Energy as an Entanglement Witnesses for One Dimensional XYZ Heisenberg Lattice: Optimization Approach


If energy ensemble average is less than the minimum energy of separable states, the system is entangled. In this study, we consider energy as an entanglement witness for one dimensional XYZ Heisenberg lattice up to ten qubits analytically. We find minimum of energy using Lagrange undetermined multipliers to construct the entanglement witness. We also find threshold temperature and magnetic field, for which below them the system is entangled. The results are in good agreement with the literature. For systems with more than six qubits, the results show temperature gets a stable value for a zero magnetic field.

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  1. 1.

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

    Google Scholar 

  2. 2.

    Arnesen, M.C., Bose, S., Vedral, V.: Natural thermal and magnetic entanglement in 1D Heisenberg model. Phys. Rev. Lett. 87, 017901 (2001)

    ADS  Article  Google Scholar 

  3. 3.

    Gühne, O., Tóth, G., Briegel, H.J.: Multipartite entanglement in spin chains. New J. Phys. 7, 229 (2005)

    Article  Google Scholar 

  4. 4.

    Yeo, Y.: Studying the thermally entangled state of a three-qubit Heisenberg \( XX \) ring via quantum teleportation. Phys. Rev. A 68, 022316 (2003)

    ADS  Article  Google Scholar 

  5. 5.

    Bose, S.: Quantum communication through an unmodulated spin chain. Phys. Rev. Lett. 91, 207901 (2003)

    ADS  Article  Google Scholar 

  6. 6.

    Barjaktarevic, J.P., et al.: Measurement-based teleportation along quantum spin chains. Phys. Rev. Lett. 95, 230501 (2005)

    ADS  Article  Google Scholar 

  7. 7.

    Qiu, L., Wang, A.M., Ma, X.S.: Optimal dense coding with thermal entangled states. Physica A 383, 325–330 (2007)

    ADS  Article  Google Scholar 

  8. 8.

    Xian He, X., He, J., Zheng, J.: Thermal entangled quantum heat engine. Physica A 391, 6594–6600 (2012)

    ADS  Article  MATH  Google Scholar 

  9. 9.

    He, J.Z., He, X., Zheng, J.: Thermal entangled quantum heat engine working with a three-qubit Heisenberg \( XX \) model. Int. J. Theor. Phys 51, 2066–2076 (2012)

    Article  MATH  Google Scholar 

  10. 10.

    Audretsch, J.: Entangled Systems: New Directions in Quantum Physics. Wiley, Weinheim (2008)

    Google Scholar 

  11. 11.

    Amico, L., et al.: Entanglement in many-body systems. Rev. Mod. Phys. 80, 517–576 (2008)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Rieper, E., Anders, J., Vedral, V.: Entanglement at the quantum phase transition in a harmonic lattice. New J. Phys. 12, 025017 (2010)

    ADS  Article  Google Scholar 

  13. 13.

    Androvitsaneas, P., Paspalakis, E., Terzis, A.F.: A quantum Monte Carlo study of the localizable entanglement in anisotropic ferromagnetic Heisenberg chains. Ann. Phys. 327, 212–223 (2012)

    ADS  Article  MATH  Google Scholar 

  14. 14.

    Ghio, M., et al.: Multipartite entanglement detection for hypergraph states. J. Phys. A 51, 045302 (2018)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Chen, L., Han, K.H., Kye, S.H.: Separability criterion for three-qubit states with a four dimensional norm. J. Phys. A 50, 345303 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Han, K.H., Kye, S.H.: Construction of multi-qubit optimal genuine entanglement witnesses. J. Phys. A 49, 175303 (2016)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Soltani, M.R., Vahedi, J., Mahdavifar, S.: Quantum correlations in the \( 1D \) spin-\( 1/2 \) Ising model with added Dzyaloshinskii–Moriya interaction. Physica A 416, 321–330 (2014)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Rutkowski, A., Horodecki, P.: Tensor product extension of entanglement witnesses and their connection with measurement-device-independent entanglement witnesses. Phys. Lett. A 378, 2043–2047 (2014)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Aolita, L., de Melo, F., Davidovich, L.: Open-system dynamics of entanglement:a key issues review. Rep. Prog. Phys 78, 042001 (2015)

    ADS  Article  Google Scholar 

  20. 20.

    Anders, J., et al.: Detecting entanglement with a thermometer. New J. Phys. 8, 140 (2006)

    ADS  Article  Google Scholar 

  21. 21.

    Terhal, B.M.: Bell inequalities and the separability criterion. Phys. Lett. A 271, 319 (2000)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Lewenstein, M., et al.: Optimization of entanglement witnesses. Phys. Rev. A 62, 052310 (2000)

    ADS  Article  Google Scholar 

  23. 23.

    Horodecki, M., Horodecki, P., Horodeck, R.: Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps. Phys. Lett. A 283, 1 (2001)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Horodecki, M., Horodecki, P., Horodeck, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1–8 (1996)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Acín, A., et al.: Classification of mixed three-qubit states. Phys. Rev. Lett. 87, 040401 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  26. 26.

    Koh, C.Y., Kwek, L.C.: Entanglement witness for spin glass. Physica A 420, 324–330 (2015)

    ADS  Article  Google Scholar 

  27. 27.

    Töth, G.: Entanglement witnesses in spin models. Phys. Rev. A 71, 010301 (2005)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Gühne, O., Tóth, G.: Energy and multipartite entanglement in multidimensional and frustrated spin models. Phys. Rev. A 73(5), 052319 (2006)

    ADS  Article  Google Scholar 

  29. 29.

    Wieśniak, M., et al.: Magnetic susceptibility as a macroscopic entanglement witness. New J. Phys. 7, 258 (2005)

    Article  Google Scholar 

  30. 30.

    Ghosh, S., et al.: Entangled quantum state of magnetic dipoles. Nature 425, 48 (2003)

    ADS  Article  Google Scholar 

  31. 31.

    Souza, A.M., et al.: Experimental determination of thermal entanglement in spin clusters using magnetic susceptibility measurements. Phys. Rev. B 77, 104402 (2008)

    ADS  Article  Google Scholar 

  32. 32.

    Soares-Pinto, D.O., et al.: Entanglement temperature in molecular magnets composed of S-spin dimers. Europhys. Lett. 87, 40008 (2009)

    ADS  Article  Google Scholar 

  33. 33.

    Smirnov, A.Y., Amin, M.H.: Ground-state entanglement in coupled qubits. Phys. Rev. A 88, 022329 (2013)

    ADS  Article  Google Scholar 

  34. 34.

    Singh, H., et al.: Experimental quantification of entanglement through heat capacity. New J. Phys. 15, 113001 (2013)

    ADS  Article  Google Scholar 

  35. 35.

    Arian Zad, H.: Entanglement detection in the mixed-spin using \( XY \) model. Chin. Phys. B 25, 030303 (2016)

    Article  Google Scholar 

  36. 36.

    Bäuml, S., et al.: Witnessing entanglement by proxy. New J. Phys. 18, 015002 (2016)

    Article  Google Scholar 

  37. 37.

    Tura, J., et al.: Energy as a detector of nonlocality of many-body spin systems. Phys. Rev. X 7, 021005 (2017)

    Google Scholar 

  38. 38.

    Hu, M., Fan, H.: Quantum-memory-assisted entropic uncertainty principle, teleportation, and entanglement witness in structured reservoirs. Phys. Rev. A 86, 032338 (2012)

    ADS  Article  Google Scholar 

  39. 39.

    Gühne, O., Töth, G.: Entanglement detection. Phys. Rep. 474, 1–75 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  40. 40.

    Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, Cambridge (1982)

    Google Scholar 

  41. 41.

    Ravindran, A., Ragsdell, K.M., Reklaitis, G.V.: Engineering Optimization: Methods and Applications, 2nd edn. Wiley, Hoboken (2006)

    Google Scholar 

  42. 42.

    Wang, X.: Threshold temperature for pairwise and many-particle thermal entanglement in the isotropic Heisenberg model. Phys. Rev. A 66, 044305 (2002)

    ADS  Article  Google Scholar 

  43. 43.

    Wang, X., Zanardi, P.: Quantum entanglement and Bell inequalities in Heisenberg spin chains. Phys. Lett. A 301, 1–6 (2002)

    ADS  MathSciNet  Article  MATH  Google Scholar 

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Correspondence to K. Aghayar.

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Homayoun, T., Aghayar, K. Energy as an Entanglement Witnesses for One Dimensional XYZ Heisenberg Lattice: Optimization Approach. J Stat Phys 176, 85–93 (2019).

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  • Entanglement witness
  • Heisenberg lattice
  • Lagrange undetermined multipliers