Quantum Correlations and Synchronization Measures

  • Fernando GalveEmail author
  • Gian Luca Giorgi
  • Roberta ZambriniEmail author
Part of the Quantum Science and Technology book series (QST)


The phenomenon of spontaneous synchronization is universal and only recently advances have been made in the quantum domain. Being synchronization a kind of temporal correlation among systems, it is interesting to understand its connection with other measures of quantum correlations. We review here what is known in the field, putting emphasis on measures and indicators of synchronization which have been proposed in the literature, and comparing their validity for different dynamical systems, highlighting when they give similar insights and when they seem to fail.



Funding from EU project QuProCS (Grant Agreement No. 641277), MINECO and FEDER/AEI (NOMAQ FIS2014-60343-P and QuStruct FIS2015-66860-P), and “Vicerectorat d’Investigació i Postgrau” of the UIB are acknowledged.


  1. 1.
    A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge edition, 2001)Google Scholar
  2. 2.
    S.H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview p edition, 2001)Google Scholar
  3. 3.
    R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    K. Modi, A. Brodutch, H. Cable, T. Paterek, V. Vedral, The classical-quantum boundary for correlations: discord and related measures. Rev. Mod. Phys. 84, 1655–1707 (2012)ADSCrossRefGoogle Scholar
  5. 5.
    G. Adesso, T.R. Bromley, M. Cianciaruso, Measures and applications of quantum correlations. J. Phys. A: Math. Theor. 49, 473001 (2016)Google Scholar
  6. 6.
    S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, C.S. Zhou, The synchronization of chaotic systems. Phys. Rep. 366, 1–101 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    I. Goychuk, J. Casado-Pascual, M. Morillo, J. Lehmann, P. Hänggi, Quantum stochastic synchronization. Phys. Rev. Lett. 97(21), 210601 (2006)ADSCrossRefGoogle Scholar
  8. 8.
    O. Zhirov, D. Shepelyansky, Synchronization and bistability of a qubit coupled to a driven dissipative oscillator. Phys. Rev. Lett. 100(1), 014101 (2008)ADSCrossRefGoogle Scholar
  9. 9.
    O. Zhirov, D. Shepelyansky, Quantum synchronization and entanglement of two qubits coupled to a driven dissipative resonator. Phys. Rev. B 80(1), 014519 (2009)ADSCrossRefGoogle Scholar
  10. 10.
    T.E. Lee, H.R. Sadeghpour, Quantum synchronization of quantum van der Pol oscillators with trapped Ions. Phys. Rev. Lett. 111, 234101 (2013)Google Scholar
  11. 11.
    S. Walter, A. Nunnenkamp, C. Bruder, Quantum synchronization of a driven self-sustained oscillator. Phys. Rev. Lett. 112(9), 094102 (2014)ADSCrossRefGoogle Scholar
  12. 12.
    P.P. Orth, D. Roosen, W. Hofstetter, K. Le Hur, Dynamics, synchronization, and quantum phase transitions of two dissipative spins. Phys. Rev. B 82, 144423 (2010)Google Scholar
  13. 13.
    G. Heinrich, M. Ludwig, J. Qian, B. Kubala, F. Marquardt, Collective dynamics in optomechanical arrays. Phys. Rev. Lett. 107(4), 043603 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    C.A. Holmes, C.P. Meaney, G.J. Milburn, Synchronization of many nanomechanical resonators coupled via a common cavity field. Phys. Rev. E 85, 066203 (2012)Google Scholar
  15. 15.
    G.L. Giorgi, F. Galve, G. Manzano, P. Colet, R. Zambrini, Quantum correlations and mutual synchronization. Phys. Rev. A 85, 052101 (2012)Google Scholar
  16. 16.
    G. Manzano, F. Galve, G.L. Giorgi, E. Hernández-García, R. Zambrini, Synchronization, quantum correlations and entanglement in oscillator networks. Sci. Rep. 3, 1439 (2013)CrossRefGoogle Scholar
  17. 17.
    A. Mari, A. Farace, N. Didier, V. Giovannetti, R. Fazio, Measures of quantum synchronization in continuous variable systems. Phys. Rev. Lett. 111(10), 103605 (2013)ADSCrossRefGoogle Scholar
  18. 18.
    S.C. Manrubia, A.S. Mikhailov, D.H. Zanette, Emergence of Dynamical Order. Synchronization Phenomena in Complex Systems (World Scientific Publishing Co., Singapore, 2004). Lecture no editionCrossRefzbMATHGoogle Scholar
  19. 19.
    A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Synchronization in complex networks. Phys. Rep. 469(3), 93–153 (2008)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    J. Pantaleone, Synchronization of metronomes. Am. J. Phys. 70(10), 992 (2002)ADSCrossRefGoogle Scholar
  21. 21.
    M. Zhang, S. Shah, J. Cardenas, M. Lipson, Synchronization and phase noise reduction in micromechanical oscillator arrays coupled through light. Phys. Rev. Lett. 115, 163902 (2015)Google Scholar
  22. 22.
    M. Aspelmeyer, T.J. Kippenberg, F. Marquardt, Cavity optomechanics. Rev. Mod. Phys. 86, 1391 (2014)Google Scholar
  23. 23.
    M. Ludwig, F. Marquardt, Quantum many-body dynamics in optomechanical arrays. Phys. Rev. Lett. 111(7), 073603 (2013)ADSCrossRefGoogle Scholar
  24. 24.
    M. Zhang, G.S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, M. Lipson, Synchronization of micromechanical oscillators using light. Phys. Rev. Lett. 109(23), 233906 (2012)ADSCrossRefGoogle Scholar
  25. 25.
    M. Bagheri, M. Poot, L. Fan, F. Marquardt, H.X. Tang, Photonic cavity synchronization of nanomechanical oscillators. Phys. Rev. Lett. 111, 213902 (2013)Google Scholar
  26. 26.
    S.Y. Shah, M. Zhang, R. Rand, M. Lipson, Master-slave locking of optomechanical oscillators over a long distance. Phys. Rev. Lett. 114, 113602 (2015)Google Scholar
  27. 27.
    W. Li, F. Zhang, C. Li, H. Song, Quantum synchronization in a star-type cavity QED network. Commun. Nonlinear Sci. Numer. Simul. 42, 121–131 (2017)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    D.K. Agrawal, J. Woodhouse, A.A. Seshia, Observation of locked phase dynamics and enhanced frequency stability in synchronized micromechanical oscillators. Phys. Rev. Lett. 111, 084101 (2013)Google Scholar
  29. 29.
    M.H. Matheny, M. Grau, L.G. Villanueva, R.B. Karabalin, M.C. Cross, M.L. Roukes, Phase synchronization of two anharmonic nanomechanical oscillators. Phys. Rev. Lett. 112, 014101 (2014)Google Scholar
  30. 30.
    T.E. Lee, C.-K. Chan, S. Wang, Entanglement tongue and quantum synchronization of disordered oscillators. Phys. Rev. E 89(2), 022913 (2014)Google Scholar
  31. 31.
    S. Walter, A. Nunnenkamp, C. Bruder, Quantum synchronization of two Van der Pol oscillators. Annalen der Physik 527(1–2), 131–138 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    V. Ameri, M. Eghbali-Arani, A. Mari, A. Farace, F. Kheirandish, V. Giovannetti, R. Fazio, Mutual information as an order parameter for quantum synchronization. Phys. Rev. A 91(1), 012301 (2015)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    H. Carmichael, An Open Systems Approach to Quantum Optics: Lectures Presented at the Université Libre de Bruxelles, October 28 to November 4, 1991, Lecture Notes in Physics Monographs (Springer, Berlin, 2009)Google Scholar
  34. 34.
    G. Manzano, F. Galve, R. Zambrini, Avoiding dissipation in a system of three quantum harmonic oscillators. Phys. Rev. A 87(3), 032114 (2013)Google Scholar
  35. 35.
    G.M. Xue, M. Gong, H.K. Xu, W.Y. Liu, H. Deng, Y. Tian, H.F. Yu, Y. Yu, D.N. Zheng, S.P. Zhao, S. Han, Observation of quantum stochastic synchronization in a dissipative quantum system. Phys. Rev. B 90, 224505 (2014)ADSCrossRefGoogle Scholar
  36. 36.
    G.L. Giorgi, F. Plastina, G. Francica, R. Zambrini, Spontaneous synchronization and quantum correlation dynamics of open spin systems. Phys. Rev. A 88(4), 042115 (2013)Google Scholar
  37. 37.
    D. Viennot, L. Aubourg, Quantum chimera states. Phys. Lett. A. 380(5–6), 678–683 (2016)Google Scholar
  38. 38.
    X. Minghui, D.A. Tieri, E.C. Fine, J.K. Thompson, M.J. Holland, Synchronization of two ensembles of atoms. Phys. Rev. Lett. 113, 154101 (2014)ADSCrossRefGoogle Scholar
  39. 39.
    B. Zhu, J. Schachenmayer, M. Xu, F. Herrera, J.G. Restrepo, M.J. Holland, A.M. Rey, Synchronization of interacting quantum dipoles. New J. Phys. 17, 083063 (2015)Google Scholar
  40. 40.
    C. Deutsch, F. Ramirez-Martinez, C. Lacroûte, F. Reinhard, T. Schneider, J.N. Fuchs, F. Piéchon, F. Laloë, J. Reichel, P. Rosenbusch, Spin self-rephasing and very long coherence times in a trapped atomic ensemble. Phys. Rev. Lett. 105(2), 020401 (2010)ADSCrossRefGoogle Scholar
  41. 41.
    Y. Liu, F. Piéchon, J.N. Fuchs, Quantum loss of synchronization in the dynamics of two spins. EPL (Europhys. Lett.) 103(1), 17007 (2013)ADSCrossRefGoogle Scholar
  42. 42.
    M.R. Hush, W. Li, S. Genway, I. Lesanovsky, A.D. Armour, Spin correlations as a probe of quantum synchronization in trapped-ion phonon lasers. Phys. Rev. A 91, 061401(R) (2015)Google Scholar
  43. 43.
    A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C.R. Mirasso, L. Pesquera, K. Alan Shore, Chaos-based communications at high bit rates using commercial fibre-optic links. Nature 438(7066), 343–346 (2005)ADSCrossRefGoogle Scholar
  44. 44.
    G.L. Giorgi, F. Galve, R. Zambrini, Probing the spectral density of a dissipative qubit via quantum synchronization. Phys. Rev. A 94, 052121 (2016)Google Scholar
  45. 45.
    H. Ollivier, W.H. Zurek, Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)ADSCrossRefzbMATHGoogle Scholar
  46. 46.
    L. Henderson, V. Vedral, Classical, quantum and total correlations. J. Phys. A: Math. Gen. 34(35), 6899 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    V.M. Bastidas, I. Omelchenko, A. Zakharova, E. Schöll, T. Brandes, Quantum signatures of chimera states. Phys. Rev. E 92, 062924 (2015)Google Scholar
  48. 48.
    A.E. Motter, Nonlinear dynamics: spontaneous synchrony breaking. Nat. Phys. 6(3), 164–165 (2010)CrossRefGoogle Scholar
  49. 49.
    W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998)ADSCrossRefGoogle Scholar
  50. 50.
    T.E. Lee, M.C. Cross, Quantum-classical transition of correlations of two coupled cavities. Phys. Rev. A 88, 013834 (2013)Google Scholar
  51. 51.
    Y. Kuramoto, International Symposium on Mathematical Problems in Theoretical Physics, vol. 39 (Springer, New York, 1975)CrossRefGoogle Scholar
  52. 52.
    J.a. Acebrón, L.L. Bonilla, C.J. Pérez Vicente, F. Ritort, R. Spigler, The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77(1), 137–185 (2005)Google Scholar
  53. 53.
    I.H. de Mendoza, L.a. Pachón, J. Gómez-Gardeñes, D. Zueco, Synchronization in a semiclassical Kuramoto model. Phys. Rev. E 90(5), 052904 (2014)Google Scholar
  54. 54.
    K. Shlomi, D. Yuvaraj, I. Baskin, O. Suchoi, R. Winik, E. Buks, Synchronization in an optomechanical cavity. Phys. Rev. E 91, 032910 (2015)Google Scholar
  55. 55.
    C. Benedetti, F. Galve, A. Mandarino, M.G.A. Paris, R. Zambrini, Minimal model for spontaneous synchronization. Phys. Rev. A 94, 052118 (2016)Google Scholar
  56. 56.
    F. Galve, A. Mandarino, M.G.A. Paris, C. Benedetti, R. Zambrini, Microscopic description for the emergence of collective dissipation in extended quantum systems. Sci. Reps. 7, 42050 (2017)Google Scholar
  57. 57.
    A. Ferraro, S. Olivares, M.G.a. Paris, Gaussian states in continuous variable quantum information. (Bibliopolis, Napoli 2005; ISBN 88-7088-483-X)Google Scholar
  58. 58.
    G. Adesso, F. Illuminati, Entanglement in continuous-variable systems: recent advances and current perspectives. J. Phys. A 40, 7821 (2007)Google Scholar
  59. 59.
    E. Lieb, T. Schultz, D. Mattis, Two soluble models of an antiferromagnetic chain. Ann. Phys. 16(3), 407–466 (1961)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    H.P. Breuer, F. Petruccione, The Theory of Open Quantum Systems. (OUP Oxford, Oxford, 2007)Google Scholar
  61. 61.
    B. Bellomo, G.L. Giorgi, G.M. Palma, R. Zambrini, Quantum synchronization as a local signature of super and subradiance. arXiv:1612.07134
  62. 62.
    W. Li, C. Li, H. Song, Criterion of quantum synchronization and controllable quantum synchronization based on an optomechanical system. J. Phys. B: At. Mol. Opt. Phys. 48(3), 035503 (2015)ADSCrossRefGoogle Scholar
  63. 63.
    P.D. Drummond, M.D. Reid, Correlations in nondegenerate parametric oscillation. II. below threshold results. Phys. Rev. A 41(7), 3930–3949 (1990)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.IFISC (UIB-CSIC), Instituto de Física Interdisciplinar y Sistemas ComplejosPalma de MallorcaSpain

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