Quantum Information Processing

, Volume 13, Issue 2, pp 201–225 | Cite as

Systems with stationary distribution of quantum correlations: open spin-1/2 chains with \(XY\) interaction

  • E. B. Fel’dman
  • A. I. Zenchuk


Although quantum correlations in a quantum system are characterized by the evolving quantities (which are entanglement and discord usually), we reveal such basis (i.e. the set of virtual particles) for the representation of the density matrix that the entanglement and/or discord between any two virtual particles in such representation are stationary. In particular, dealing with the nearest neighbor approximation, this system of virtual particles is represented by the \(\beta \)-fermions of the Jordan–Wigner transformation. Such systems are important in quantum information devices because the evolution of quantum entanglement/discord leads to the problems of realization of quantum operations. The advantage of stationary entanglement/discord is that they are completely defined by the initial density matrix and by the Hamiltonian governing the quantum dynamics in the system under consideration. Moreover, using the special initial condition together with the special system’s geometry, we construct large cluster of virtual particles with the same pairwise entanglement/discord. In other words, the measure of quantum correlations is stationary in this system and correlations are uniformly “distributed” among all virtual particles. As examples, we use both homogeneous and non-homogeneous spin-1/2 open chains with XY-interaction although other types of interactions might be also of interest.


Stationary quantum discord Quantum entanglement Spin chain  XY Hamiltonian Quantum register 



This work is supported by the Program of the Presidium of RAS No. 8 ”Development of methods of obtaining chemical compounds and creation of new materials” and by the Russian Foundation for Basic Research, Grant No. 13-03-00017.


  1. 1.
    Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)CrossRefADSGoogle Scholar
  2. 2.
    Hill, S., Wootters, W.K.: Entanglement of a pair of quantum bits. Phys. Rev. Lett. 78, 5022 (1997)CrossRefADSGoogle Scholar
  3. 3.
    Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413 (1996)MathSciNetCrossRefADSMATHGoogle Scholar
  4. 4.
    Amico, L., Fazio, R., Osterloh, A., Vedral, V.: Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008)MathSciNetCrossRefADSMATHGoogle Scholar
  5. 5.
    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. JETP Lett. 85, 519 (2007)CrossRefGoogle Scholar
  6. 6.
    Zurek, W.H.: Einselection and decoherence from an information theory perspective. Ann. Phys. (Leipzig) 9, 855 (2000)MathSciNetCrossRefADSGoogle Scholar
  7. 7.
    Henderson, L., Vedral, V.: Classical, quantum and total correlations. J. Phys. A Math. Gen. 34, 6899 (2001)MathSciNetCrossRefADSMATHGoogle Scholar
  8. 8.
    Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)CrossRefADSMATHGoogle Scholar
  9. 9.
    Zurek, W.H.: Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715 (2003)MathSciNetCrossRefADSMATHGoogle Scholar
  10. 10.
    Datta, A., Shaji, A., Caves, C.M.: Quantum discord and the power of one qubit. Phys. Rev. Lett. 100, 050502 (2008)CrossRefADSGoogle Scholar
  11. 11.
    Luo, S.: Quantum discord for two-qubit systems. Phys. Rev. A 77, 042303 (2008)CrossRefADSGoogle Scholar
  12. 12.
    Ali, M., Rau, A.R.P., Alber, G.: Quantum discord for two-qubit X states. Phys. Rev. A 81, 042105 (2010); Erratum: Phys. Rev. A 82, 069902(E) (2010)Google Scholar
  13. 13.
    Xu, J.-W.: Geometric measure of quantum discord over two-sided projective measurements. arXiv:1101.3408 [quant-ph] (2011)Google Scholar
  14. 14.
    Bang-Fu, D., Xiao-Yun, W., He-Ping, Z.: Quantum and classical correlations for a two-qubit X structure density matrix. Chin. Phys. B 20(10), 100302 (2011)CrossRefGoogle Scholar
  15. 15.
    Goldman, M.: Spin Temperature and Nuclear Magnetic Resonance in Solids. Clarendon Press, Oxford (1970)Google Scholar
  16. 16.
    Kuznetsova, E.I., Zenchuk, A.I.: Quantum discord versus second-order MQ NMR coherence intensity in dimers. Phys. Lett. A 376, 1029 (2012)CrossRefADSMATHGoogle Scholar
  17. 17.
    Bose, S.: Quantum communication through an unmodulated spin chain. Phys. Rev. Lett. 91, 207901 (2003)CrossRefADSGoogle Scholar
  18. 18.
    Zhang, S., Meier, B.H., Ernst, R.R.: Polarization echoes in NMR. Phys. Rev. Lett. 69, 2149 (1992)CrossRefADSGoogle Scholar
  19. 19.
    Fel’dman, E.B., Brüschweiler, R., Ernst, R.R.: From regular to erratic quantum dynamics in long spin 1/2 chains with an XY hamiltonian. Chem. Phys. Lett. 294, 297 (1998)CrossRefADSGoogle Scholar
  20. 20.
    Fel’dman, E.B., Zenchuk, A.I.: Quantum correlations in different density-matrix representations of spin-1/2 open chain. Phys. Rev. A 86, 012303 (2012)CrossRefADSGoogle Scholar
  21. 21.
    Christandl, M., Datta, N., Ekert, A., Landahl, A.J.: Perfect state transfer in quantum spin networks. Phys. Rev. Lett. 92, 187902 (2004)CrossRefADSGoogle Scholar
  22. 22.
    Albanese, C., Christandl, M., Datta, N., Ekert, A.: Mirror inversion of quantum states in linear registers. Phys. Rev. Lett. 93, 230502 (2004)MathSciNetCrossRefADSGoogle Scholar
  23. 23.
    Karbach, P., Stolze, J.: Spin chains as perfect quantum state mirrors. Phys. Rev. A 72, 030301(R) (2005)MathSciNetCrossRefADSGoogle Scholar
  24. 24.
    Gualdi, G., Kostak, V., Marzoli, I., Tombesi, P.: Perfect state transfer in long-range interacting spin chains. Phys. Rev. A 78, 022325 (2008)CrossRefADSGoogle Scholar
  25. 25.
    Fel’dman, E.B., Kuznetsova, E.I., Zenchuk, A.I.: High-probability state transfer in spin-1/2 chains: analytical and numerical approaches. Phys. Rev. A 82, 022332 (2010)CrossRefADSGoogle Scholar
  26. 26.
    Zenchuk, A.I.: Unitary invariant discord as a measures of bipartite quantum correlations in an N-qubit quantum system. Quant. Inf. Proc. 11(6), 1551 (2012)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Jordan, P., Wigner, E.: Über das paulische äquivalenzverbot. Z. Phys. 47, 631 (1928)CrossRefADSMATHGoogle Scholar
  28. 28.
    Dugić, M., Jeknić, J.: What is “system”: some decoherence-theory arguments. Int. J. Theor. Phys. 45(12), 2249 (2006)MathSciNetMATHGoogle Scholar
  29. 29.
    Dugić, M., Jeknić-Dugić, J.: What is “system”: the information-theoretic arguments. Int. J. Theor. Phys. 47, 805 (2008)CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Arsenijevic, M., Jeknic-Dugic, J., Dugic, M.: Asymptotic dynamics of the alternate degrees of freedom for a two-mode system: an analytically solvable model. Chin. Phys. B 22, 020302 (2013)CrossRefADSGoogle Scholar
  31. 31.
    Dugic, M., Arsenijevic, M., Jeknic-Dugic, J.: Quantum correlations relativity for continuous variable systems. arXiv:1112.5797 [quant-ph] (2011)Google Scholar
  32. 32.
    Arsenijevic, M., Jeknic-Dugic, J., Dugic, M.: A limitation of the Nakajima–Zwanzig projection method. arXiv:1301.1005 [quant-ph] (2013)Google Scholar
  33. 33.
    Lychkovskiy, O.: Dependence of decoherence-assisted classicality on the way a system is partitioned into subsystems. Phys. Rev. A 87, 022112 (2013)CrossRefADSGoogle Scholar
  34. 34.
    Doronin, S.I., Fel’dman, E.B., Zenchuk, A.I.: Relationship between probabilities of the state transfers and entanglements in spin systems with simple geometrical configurations. Phys. Rev. A 79, 042310 (2009)CrossRefADSGoogle Scholar
  35. 35.
    Huang, Y.: Quantum discord for two-qubit X states: analytical formula with very small worst-case error. Phys. Rev. A 88, 014302 (2013)CrossRefADSGoogle Scholar
  36. 36.
    Fel’dman, E.B., Zenchuk, A.I.: Asymmetry of bipartite quantum discord. JETP Lett. 93, 459 (2011)CrossRefADSGoogle Scholar
  37. 37.
    Fel’dman, E.B., Rudavets, M.G.: Exact results on spin dynamics and multiple quantum NMR dynamics in alternating spin-1/2 chains with XY Hamiltonian at high temperatures. JETP Lett. 81, 47 (2005)CrossRefADSGoogle Scholar
  38. 38.
    Pachos, J.K., Knight, P.L.: Quantum computation with a one-dimensional optical lattice. Phys. Rev. Lett. 91, 107902 (2003)CrossRefADSGoogle Scholar
  39. 39.
    Doronin, S.I., Zenchuk, A.I.: High-probability state transfers and entanglements between different nodes of the homogeneous spin-1/2 chain in an inhomogeneous external magnetic field. Phys. Rev. A 81, 022321 (2010)CrossRefADSGoogle Scholar
  40. 40.
    Kuznetsova, E.I., Fel’dman, E.B.: Exact solutions in the dynamics of alternating open chains of spins \(\text{ s } = 1/2\) with the XY Hamiltonian and their application to problems of multiple-quantum dynamics and quantum information theory. J. Exp. Theor. Phys. 102, 882 (2006)CrossRefADSGoogle Scholar
  41. 41.
    Feldman, K.E.: Exact diagonalization of the XY-Hamiltonian of open linear chains with periodic coupling constants and its application. J. Phys. A: Math. Gen. 39(5), 1039 (2006)CrossRefADSMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute of Problems of Chemical PhysicsRASChernogolovka, Moscow RegionRussia

Personalised recommendations