglement 

Notes

Acknowledgements

This work is partially supported by CSIR (Council of Scientific and Industrial Research) Grant No. 25(0210)/13/EMR-II, New Delhi, India. SB wishes to thank Supriyo Dutta for his help in carefully reading the manuscript and suggesting a number of changes.

References

  1. 1.
    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935)ADSCrossRefMATHGoogle Scholar
  2. 2.
    Bell, J.S.: On the Einstein–Podolsky–Rosen paradox. Physics (Long Island City, N. Y.) 1, 195–200 (1964)Google Scholar
  3. 3.
    Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998)ADSCrossRefGoogle Scholar
  4. 4.
    Miyake, A.: Classification of multipartite entangled states by multidimensional determinants. Phys. Rev. A 67(012108), 1–10 (2003)MathSciNetGoogle Scholar
  5. 5.
    Sunada, T.: A discrete analogue of periodic magnetic Schr\(\ddot{o}\)dinger operators, Geometry of the spectrum, Contemp. Math., Amer. Math. Soc., Providence, RI (Seattle, WA, 1993), 173 (1994) 283–299Google Scholar
  6. 6.
    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, 291–317 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ali, Hassan, Saif, M., Pramod, S., Joag, A.: combinatorial approach to multipartite quantum systems: basic formulation. J. Phys. A Math. Theor. 40(33), 10251 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chai Wah, Wu.: Multipartite separability of Laplacian matrices of graphs. Electron. J. Combin. 16(1) R61:(2009)Google Scholar
  9. 9.
    Dutta, Supriyo, Adhikari, Bibhas, Banerjee, Subhashish: A graph theoretical approach to states and unitary operations. Quantum Inf. Process. 15(5), 2193–2212 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dutta, S., Supriyo, B., Banerjee, S., Srikanth, R.: Bipartite separability and non-local quantum operations on graphs. Phys. Rev. 94, 012306 (2016)ADSCrossRefGoogle Scholar
  11. 11.
    Reff, Nathan: Spectral properties of complex unit gain graphs. Linear Algebra Appl. 436(9), 3165–3176 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bapat, R.B.: Graphs and Matrices, Ist Edition edn. Hindustan Book Agency, New Delhi, India (2011)MATHGoogle Scholar
  13. 13.
    Bapat, R.B., Kalita, D., Pati, S.: On weighted directed graphs. Linear Algebra Appl. 436(1), 99–111 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Drago, Cvetkovic, Rowlinson, Peter, Simic, Slobodan K.: Signless Laplacians of finite graphs. Linear Algebra Appl. 423(1), 155–171 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Chai Wah, Wu.: Conditions for separability in generalized Laplacian matrices and diagonally dominant matrices as density matrices, IBM Research Report RC23758(W0508-118)(2005)Google Scholar
  16. 16.
    Greenberger, D.M., Horne, M.A., Shimony, A., Zeilinger, A.: Bell’s theorem without inequalities. A. J. Phys. 58, 1131–1143 (1990)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Dür, W., Vidal G., Cirac, J. I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A. 62(6), 062314 (2000)Google Scholar
  18. 18.
    Briegel, H.J., Raussendorf, R.: Persistent entanglement in arrays of interacting particles. Phys. Rev. Lett. 86, 910–913 (2001)ADSCrossRefGoogle Scholar
  19. 19.
    Yeo, Y., Chua, W.K.: Teleportation and dense coding with genuine multipartite entanglement. Phys. Rev. Lett. 96, 060502(1)–060502(4) (2006)ADSCrossRefGoogle Scholar
  20. 20.
    Brown, I.D.K., Stepney, S., Sudbery, A., Braunstein, S.L.: Searching for highly entangled multi-qubit states. J. Phys. A 38, 1119–1131 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Man, Z.X., Xia, Y.J., An, N.Ba: Genuine multiqubit entanglement and controlled teleportation. Phys. Rev. A 75, 05306(1)–05306(5) (2006)Google Scholar
  22. 22.
    Ozols, M., Mancinska, L.: Generalized Bloch vector and the eigenvalues of a density matrix. http://home.lu.lv/~sd20008/papers/Bloch%20Vectors%20and%20Eigenvalues

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsIIT KharagpurKharagpurIndia
  2. 2.Department of PhysicsIIT JodhpurJodhpurIndia
  3. 3.Department of MathematicsBIT MesraRanchiIndia
  4. 4.Department of ChemistryIIT JodhpurJodhpurIndia

Personalised recommendations