Quantum Information Processing

, Volume 14, Issue 6, pp 1973–1996 | Cite as

Thermal entanglement of a coupled electronic spins system: interplay between an external magnetic field, nuclear field and spin–orbit interaction

  • Roberto J. Guerrero M.
  • F. Rojas


We have studied the thermal entanglement as a function of the temperature for a two-qubits Heisenberg spins system; we have included Dzyaloshinskii–Moriya interaction (DM), an external magnetic field (EMF) and hyperfine interaction due to the nuclear field of the surrounding nuclei. A critical value for the EMF was found, around \(B^{(c)}_{\mathrm{ext},z} \sim 39\) mT, which characterizes two regimes of behavior of the thermal entanglement. Our results show that the DM term acts as a facilitator for the entanglement because it prolongs the nonzero thermal entanglement for larger temperatures. We found that the concurrence as a function of the temperature has a local maximum, for values of the magnetic field larger than the critical field. We also show that the critical temperature \(T_\mathrm{c}\) follows a polynomial growth as a function of the DM term, with characteristic behavior \(T_{\mathrm{c}} \sim \beta _{0}^{2}\), and the hyperfine field implies a critical temperature as a function of the field variance, \(\sigma \) of the form \(T_{\mathrm{c}} \sim \sigma ^{2}\). We show that in this system, the entanglement measure by the concurrence and the one-spin polarization observable exhibit opposite behavior, providing a method to obtain the entanglement from the measurement of an observable.


Quantum computation Quantum entaglement Entaglement production Thermal entaglement Quantum dots 



We would like to thanks DGAPA-UNAM for support with the Project IN112012 and R.G. thanks CONACYT and CICESE for financial aid with a PhD scholarship.


  1. 1.
    Bennett, C.H.: Quantum cryptography using any two nonorthogonal states. Phys. Rev. Lett. 68, 3121 (1992)CrossRefADSzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bennett, C.H., et al.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895 (1993)CrossRefADSzbMATHMathSciNetGoogle Scholar
  3. 3.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  4. 4.
    Ekert, A.K.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661 (1991)CrossRefADSzbMATHMathSciNetGoogle Scholar
  5. 5.
    Loss, D., DiVicenzo, D.P.: Quantum computation with quantum dots. Phys. Rev. A 57, 120 (1998)CrossRefADSGoogle Scholar
  6. 6.
    Peta, J.R., et al.: Double quantum dot as a quantum bit. Science 309, 2180 (2005)CrossRefADSGoogle Scholar
  7. 7.
    Kane, B.E.: A silicon-based nuclear spin quantum computer. Nature 393, 133 (1998)CrossRefADSGoogle Scholar
  8. 8.
    Imamoglu, A.: Quantum information processing using quantum dot spins and cavity QED. Phys. Rev. Lett. 83, 4204 (1999)CrossRefADSGoogle Scholar
  9. 9.
    Zheng, S.B., Guo, G.C.: Efficient scheme for two-atom entanglement and quantum information processing in cavity QED. Phys. Rev. Lett. 85, 2392 (2000)CrossRefADSGoogle Scholar
  10. 10.
    Wang, X.: Entanglement in the quantum Heisenberg XY model. Phys. Rev. A 64, 012313 (2001)CrossRefADSGoogle Scholar
  11. 11.
    Arnesen, M.C., et al.: Natural thermal and magnetic entanglement in the 1D Heisenberg model. Phys. Rev. Lett. 87, 017901 (2001)CrossRefADSGoogle Scholar
  12. 12.
    Yi, X.X., et al.: Entanglement induced in spin-1/2 particles by a spin chain near its critical points. Phys. Rev. A 74, 054102 (2006)CrossRefADSGoogle Scholar
  13. 13.
    Porras, D., Cirac, J.I.: Effective quantum spin systems with trapped ions. Phys. Rev. Lett. 92, 207901 (2004)CrossRefADSGoogle Scholar
  14. 14.
    Wang, F., et al.: Anisotropy and magnetic field effects on entanglement of a two-spin (1/2, 1) mixed-spin Heisenberg XY chain. Commun. Theor. Phys. 50, 341 (2008)CrossRefADSGoogle Scholar
  15. 15.
    Wang, X.: Effects of anisotropy on thermal entanglement. Phys. Lett. A 281, 101 (2001)CrossRefADSzbMATHMathSciNetGoogle Scholar
  16. 16.
    Zhang, G.: Thermal entanglement and teleportation in a two-qubit Heisenberg chain with Dzyaloshinski–Moriya anisotropic antisymmetric interaction. Phys. Rev. A 75, 034304 (2007)CrossRefADSGoogle Scholar
  17. 17.
    Ma, X.S.: Thermal entanglement of a two-qutrit XX spin chain with Dzialoshinski–Moriya interaction. Opt. Commun. 281, 484 (2008)CrossRefADSGoogle Scholar
  18. 18.
    Kheirandish, F., et al.: Effect of spin–orbit interaction on entanglement of two-qubit Heisenberg XYZ systems in an inhomogeneous magnetic field. Phys. Rev. A 77, 042309 (2009)CrossRefADSGoogle Scholar
  19. 19.
    Akyüz, C., et al.: Thermal entanglement of a two-qutrit Ising system with Dzialoshinski–Moriya interaction. Opt. Commun. 281, 5271 (2008)CrossRefADSGoogle Scholar
  20. 20.
    li, D.: Thermal entanglement in the anisotropic Heisenberg XXZ model with the Dzyaloshinskii–Moriya interaction. J. Phys.: Condens. Matter 20, 325229 (2008)Google Scholar
  21. 21.
    Gunlycke, D., et al.: Thermal concurrence mixing in a one-dimensional Ising model. Phys. Rev. A 64, 042302 (2001)CrossRefADSGoogle Scholar
  22. 22.
    Kamta, G.L., Starace, A.F.: Anisotropy and magnetic field effects on the entanglement of a two qubit Heisenberg XY chain. Phys. Rev. Lett. 88, 107901 (2002)CrossRefADSGoogle Scholar
  23. 23.
    Zhou, L., et al.: Enhanced thermal entanglement in an anisotropic Heisenberg XYZ chain. Phys. Rev. A 68, 024301 (2003)CrossRefADSGoogle Scholar
  24. 24.
    Sun, Y., et al.: Thermal entanglement in the two-qubit Heisenberg XY model under a nonuniform external magnetic field. Phys. Rev. A 68, 044301 (2003)CrossRefADSGoogle Scholar
  25. 25.
    Asoudeh, M., Karimipour, V.: Thermal entanglement of spins in the Heisenberg model at low temperatures. Phys. Rev. A 70, 052307 (2004)CrossRefADSGoogle Scholar
  26. 26.
    Asoudeh, M., Karimipour, V.: Thermal entanglement of spins in an inhomogeneous magnetic field. Phys. Rev. A 71, 022308 (2005)CrossRefADSGoogle Scholar
  27. 27.
    Chruscinskim, D., Pytel, J.: Constructing optimal entanglement witnesses. II. Witnessing entanglement in 4N\(\times \)4N systems. Phys. Rev. A 82, 052310 (2010)CrossRefADSGoogle Scholar
  28. 28.
    Bennet, A., et al.: Experimental semi-device-independent certification of entangled measurements. Phys. Rev. Lett. 113, 080405 (2014)CrossRefADSGoogle Scholar
  29. 29.
    Vedral, V., Plenio, M.B.: Entanglement measures and purification procedures. Phys. Rev. A 57, 1619 (1998)CrossRefADSGoogle Scholar
  30. 30.
    Zyczkowski, K., Horodecki, P.: Volume of the set of separable states. Phys. Rev. A 58, 883 (1998)CrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Guerrero, R.J., Rojas, F.: Effect of the Dzyaloshinski–Moriya term in the quantum (SWAP)a gate produced with exchange coupling. Phys. Rev. A 77, 012331 (2007)CrossRefADSGoogle Scholar
  32. 32.
    Elzerman, J.M., et al.: Single-shot read-out of an individual electron spin in a quantum dot. Nature 430, 431 (2006)CrossRefADSGoogle Scholar
  33. 33.
    Merkulov, I.A., et al.: Electron spin relaxation by nuclei in semiconductor quantum dots. Phys. Rev. B 65, 205309 (2002)CrossRefADSGoogle Scholar
  34. 34.
    Khaetskii, A.V., et al.: Electron spin decoherence in quantum dots due to interaction with nuclei. Phys. Rev. Lett. 88, 186802 (2002)CrossRefADSGoogle Scholar
  35. 35.
    Khaetskii, A.V., et al.: Electron spin evolution induced by interaction with nuclei in a quantum dot. Phys. Rev. Lett. 67, 195329 (2003)ADSGoogle Scholar
  36. 36.
    Coish, W.A., Loss, D.: Hyperfine interaction in a quantum dot: non-Markovian electron spin dynamics. Phys. Rev. B 70, 195340 (2004)CrossRefADSGoogle Scholar
  37. 37.
    Johnson, A.C., et al.: Triplet-singlet spin relaxation via nuclei in a double quantum dot. Nature 435, 925 (2005)CrossRefADSGoogle Scholar
  38. 38.
    Coish, W.A., Loss, D.: Singlet-triplet decoherence due to nuclear spins in a double quantum dot. Phys. Rev. B 72, 125337 (2005)CrossRefADSGoogle Scholar
  39. 39.
    Klauser, D., et al.: Nuclear spin state narrowing via gate-controlled Rabi oscillations in a double quantum dot. Phys. Rev. B 73, 205302 (2006)CrossRefADSGoogle Scholar
  40. 40.
    Chuntia, S., et al.: Detection and measurement of the Dzyaloshinskii–Moriya interaction in double quantum dot system. Phys. Rev. B 73, 241304 (2006)CrossRefADSGoogle Scholar
  41. 41.
    Taylor, J.M., et al.: Relaxation, dephasing, and quantum control of electron spins in double quantum dots. Phys. Rev. B 76, 035315 (2007)CrossRefADSGoogle Scholar
  42. 42.
    Golovach, V.N., et al.: Spin relaxation at the singlet-triplet crossing in a quantum dot. Phys. Rev. B 77, 045328 (2008)CrossRefADSGoogle Scholar
  43. 43.
    Petta, J.R., et al.: Dynamic nuclear polarization with single electron spins. Phys. Rev. Lett. 100, 067601 (2008)CrossRefADSGoogle Scholar
  44. 44.
    Erlingsson, S.I., et al.: Radiatively limited dephasing in InAs quantum dots. Phys. Rev. B 70, 033301 (2005)CrossRefADSGoogle Scholar
  45. 45.
    Erlingsson, S.I., Nazarov, Y.V.: Evolution of localized electron spin in a nuclear spin environment. Phys. Rev. B 70, 205327 (2004)CrossRefADSGoogle Scholar
  46. 46.
    Dzyaloshinskii, I.: A thermodynamics theory of “weak” ferromagnetirm of antiferromagnetics; anisotropic superexchange interaction and weak ferromagnetism. J. Chem. Solids 4, 241 (1958)CrossRefADSGoogle Scholar
  47. 47.
    Moriya, T.: Anisotropic exchange interaction of localized conduction-band electrons in semiconductors. Phys. Rev. 120, 91 (1960)CrossRefADSGoogle Scholar
  48. 48.
    Kavokin, K.V.: Anisotropic exchange of localized conduction-band electrons in semiconductor. Phys. Rev. B 64, 075305 (2001)CrossRefADSGoogle Scholar
  49. 49.
    Abragam, A.: The Principles of Nuclear Megnetism. Oxford University Press, Oxford (1961)Google Scholar
  50. 50.
    Abragam, A., Bleaney, B.: Entanglement of Formation of an Arbitrary State of Two Qubits. Dover, New York (1986)Google Scholar
  51. 51.
    Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Centro de Ingeniería y TecnologíaUniversidad Autónoma de Baja CaliforniaTijuanaMexico
  2. 2.Departamento de Física Teórica, Centro Nanociencias y NanotecnologiaUniversidad Nacional Autónoma de MéxicoEnsenadaMexico

Personalised recommendations