Quantum and classical correlations in three-qubit spin

  • G. B. FurmanEmail author
  • S. D. Goren
  • V. M. Meerovich
  • V. L. Sokolovsky
  • A. B. Kozyrev


The Hamiltonian and the spin operators for a spin \( \tfrac{7}{2} \) are represented in the basis formed by the Kronecker products of the Pauli matrices. This allows us to represent eight quantum states of the spin 7/2 as the states of three coupled fictitious spins \( \tfrac{1}{2}, \) which can be considered as a system of three coupling qubits. The Hamiltonian for the three-spin system contains terms describing bi- and tripartite interactions with the strengths depending on the asymmetry parameter of the electric field gradient and the applied magnetic field. This leads to unusual magnetic field dependences of the classical and quantum correlations between the fictitious spins. It is shown that, unlike the predictions of the Ising, Heisenberg, and dipole–dipole coupling spin models, the quantum mutual information, classical correlations, entanglement, and quantum discords between the fictitious spins do not vanish with an increase in magnetic field. (The correlations tend to their limit values in a high field.) All the correlations possess the minima in the field dependences. The tripartite concurrence can achieve the maximal concurrence in a three-spin system in the pure state. The proposed approach may be useful for analysis of properties of particles with larger angular momentum and the many-body interactions.


Spin \( \tfrac{7}{2} \) Quadrupole interaction Quantum and classical correlations Fictitious spins \( \tfrac{1}{2} \) 



ABK thanks the Ministry of Education and Science of the Russian Federation for support within the framework of Research and development in priority areas of advancement of the Russian Scientific and Technological Complex for 2014–2020, Agreement No. 14.608.21.0002 of 27.10.2015 (Unique Number of Agreement RFMEFI60815X0002).


  1. 1.
    Bloom, M., Herzog, B., Hahn, E.L.: Free magnetic induction in nuclear quadrupole resonance. Phys. Rev. 97, 1699 (1955)ADSCrossRefGoogle Scholar
  2. 2.
    Das, T.P., Hahn, E.L.: Nuclear quadrupole resonance spectroscopy. In: Seitz, F., Turnbull, D. (eds.) Solid State Physics, Suppl. I. Academic Press Inc., New York (1957)Google Scholar
  3. 3.
    Kessel, A.R.: Analog of the Bloch equations for spin > 1/2. Fiz. Tverd. Tela (Leningrad) 5, 1055 (1963) [Sov. Phys. Solid State 5, 934 (1963)]Google Scholar
  4. 4.
    Leppelmeier, G.W., Hanh, E.L.: Zero-field nuclear quadrupole spin-lattice relaxation in the rotating frame. Phys. Rev. 142, 179 (1966)ADSCrossRefGoogle Scholar
  5. 5.
    Cohen-Tannoudji, C., Diu, B., Laloe, F.: Quantum Mechanics, vol. 1. Wiley, New York (1977)zbMATHGoogle Scholar
  6. 6.
    Vega, S., Pines, A.: Operator formalism for double quantum NMR. J. Chem. Phys. 66, 5624 (1977)ADSCrossRefGoogle Scholar
  7. 7.
    Vega, S.: Fictitious spin 1/2 operator formalism for multiple quantum NMR. J. Chem. Phys. 68, 5518 (1978)ADSCrossRefGoogle Scholar
  8. 8.
    Vega, S., Naor, Y.: Triple quantum NMR on spin systems with I = 3/2 in solids. J. Chem. Phys. 75, 75 (1981)ADSCrossRefGoogle Scholar
  9. 9.
    Ainbinder, N.E., Furman G.B.: Theory of multipulse averaging for spin system with arbitrary nonequidistant spectra. Zh. Eksp. Teor. Fiz. 85, 988 (1983) [Sov. Phys. JETP 58, 575 (1983)]Google Scholar
  10. 10.
    Goldman, M.: Spin-1/2 description of spin-3/2. Adv. Magn. Reson. 14, 59–74 (1990)CrossRefGoogle Scholar
  11. 11.
    Furman, G.B., Goren, S.D., Meerovich, V.M., Sokolovsky, V.L.: Fictitious spin-1/2 operators and correlations in quadrupole nuclear spin system. Int. J. Quantum Inf. 16, 1850008 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Petit, D., Korb, J.-P.: Fictitious spin-1/2 operators and multitransition nuclear relaxation in solids: general theory. Phys. Rev. B 37, 5761 (1988)ADSCrossRefGoogle Scholar
  13. 13.
    Kessel, A.R., Ermakov, V.L.: Multiqubit spin. JETP Lett. 70, 61 (1999)ADSCrossRefGoogle Scholar
  14. 14.
    Khitrin, A.K., Fung, B.M.: Nuclear magnetic resonance quantum logic gates using quadrupolar nuclei. J. Chem. Phys. 112, 6963 (2000)ADSCrossRefGoogle Scholar
  15. 15.
    Furman, G.B., Goren, S.D.: Pure NQR quantum computing. Z. Naturforsch. 57a, 315 (2002)ADSGoogle Scholar
  16. 16.
    Furman, G.B., Goren, S.D., Meerovich, V.M., Sokolovsky, V.L.: Two qubits in pure nuclear quadrupole resonance. J. Phys. Condens. Matter 14, 8715 (2002)ADSCrossRefGoogle Scholar
  17. 17.
    Furman, G.B., Meerovich, V.M., Sokolovsky, V.L.: Entanglement in nuclear quadrupole resonance. Hyperfine Interact. 198, 153 (2010)ADSCrossRefGoogle Scholar
  18. 18.
    Khitrin, A., Song, H., Fung, B.M.: Method of multifrequency excitation for creating pseudopure states for NMR quantum computing. Phys. Rev. A 63, 020301 (2001)ADSCrossRefGoogle Scholar
  19. 19.
    Furman, G.B., Meerovich, V.M., Sokolovsky, V.L.: Single-spin entanglement. Quantum Inf. Process. 16, 206 (2017)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Abragam, A.: The Principles of Nuclear Magnetism. Clarendon, Oxford (1961)Google Scholar
  21. 21.
    Kramers, H.A.: Theorie generale de la rotation paramagnetique dans les cristaux. Proc. Amst. Acad. 33, 959 (1930)zbMATHGoogle Scholar
  22. 22.
    Modi, K., Brodutch, A., Cable, H., Paterek, T., Vedral, V.: The classical-quantum boundary for correlations: discord and related measures. Rev. Mod. Phys. 84, 1655 (2012)ADSCrossRefGoogle Scholar
  23. 23.
    Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)ADSCrossRefGoogle Scholar
  24. 24.
    Carvalho, A.R.R., Mintert, F., Buchleitner, A.: Decoherence and multipartite entanglement. Phys. Rev. Lett. 93, 230501 (2004)ADSCrossRefGoogle Scholar
  25. 25.
    Mintert, F., Kus, M., Buchleitner, A.: Concurrence of mixed multipartite quantum states. Phys. Rev. Lett. 95, 260502 (2005)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Dur, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Horodecki, M.: Simplifying monotonicity conditions for entanglement measures. Open Syst. Inf. Dyn. 12, 231 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Walter, M., Gross, D., Eisert, J.: Multi-Partite Entanglement. arXiv:1612.02437
  29. 29.
    Chitambar, E., Leung, D., Mančinska, L., Ozols, M., Winter, A.: Everything you always wanted to know about LOCC. Commun. Math. Phys. 328(1), 303–326 (2014)ADSCrossRefGoogle Scholar
  30. 30.
    Xu, Y.-L., Kong, X.-M., Liu, Z.-Q., Wang, C.-Y.: Quantum entanglement and quantum phase transition for the Ising model on a two-dimension square lattice. Phys. A 446, 217 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Furman, G., Goren, S., Meerovich, V., Sokolovsky, V.: Nuclear quadrupole resonance of spin 3/2 and entangled two-qubit states. Phys. Scr. 90, 105301 (2015)ADSCrossRefGoogle Scholar
  32. 32.
    Furman, G.B., Meerovich, V.M., Sokolovsky, V.L.: Dynamics of entanglement in a one-dimensional Ising chain. Phys. Rev. A 77, 062330 (2008)ADSCrossRefGoogle Scholar
  33. 33.
    Yureishchev, M.A.: Entanglement entropy fluctuations in quantum Ising chains. J. Exp. Theor. Phys. 111, 525 (2010)ADSCrossRefGoogle Scholar
  34. 34.
    Doronin, S.I., Pyrkov, A.N., Fel’dman, E.B.: Entanglement in alternating open chains of nuclear spins s = 1/2 with the XY Hamiltonian. J. Exp. Theor. Phys. Lett. 85, 519 (2007)CrossRefGoogle Scholar
  35. 35.
    Furman, G.B., Meerovich, V.M., Sokolovsky, V.L.: Nuclear polarization and entanglement in spin systems. Quantum Inf. Process. 8, 283–291 (2009)CrossRefGoogle Scholar
  36. 36.
    Furman, G.B., Meerovich, V.M., Sokolovsky, V.L.: Entanglement and multiple quantum coherence dynamics in spin clusters. Quantum Inf. Process. 8, 379–386 (2009)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Furman, G.B., Meerovich, V.M., Sokolovsky, V.L.: Entanglement of dipolar coupling spins. Quantum Inf. Process. 10, 307 (2011)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Furman, G.B., Meerovich, V.M., Sokolovsky, V.L.: Entanglement in dipolar coupling spin system in equilibrium state Quantum Inf. Process 11, 1603–1617 (2012)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Kuznetsova, E.I., Yurischev, M.A.: Quantum discord in spin systems with dipole–dipole interaction. Quantum Inf. Process. 12, 3587–3605 (2013)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Amico, L., Fazio, R., Osterloh, A., Vedral, V.: Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 885 (2009)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Hill, S., Wootters, W.K.: Entanglement of a pair of quantum bits. Phys. Rev. Lett. 78, 5022 (1997)ADSCrossRefGoogle Scholar
  43. 43.
    Ali, M., Rau, A.R.P., Alber, G.: Quantum discord for two-qubit X states. Phys. Rev. A 81, 042105 (2010)ADSCrossRefGoogle Scholar
  44. 44.
    Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)ADSCrossRefGoogle Scholar
  45. 45.
    Henderson, L., Vedral, V.: Classical, quantum and total correlations. J. Phys. A 34, 6899 (2001)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    Vedral, V.: Classical correlations and entanglement in quantum measurements. Phys. Rev. Lett. 90, 050401 (2003)ADSCrossRefGoogle Scholar
  47. 47.
    Furman, G.B., Goren, S.D., Meerovich, V.M., Sokolovsky, V.L.: Generation of quantum correlations at adiabatic demagnetization. J. Phys. Commun. 1, 045009 (2017)CrossRefGoogle Scholar
  48. 48.
    Brodutch, A., Terno, D.R.: Quantum discord, local operations, and Maxwell’s demons. Phys. Rev. A 81, 062103 (2010)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Dakic, B., Vedral, V., Brukner, C.: Necessary and sufficient condition for nonzero quantum discord. Phys. Rev. Lett. 105, 190502 (2010)ADSCrossRefGoogle Scholar
  50. 50.
    YurishchevM, A.: NMR dynamics of quantum discord for spin-carrying gas molecules in a closed nanopore. J. Exp. Theor. Phys. 119, 828–837 (2014)ADSCrossRefGoogle Scholar
  51. 51.
    Ali, M., Rau, A.R.P., Alber, G.: Quantum discord for two-qubit X states. Erratum Phys. Rev. A 82, 069902 (2010), Phys. Rev. A 82, 069902(E) (2010)Google Scholar
  52. 52.
    Huang, Y.: Quantum discord for two-qubit X states: analytical formula with very small worst-case error. Phys. Rev. A 88, 014302 (2013)ADSCrossRefGoogle Scholar
  53. 53.
    Ramanathan, R., Kurzynski, P., Chuan, T.K., Santos, M.F., Kaszlikowski, D.: Criteria for two distinguishable fermions to form a boson. Phys. Rev. A 84, 034304 (2011)ADSCrossRefGoogle Scholar
  54. 54.
    Yang, J., Huang, Y.: Tripartite and bipartite quantum correlations in the XXZ spin chain with three-site interaction. Quantum Inf. Process. 16, 281 (2017)CrossRefGoogle Scholar
  55. 55.
    Dakic, B., Lipp, Y.O., Ma, X., Ringbauer, M., Kropatschek, S., Barz, S., Paterek, T., Vedral, V., Zeilinger, A., Brukner, C., Walther, P.: Quantum discord as resource for remote state preparation. Nat. Phys. 8, 666–670 (2012)CrossRefGoogle Scholar
  56. 56.
    Kessel’, A.R., Ermakov, V.L.: Physical implementation of three-qubit gates on a separate quantum particle. J. Exp. Theor. Phys. Lett. 71, 307–309 (2000)CrossRefGoogle Scholar
  57. 57.
    Khitrin, A.K., Fung, B.M.: NMR simulation of an eight-state quantum system. Phys. Rev. A 64, 032306 (2001)ADSCrossRefGoogle Scholar
  58. 58.
    Joyia, W., Khan, K.: Exploring the tripartite entanglement and quantum phase transition in the XXZ + h model. Quantum Inf. Process. 16, 243 (2017)ADSMathSciNetCrossRefGoogle Scholar
  59. 59.
    Jones, J.A., Hansen, R.H., Mosca, M.: Quantum logic gates and nuclear magnetic resonance pulse sequences. JMR 135, 353 (1998)Google Scholar
  60. 60.
    Jones, J.A., Hansen, R.H., Mosca, M.: Implementation of a quantum search algorithm on a quantum computer. Nature 393, 344–346 (1998)ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Physics DepartmentBen Gurion University of the NegevBeer ShevaIsrael
  2. 2.Dagestan University of National EconomyMakhachkalaRussia
  3. 3.Saint-Petersburg Electrotechnical University LETISaint-PetersburgRussia

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