Skip to main content

Linear growth of the entanglement entropy and the Kolmogorov-Sinai rate

A preprint version of the article is available at arXiv.


The rate of entropy production in a classical dynamical system is characterized by the Kolmogorov-Sinai entropy rate hKS given by the sum of all positive Lyapunov exponents of the system. We prove a quantum version of this result valid for bosonic systems with unstable quadratic Hamiltonian. The derivation takes into account the case of time-dependent Hamiltonians with Floquet instabilities. We show that the entanglement entropy SA of a Gaussian state grows linearly for large times in unstable systems, with a rate ΛAhKS determined by the Lyapunov exponents and the choice of the subsystem A. We apply our results to the analysis of entanglement production in unstable quadratic potentials and due to periodic quantum quenches in many-body quantum systems. Our results are relevant for quantum field theory, for which we present three applications: a scalar field in a symmetry-breaking potential, parametric resonance during post-inflationary reheating and cosmological perturbations during inflation. Finally, we conjecture that the same rate ΛA appears in the entanglement growth of chaotic quantum systems prepared in a semiclassical state.


  1. A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalattore, Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83 (2011) 863 [arXiv:1007.5331] [INSPIRE].

    ADS  Article  Google Scholar 

  2. C. Gogolin and J. Eisert, Equilibration, thermalisation and the emergence of statistical mechanics in closed quantum systems, Rept. Prog. Phys. 79 (2016) 056001 [arXiv:1503.07538] [INSPIRE].

  3. L. D’Alessio, Y. Kafri, A. Polkovnikov and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65 (2016) 23 [arXiv:1509.06411] [INSPIRE].

    Google Scholar 

  4. W.H. Zurek and J.P. Paz, Decoherence, chaos and the second law, Phys. Rev. Lett. 72 (1994) 2508 [gr-qc/9402006] [INSPIRE].

  5. P.A. Miller and S. Sarkar, Signatures of chaos in the entanglement of two coupled quantum kicked tops, Phys. Rev. E 60 (1999) 1542.

  6. A.K. Pattanayak, Lyapunov exponents, entropy production, and decoherence, Phys. Rev. Lett. 83 (1999) 4526 [chao-dyn/9911017].

  7. D. Monteoliva and J.P. Paz, Decoherence and the rate of entropy production in chaotic quantum systems, Phys. Rev. Lett. 85 (2000) 3373 [quant-ph/0007052].

  8. A. Tanaka, H. Fujisaki and T. Miyadera, Saturation of the production of quantum entanglement between weakly coupled mapping systems in a strongly chaotic region, Phys. Rev. E 66 (2002) 045201 [quant-ph/0209086].

  9. H. Kim and D.A. Huse, Ballistic spreading of entanglement in a diffusive nonintegrable system, Phys. Rev. Lett. 111 (2013) 127205.

    ADS  Article  Google Scholar 

  10. P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].

  11. P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].

    MATH  Google Scholar 

  12. J.S. Cotler, M.P. Hertzberg, M. Mezei and M.T. Mueller, Entanglement Growth after a Global Quench in Free Scalar Field Theory, JHEP 11 (2016) 166 [arXiv:1609.00872] [INSPIRE].

  13. V. Balasubramanian et al., Thermalization of Strongly Coupled Field Theories, Phys. Rev. Lett. 106 (2011) 191601 [arXiv:1012.4753] [INSPIRE].

    ADS  Article  Google Scholar 

  14. V. Balasubramanian et al., Holographic Thermalization, Phys. Rev. D 84 (2011) 026010 [arXiv:1103.2683] [INSPIRE].

  15. T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  16. H. Liu and S.J. Suh, Entanglement Tsunami: Universal Scaling in Holographic Thermalization, Phys. Rev. Lett. 112 (2014) 011601 [arXiv:1305.7244] [INSPIRE].

  17. H. Liu and S.J. Suh, Entanglement growth during thermalization in holographic systems, Phys. Rev. D 89 (2014) 066012 [arXiv:1311.1200] [INSPIRE].

  18. B. Müller and A. Schäfer, Entropy Creation in Relativistic Heavy Ion Collisions, Int. J. Mod. Phys. E 20 (2011) 2235 [arXiv:1110.2378] [INSPIRE].

  19. T. Kunihiro, B. Müller, A. Ohnishi, A. Schäfer, T.T. Takahashi and A. Yamamoto, Chaotic behavior in classical Yang-Mills dynamics, Phys. Rev. D 82 (2010) 114015 [arXiv:1008.1156] [INSPIRE].

  20. K. Hashimoto, K. Murata and K. Yoshida, Chaos in chiral condensates in gauge theories, Phys. Rev. Lett. 117 (2016) 231602 [arXiv:1605.08124] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  21. Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].

  22. M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  23. E. Bianchi and R.C. Myers, On the Architecture of Spacetime Geometry, Class. Quant. Grav. 31 (2014) 214002 [arXiv:1212.5183] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  24. E. Bianchi, L. Hackl and N. Yokomizo, Entanglement time in the primordial universe, Int. J. Mod. Phys. D 24 (2015) 1544006 [arXiv:1512.08959] [INSPIRE].

  25. L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  26. R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  27. S. Chapman, M.P. Heller, H. Marrochio and F. Pastawski, Towards Complexity for Quantum Field Theory States, arXiv:1707.08582 [INSPIRE].

  28. V. Latora and M. Baranger, Kolmogorov-sinai entropy rate versus physical entropy, Phys. Rev. Lett. 82 (1999) 520 [chao-dyn/9806006].

  29. M. Falcioni, L. Palatella and A. Vulpiani, Production rate of the coarse-grained gibbs entropy and the kolmogorov-sinai entropy: A real connection?, Phys. Rev. E 71 (2005) 016118 [nlin/0407056].

  30. A.N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in lebesgue spaces, Dokl. Akad. Nauk SSSR 119 (1958) 861.

    MathSciNet  MATH  Google Scholar 

  31. Y. Sinai, Kolmogorov-Sinai entropy, Scholarpedia 4 (2009) 2034 [revision 91406].

  32. G.M. Zaslavsky, Hamiltonian chaos and fractional dynamics, Oxford University Press (2008).

  33. M. Cencini, F. Cecconi and A. Vulpiani, Chaos: from simple models to complex systems, vol. 17, World Scientific (2010).

  34. T. Kunihiro, B. Müller, A. Ohnishi and A. Schäfer, Towards a Theory of Entropy Production in the Little and Big Bang, Prog. Theor. Phys. 121 (2009) 555 [arXiv:0809.4831] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  35. C.T. Asplund and D. Berenstein, Entanglement entropy converges to classical entropy around periodic orbits, Annals Phys. 366 (2016) 113 [arXiv:1503.04857] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  36. E. Bianchi, L. Hackl and N. Yokomizo, Entanglement entropy of squeezed vacua on a lattice, Phys. Rev. D 92 (2015) 085045 [arXiv:1507.01567] [INSPIRE].

  37. L. Vidmar, L. Hackl, E. Bianchi and M. Rigol, Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians, Phys. Rev. Lett. 119 (2017) 020601 [arXiv:1703.02979] [INSPIRE].

  38. A.S. Holevo, Probabilistic and statistical aspects of quantum theory, vol. 1, Springer (2011).

  39. V. Vedral, The role of relative entropy in quantum information theory, Rev. Mod. Phys. 74 (2002) 197 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  40. M. Ohya and D. Petz, Quantum entropy and its use, Springer Science & Business Media (2004).

  41. L. Hackl, E. Bianchi, R. Modak and M. Rigol, Entanglement production in bosonic systems: Linear and logarithmic growth, arXiv:1710.04279 [INSPIRE].

  42. G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Annales Sci. École Norm. Sup. 12 (1883) 47.

  43. C. Chicone, Ordinary Differential Equations with Applications, Springer (1999).

  44. A. Ashtekar and A. Magnon-Ashtekar, A geometrical approach to external potential problems in quantum field theory, Gen. Rel. Grav. 12 (1980) 205 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  45. A. Ashtekar and A. Magnon, Quantum Fields in Curved Space-Times, Proc. Roy. Soc. Lond. A 346 (1975) 375 [INSPIRE].

  46. R.M. Wald, Quantum field theory in curved spacetime and black hole thermodynamics, University of Chicago Press (1994).

  47. R. Haag, Local quantum physics: Fields, particles, algebras, Springer (2012).

  48. D. Shale, Linear symmetries of free boson fields, Trans. Am. Math. Soc. 103 (1962) 149.

    MathSciNet  Article  MATH  Google Scholar 

  49. D. Shale and W.F. Stinespring, States of the clifford algebra, Ann. Math. 80 (1964) 365.

    MathSciNet  Article  MATH  Google Scholar 

  50. J.T. Ottesen, Infinite dimensional groups and algebras in quantum physics, vol. 27, Springer (2008).

  51. F. Berezin, The method of second quantization, Pure and Applied Physics, Academic Press (1966).

  52. N.D. Birrell and P.C.W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, U.K. (1984).

  53. L. Parker and D. Toms, Quantum field theory in curved spacetime: quantized fields and gravity, Cambridge University Press (2009).

  54. S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].

  55. J.S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951) 664 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  56. W. Greiner, B. Muller and J. Rafelski, Quantum electrodynamics of strong fields, Springer (1985).

  57. H.B. Casimir, On the attraction between two perfectly conducting plates, Kon. Ned. Akad. Wetensch. Proc. 51 (1948) 793 [INSPIRE].

    MATH  Google Scholar 

  58. D. Klemm and O. Vaughan, Nonextremal black holes in gauged supergravity and the real formulation of special geometry II, Class. Quant. Grav. 30 (2013) 065003 [arXiv:1211.1618] [INSPIRE].

  59. C.G. Torre and M. Varadarajan, Functional evolution of free quantum fields, Class. Quant. Grav. 16 (1999) 2651 [hep-th/9811222] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  60. I. Agullo and A. Ashtekar, Unitarity and ultraviolet regularity in cosmology, Phys. Rev. D 91 (2015) 124010 [arXiv:1503.03407] [INSPIRE].

  61. R.D. Sorkin, 1983 paper on entanglement entropy: “On the Entropy of the Vacuum outside a Horizon”, arXiv:1402.3589 [INSPIRE].

  62. M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  63. J. Eisert, M. Cramer and M.B. Plenio, Area laws for the entanglement entropy — a review, Rev. Mod. Phys. 82 (2010) 277 [arXiv:0808.3773] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  64. S. Hollands and K. Sanders, Entanglement measures and their properties in quantum field theory, arXiv:1702.04924 [INSPIRE].

  65. L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A Quantum Source of Entropy for Black Holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].

  66. H. Casini and M. Huerta, Remarks on the entanglement entropy for disconnected regions, JHEP 03 (2009) 048 [arXiv:0812.1773] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  67. E. Bianchi, T. De Lorenzo and M. Smerlak, Entanglement entropy production in gravitational collapse: covariant regularization and solvable models, JHEP 06 (2015) 180 [arXiv:1409.0144] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  68. C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].

  69. E. Bianchi and A. Satz, Entropy of a subalgebgra of observables and the geometric entanglement entropy, to appear (2018).

  70. S. Weinberg, The quantum theory of fields, vol. 2, Cambridge University Press (1995).

  71. F. Strocchi, Symmetry breaking, vol. 643, Springer (2005).

  72. E. Calzetta and B.L. Hu, Nonequilibrium Quantum Fields: Closed Time Path Effective Action, Wigner Function and Boltzmann Equation, Phys. Rev. D 37 (1988) 2878 [INSPIRE].

  73. J. Berges, Nonequilibrium Quantum Fields: From Cold Atoms to Cosmology, arXiv:1503.02907 [INSPIRE].

  74. J. Berges and J. Serreau, Parametric resonance in quantum field theory, Phys. Rev. Lett. 91 (2003) 111601 [hep-ph/0208070] [INSPIRE].

  75. L. Kofman, A.D. Linde and A.A. Starobinsky, Reheating after inflation, Phys. Rev. Lett. 73 (1994) 3195 [hep-th/9405187] [INSPIRE].

    ADS  Article  Google Scholar 

  76. L. Kofman, A.D. Linde and A.A. Starobinsky, Reheating after inflation, Phys. Rev. Lett. 73 (1994) 3195 [hep-th/9405187] [INSPIRE].

    ADS  Article  Google Scholar 

  77. R. Allahverdi, R. Brandenberger, F.-Y. Cyr-Racine and A. Mazumdar, Reheating in Inflationary Cosmology: Theory and Applications, Ann. Rev. Nucl. Part. Sci. 60 (2010) 27 [arXiv:1001.2600] [INSPIRE].

    ADS  Article  Google Scholar 

  78. M.A. Amin, M.P. Hertzberg, D.I. Kaiser and J. Karouby, Nonperturbative Dynamics Of Reheating After Inflation: A Review, Int. J. Mod. Phys. D 24 (2014) 1530003 [arXiv:1410.3808] [INSPIRE].

  79. S. Mrówczynski and B. Müller, Reheating after supercooling in the chiral phase transition, Phys. Lett. B 363 (1995) 1 [nucl-th/9507033] [INSPIRE].

  80. X. Busch, R. Parentani and S. Robertson, Quantum entanglement due to a modulated dynamical Casimir effect, Phys. Rev. A 89 (2014) 063606 [arXiv:1404.5754] [INSPIRE].

  81. P.O. Fedichev and U.R. Fischer, ’Cosmological’ quasiparticle production in harmonically trapped superfluid gases, Phys. Rev. A 69 (2004) 033602 [cond-mat/0303063] [INSPIRE].

  82. I. Carusotto, R. Balbinot, A. Fabbri and A. Recati, Density correlations and dynamical Casimir emission of Bogoliubov phonons in modulated atomic Bose-Einstein condensates, Eur. Phys. J. D 56 (2010) 391 [arXiv:0907.2314] [INSPIRE].

  83. J.C. Jaskula et al., An acoustic analog to the dynamical Casimir effect in a Bose-Einstein condensate, Phys. Rev. Lett. 109 (2012) 220401 [arXiv:1207.1338] [INSPIRE].

    ADS  Article  Google Scholar 

  84. J. Steinhauer, Observation of quantum Hawking radiation and its entanglement in an analogue black hole, Nature Phys. 12 (2016) 959 [arXiv:1510.00621] [INSPIRE].

    ADS  Article  Google Scholar 

  85. D. Campo and R. Parentani, Inflationary spectra and partially decohered distributions, Phys. Rev. D 72 (2005) 045015 [astro-ph/0505379] [INSPIRE].

  86. D. Polarski and A.A. Starobinsky, Semiclassicality and decoherence of cosmological perturbations, Class. Quant. Grav. 13 (1996) 377 [gr-qc/9504030] [INSPIRE].

  87. C. Kiefer, D. Polarski and A.A. Starobinsky, Entropy of gravitons produced in the early universe, Phys. Rev. D 62 (2000) 043518 [gr-qc/9910065] [INSPIRE].

  88. J. Martin and V. Vennin, Quantum Discord of Cosmic Inflation: Can we Show that CMB Anisotropies are of Quantum-Mechanical Origin?, Phys. Rev. D 93 (2016) 023505 [arXiv:1510.04038] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  89. V.I. Arnold, Mathematical methods of classical mechanics, vol. 60, Springer (2013).

  90. J.P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57 (1985) 617 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  91. F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi and A. Politi, Characterizing dynamics with covariant lyapunov vectors, Phys. Rev. Lett. 99 (2007) 130601 [arXiv:0706.0510].

    ADS  Article  Google Scholar 

  92. Y.B. Pesin, Characteristic lyapunov exponents and smooth ergodic theory, Russ. Math. Surv. 32 (1977) 55.

    Article  MATH  Google Scholar 

  93. G. Bennetin, L. Galgani, A. Giorgilli and J. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems: A method for computing all of them, Meccanica 15 (1980) 9.

    ADS  Article  MATH  Google Scholar 

  94. P. Calabrese and J. Cardy, Quantum Quenches in Extended Systems, J. Stat. Mech. 0706 (2007) P06008 [arXiv:0704.1880] [INSPIRE].

  95. G. De Chiara, S. Montangero, P. Calabrese and R. Fazio, Entanglement entropy dynamics in Heisenberg chains, J. Stat. Mech. 0603 (2006) P03001 [cond-mat/0512586] [INSPIRE].

  96. M. Fagotti and P. Calabrese, Evolution of entanglement entropy following a quantum quench: Analytic results for the XY chain in a transverse magnetic field, Phys. Rev. A 78 (2008) 010306 [arXiv:0804.3559].

  97. V. Eisler and I. Peschel, Entanglement in a periodic quench, Annalen Phys. 520 (2008) 410 [arXiv:0803.2655].

    ADS  Article  MATH  Google Scholar 

  98. A.M. Läuchli and C. Kollath, Spreading of correlations and entanglement after a quench in the one-dimensional bose-hubbard model, J. Stat. Mech. 5 (2008) 05018 [arXiv:0803.2947].

  99. V. Alba and P. Calabrese, Entanglement and thermodynamics after a quantum quench in integrable systems, Proc. Nat. Acad. Sci. 114 (2017) 7947 [arXiv:1608.00614].

  100. S.L. Braunstein and P. Van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77 (2005) 513 [quant-ph/0410100].

  101. A. Ferraro, S. Olivares and M.G. Paris, Gaussian states in continuous variable quantum information, Bibliopolis, Napoli (2005) [quant-ph/0503237].

  102. C. Weedbrook et al., Gaussian quantum information, Rev. Mod. Phys. 84 (2012) 621 [arXiv:1110.3234].

    ADS  Article  Google Scholar 

  103. G. Adesso, S. Ragy and A.R. Lee, Continuous variable quantum information: Gaussian states and beyond, Open Syst. Inf. Dyn. 21 (2014) 1440001 [arXiv:1401.4679].

  104. J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.

  105. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.

  106. M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452 (2008) 854.

    ADS  Article  Google Scholar 

  107. M.C. Gutzwiller, Chaos in classical and quantum mechanics, vol. 1, Springer (2013).

  108. F. Haake, Quantum signatures of chaos, vol. 54, Springer (2013).

  109. L. Reichl, The transition to chaos: conservative classical systems and quantum manifestations, Springer (2013).

  110. T. Biro, S.G. Matinyan and B. Muller, Chaos and gauge field theory, World Sci. Lect. Notes Phys. 56 (1994) 1.

    MathSciNet  MATH  Google Scholar 

  111. C.C. Martens, R.L. Waterland and W.P. Reinhardt, Classical, semiclassical, and quantum mechanics of a globally chaotic system: Integrability in the adiabatic approximation, J. Chem. Phys. 90 (1989) 2328.

    ADS  MathSciNet  Article  Google Scholar 

  112. J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  113. D. Berenstein and A.M. Garcia-Garcia, Universal quantum constraints on the butterfly effect, arXiv:1510.08870 [INSPIRE].

  114. G.B. Folland, Harmonic Analysis in Phase Space. (AM-122), first edition, Princeton University Press (1989).

  115. M.A. de Gosson, Symplectic geometry and quantum mechanics, vol. 166, Springer (2006).

  116. P. Woit, Quantum theory, groups and representations: An introduction, Springer (2017).

  117. J. von Neumann, Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann. 104 (1931) 570.

  118. H.J. Groenewold, On the Principles of elementary quantum mechanics, Physica 12 (1946) 405 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  119. D.F. Walls and G.J. Milburn, Quantum optics, Springer (2007).

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Lucas Hackl.

Additional information

ArXiv ePrint: 1709.00427

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bianchi, E., Hackl, L. & Yokomizo, N. Linear growth of the entanglement entropy and the Kolmogorov-Sinai rate. J. High Energ. Phys. 2018, 25 (2018).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI:


  • Field Theories in Lower Dimensions
  • Lattice Quantum Field Theory
  • Quantum Dissipative Systems