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Laplacian matrices of weighted digraphs represented as quantum states

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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.

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References

  1. Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935)

    Article  ADS  MATH  Google Scholar 

  2. Bell, J.S.: On the Einstein–Podolsky–Rosen paradox. Physics (Long Island City, N. Y.) 1, 195–200 (1964)

    Google Scholar 

  3. Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998)

    Article  ADS  Google Scholar 

  4. Miyake, A.: Classification of multipartite entangled states by multidimensional determinants. Phys. Rev. A 67(012108), 1–10 (2003)

    MathSciNet  Google Scholar 

  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) 283–299

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chai Wah, Wu.: Multipartite separability of Laplacian matrices of graphs. Electron. J. Combin. 16(1) R61:(2009)

  9. Dutta, Supriyo, Adhikari, Bibhas, Banerjee, Subhashish: A graph theoretical approach to states and unitary operations. Quantum Inf. Process. 15(5), 2193–2212 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Dutta, S., Supriyo, B., Banerjee, S., Srikanth, R.: Bipartite separability and non-local quantum operations on graphs. Phys. Rev. 94, 012306 (2016)

    Article  ADS  Google Scholar 

  11. Reff, Nathan: Spectral properties of complex unit gain graphs. Linear Algebra Appl. 436(9), 3165–3176 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bapat, R.B.: Graphs and Matrices, Ist Edition edn. Hindustan Book Agency, New Delhi, India (2011)

    MATH  Google Scholar 

  13. Bapat, R.B., Kalita, D., Pati, S.: On weighted directed graphs. Linear Algebra Appl. 436(1), 99–111 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Drago, Cvetkovic, Rowlinson, Peter, Simic, Slobodan K.: Signless Laplacians of finite graphs. Linear Algebra Appl. 423(1), 155–171 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  16. Greenberger, D.M., Horne, M.A., Shimony, A., Zeilinger, A.: Bell’s theorem without inequalities. A. J. Phys. 58, 1131–1143 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Dür, W., Vidal G., Cirac, J. I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A. 62(6), 062314 (2000)

  18. Briegel, H.J., Raussendorf, R.: Persistent entanglement in arrays of interacting particles. Phys. Rev. Lett. 86, 910–913 (2001)

    Article  ADS  Google Scholar 

  19. Yeo, Y., Chua, W.K.: Teleportation and dense coding with genuine multipartite entanglement. Phys. Rev. Lett. 96, 060502(1)–060502(4) (2006)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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. Ozols, M., Mancinska, L.: Generalized Bloch vector and the eigenvalues of a density matrix. http://home.lu.lv/~sd20008/papers/Bloch%20Vectors%20and%20Eigenvalues

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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.

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Authors and Affiliations

  1. Department of Mathematics, IIT Kharagpur, Kharagpur, India

    Bibhas Adhikari

  2. Department of Physics, IIT Jodhpur, Jodhpur, India

    Subhashish Banerjee

  3. Department of Mathematics, BIT Mesra, Ranchi, India

    Satyabrata Adhikari

  4. Department of Chemistry, IIT Jodhpur, Jodhpur, India

    Atul Kumar

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  1. Bibhas Adhikari
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  2. Subhashish Banerjee
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  3. Satyabrata Adhikari
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  4. Atul Kumar
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Correspondence to Subhashish Banerjee.

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Adhikari, B., Banerjee, S., Adhikari, S. et al. Laplacian matrices of weighted digraphs represented as quantum states. Quantum Inf Process 16, 79 (2017). https://doi.org/10.1007/s11128-017-1530-1

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