Advertisement

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

  • Eugenio Bianchi
  • Lucas HacklEmail author
  • Nelson Yokomizo
Open Access
Regular Article - Theoretical Physics

Abstract

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.

Keywords

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

Notes

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.

References

  1. [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].ADSCrossRefGoogle Scholar
  2. [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. [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. [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. [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.Google Scholar
  6. [6]
    A.K. Pattanayak, Lyapunov exponents, entropy production, and decoherence, Phys. Rev. Lett. 83 (1999) 4526 [chao-dyn/9911017].
  7. [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. [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. [9]
    H. Kim and D.A. Huse, Ballistic spreading of entanglement in a diffusive nonintegrable system, Phys. Rev. Lett. 111 (2013) 127205.ADSCrossRefGoogle Scholar
  10. [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. [11]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].zbMATHGoogle Scholar
  12. [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. [13]
    V. Balasubramanian et al., Thermalization of Strongly Coupled Field Theories, Phys. Rev. Lett. 106 (2011) 191601 [arXiv:1012.4753] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    V. Balasubramanian et al., Holographic Thermalization, Phys. Rev. D 84 (2011) 026010 [arXiv:1103.2683] [INSPIRE].
  15. [15]
    T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [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. [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. [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. [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. [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].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
  22. [22]
    M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    E. Bianchi and R.C. Myers, On the Architecture of Spacetime Geometry, Class. Quant. Grav. 31 (2014) 214002 [arXiv:1212.5183] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [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. [25]
    L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    S. Chapman, M.P. Heller, H. Marrochio and F. Pastawski, Towards Complexity for Quantum Field Theory States, arXiv:1707.08582 [INSPIRE].
  28. [28]
    V. Latora and M. Baranger, Kolmogorov-sinai entropy rate versus physical entropy, Phys. Rev. Lett. 82 (1999) 520 [chao-dyn/9806006].
  29. [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. [30]
    A.N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in lebesgue spaces, Dokl. Akad. Nauk SSSR 119 (1958) 861.MathSciNetzbMATHGoogle Scholar
  31. [31]
    Y. Sinai, Kolmogorov-Sinai entropy, Scholarpedia 4 (2009) 2034 [revision 91406].Google Scholar
  32. [32]
    G.M. Zaslavsky, Hamiltonian chaos and fractional dynamics, Oxford University Press (2008).Google Scholar
  33. [33]
    M. Cencini, F. Cecconi and A. Vulpiani, Chaos: from simple models to complex systems, vol. 17, World Scientific (2010).Google Scholar
  34. [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].ADSCrossRefzbMATHGoogle Scholar
  35. [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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [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. [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. [38]
    A.S. Holevo, Probabilistic and statistical aspects of quantum theory, vol. 1, Springer (2011).Google Scholar
  39. [39]
    V. Vedral, The role of relative entropy in quantum information theory, Rev. Mod. Phys. 74 (2002) 197 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    M. Ohya and D. Petz, Quantum entropy and its use, Springer Science & Business Media (2004).Google Scholar
  41. [41]
    L. Hackl, E. Bianchi, R. Modak and M. Rigol, Entanglement production in bosonic systems: Linear and logarithmic growth, arXiv:1710.04279 [INSPIRE].
  42. [42]
    G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Annales Sci. École Norm. Sup. 12 (1883) 47.Google Scholar
  43. [43]
    C. Chicone, Ordinary Differential Equations with Applications, Springer (1999).Google Scholar
  44. [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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    A. Ashtekar and A. Magnon, Quantum Fields in Curved Space-Times, Proc. Roy. Soc. Lond. A 346 (1975) 375 [INSPIRE].
  46. [46]
    R.M. Wald, Quantum field theory in curved spacetime and black hole thermodynamics, University of Chicago Press (1994).Google Scholar
  47. [47]
    R. Haag, Local quantum physics: Fields, particles, algebras, Springer (2012).Google Scholar
  48. [48]
    D. Shale, Linear symmetries of free boson fields, Trans. Am. Math. Soc. 103 (1962) 149.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    D. Shale and W.F. Stinespring, States of the clifford algebra, Ann. Math. 80 (1964) 365.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    J.T. Ottesen, Infinite dimensional groups and algebras in quantum physics, vol. 27, Springer (2008).Google Scholar
  51. [51]
    F. Berezin, The method of second quantization, Pure and Applied Physics, Academic Press (1966).Google Scholar
  52. [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).Google Scholar
  53. [53]
    L. Parker and D. Toms, Quantum field theory in curved spacetime: quantized fields and gravity, Cambridge University Press (2009).Google Scholar
  54. [54]
    S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
  55. [55]
    J.S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951) 664 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    W. Greiner, B. Muller and J. Rafelski, Quantum electrodynamics of strong fields, Springer (1985).Google Scholar
  57. [57]
    H.B. Casimir, On the attraction between two perfectly conducting plates, Kon. Ned. Akad. Wetensch. Proc. 51 (1948) 793 [INSPIRE].zbMATHGoogle Scholar
  58. [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. [59]
    C.G. Torre and M. Varadarajan, Functional evolution of free quantum fields, Class. Quant. Grav. 16 (1999) 2651 [hep-th/9811222] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    I. Agullo and A. Ashtekar, Unitarity and ultraviolet regularity in cosmology, Phys. Rev. D 91 (2015) 124010 [arXiv:1503.03407] [INSPIRE].
  61. [61]
    R.D. Sorkin, 1983 paper on entanglement entropy: “On the Entropy of the Vacuum outside a Horizon”, arXiv:1402.3589 [INSPIRE].
  62. [62]
    M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [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].ADSCrossRefzbMATHGoogle Scholar
  64. [64]
    S. Hollands and K. Sanders, Entanglement measures and their properties in quantum field theory, arXiv:1702.04924 [INSPIRE].
  65. [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. [66]
    H. Casini and M. Huerta, Remarks on the entanglement entropy for disconnected regions, JHEP 03 (2009) 048 [arXiv:0812.1773] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  67. [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].ADSMathSciNetzbMATHGoogle Scholar
  68. [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. [69]
    E. Bianchi and A. Satz, Entropy of a subalgebgra of observables and the geometric entanglement entropy, to appear (2018).Google Scholar
  70. [70]
    S. Weinberg, The quantum theory of fields, vol. 2, Cambridge University Press (1995).Google Scholar
  71. [71]
    F. Strocchi, Symmetry breaking, vol. 643, Springer (2005).Google Scholar
  72. [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. [73]
    J. Berges, Nonequilibrium Quantum Fields: From Cold Atoms to Cosmology, arXiv:1503.02907 [INSPIRE].
  74. [74]
    J. Berges and J. Serreau, Parametric resonance in quantum field theory, Phys. Rev. Lett. 91 (2003) 111601 [hep-ph/0208070] [INSPIRE].
  75. [75]
    L. Kofman, A.D. Linde and A.A. Starobinsky, Reheating after inflation, Phys. Rev. Lett. 73 (1994) 3195 [hep-th/9405187] [INSPIRE].ADSCrossRefGoogle Scholar
  76. [76]
    L. Kofman, A.D. Linde and A.A. Starobinsky, Reheating after inflation, Phys. Rev. Lett. 73 (1994) 3195 [hep-th/9405187] [INSPIRE].ADSCrossRefGoogle Scholar
  77. [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].ADSCrossRefGoogle Scholar
  78. [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. [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. [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. [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. [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. [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].ADSCrossRefGoogle Scholar
  84. [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].ADSCrossRefGoogle Scholar
  85. [85]
    D. Campo and R. Parentani, Inflationary spectra and partially decohered distributions, Phys. Rev. D 72 (2005) 045015 [astro-ph/0505379] [INSPIRE].
  86. [86]
    D. Polarski and A.A. Starobinsky, Semiclassicality and decoherence of cosmological perturbations, Class. Quant. Grav. 13 (1996) 377 [gr-qc/9504030] [INSPIRE].
  87. [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. [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].ADSMathSciNetGoogle Scholar
  89. [89]
    V.I. Arnold, Mathematical methods of classical mechanics, vol. 60, Springer (2013).Google Scholar
  90. [90]
    J.P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57 (1985) 617 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  91. [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].ADSCrossRefGoogle Scholar
  92. [92]
    Y.B. Pesin, Characteristic lyapunov exponents and smooth ergodic theory, Russ. Math. Surv. 32 (1977) 55.CrossRefzbMATHGoogle Scholar
  93. [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.ADSCrossRefzbMATHGoogle Scholar
  94. [94]
    P. Calabrese and J. Cardy, Quantum Quenches in Extended Systems, J. Stat. Mech. 0706 (2007) P06008 [arXiv:0704.1880] [INSPIRE].
  95. [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. [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. [97]
    V. Eisler and I. Peschel, Entanglement in a periodic quench, Annalen Phys. 520 (2008) 410 [arXiv:0803.2655].ADSCrossRefzbMATHGoogle Scholar
  98. [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. [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. [100]
    S.L. Braunstein and P. Van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77 (2005) 513 [quant-ph/0410100].
  101. [101]
    A. Ferraro, S. Olivares and M.G. Paris, Gaussian states in continuous variable quantum information, Bibliopolis, Napoli (2005) [quant-ph/0503237].
  102. [102]
    C. Weedbrook et al., Gaussian quantum information, Rev. Mod. Phys. 84 (2012) 621 [arXiv:1110.3234].ADSCrossRefGoogle Scholar
  103. [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. [104]
    J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.Google Scholar
  105. [105]
    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.Google Scholar
  106. [106]
    M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452 (2008) 854.ADSCrossRefGoogle Scholar
  107. [107]
    M.C. Gutzwiller, Chaos in classical and quantum mechanics, vol. 1, Springer (2013).Google Scholar
  108. [108]
    F. Haake, Quantum signatures of chaos, vol. 54, Springer (2013).Google Scholar
  109. [109]
    L. Reichl, The transition to chaos: conservative classical systems and quantum manifestations, Springer (2013).Google Scholar
  110. [110]
    T. Biro, S.G. Matinyan and B. Muller, Chaos and gauge field theory, World Sci. Lect. Notes Phys. 56 (1994) 1.MathSciNetzbMATHGoogle Scholar
  111. [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.ADSMathSciNetCrossRefGoogle Scholar
  112. [112]
    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  113. [113]
    D. Berenstein and A.M. Garcia-Garcia, Universal quantum constraints on the butterfly effect, arXiv:1510.08870 [INSPIRE].
  114. [114]
    G.B. Folland, Harmonic Analysis in Phase Space. (AM-122), first edition, Princeton University Press (1989).Google Scholar
  115. [115]
    M.A. de Gosson, Symplectic geometry and quantum mechanics, vol. 166, Springer (2006).Google Scholar
  116. [116]
    P. Woit, Quantum theory, groups and representations: An introduction, Springer (2017).Google Scholar
  117. [117]
    J. von Neumann, Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann. 104 (1931) 570.Google Scholar
  118. [118]
    H.J. Groenewold, On the Principles of elementary quantum mechanics, Physica 12 (1946) 405 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  119. [119]
    D.F. Walls and G.J. Milburn, Quantum optics, Springer (2007).Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Institute for Gravitation and the Cosmos & Department of Physics, The Pennsylvania State University, Davey LaboratoryUniversity ParkU.S.A.
  2. 2.Departamento de Física — ICEx, Universidade Federal de Minas GeraisBelo HorizonteBrazil

Personalised recommendations