Abstract
We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent λ L ≤ 2πk B T/ℏ. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.
References
A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An Apologia for Firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].
P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
A.I. Larkin and Y.N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, JETP 28 (1969) 1200.
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
E.H. Lieb and D.W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28 (1972) 251 [INSPIRE].
M.B. Hastings and T. Koma, Spectral gap and exponential decay of correlations, Commun. Math. Phys. 265 (2006) 781 [math-ph/0507008] [INSPIRE].
M.B. Hastings, Locality in quantum systems, arXiv:1008.5137.
C. Dankert, R. Cleve, J. Emerson and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation, Phys. Rev. A 80 (2009) 012304 [quant-ph/0606161].
A.W. Harrow and R.A. Low, Random quantum circuits are approximate 2-designs, Comm. Math. Phys. 291 (2009) 257 [arXiv:0802.1919].
L. Arnaud and D. Braun, Efficiency of producing random unitary matrices with quantum circuits, Phys. Rev. A 78 (2008) 062329 [arXiv:0807.0775].
W.G. Brown and L. Viola, Convergence rates for arbitrary statistical moments of random quantum circuits, Phys. Rev. Lett. 104 (2010) 250501 [arXiv:0910.0913].
I.T. Diniz and D. Jonathan, Comment on the paper ‘Random quantum circuits are approximate 2-designs’, Comm. Math. Phys. 304 (2011) 281 [arXiv:1006.4202].
W. Brown and O. Fawzi, Scrambling speed of random quantum circuits, arXiv:1210.6644 [INSPIRE].
N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the Fast Scrambling Conjecture, JHEP 04 (2013) 022 [arXiv:1111.6580] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
S.H. Shenker and D. Stanford, Multiple Shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].
A. Kitaev, Hidden Correlations in the Hawking Radiation and Thermal Noise, talk given at Fundamental Physics Prize Symposium, November 10, 2014.
A. Kitaev, Stanford SITP seminars, November 11 and December 18, 2014.
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, Phys. Rev. Lett. 115 (2015) 131603 [arXiv:1412.5123] [INSPIRE].
S. Jackson, L. McGough and H. Verlinde, Conformal Bootstrap, Universality and Gravitational Scattering, Nucl. Phys. B 901 (2015) 382 [arXiv:1412.5205] [INSPIRE].
G. ’t Hooft, The black hole interpretation of string theory, Nucl. Phys. B 335 (1990) 138 [INSPIRE].
Y. Kiem, H.L. Verlinde and E.P. Verlinde, Black hole horizons and complementarity, Phys. Rev. D 52 (1995) 7053 [hep-th/9502074] [INSPIRE].
P. Kovtun, D.T. Son and A.O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601 [hep-th/0405231] [INSPIRE].
Y. Kats and P. Petrov, Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory, JHEP 01 (2009) 044 [arXiv:0712.0743] [INSPIRE].
M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, Viscosity Bound Violation in Higher Derivative Gravity, Phys. Rev. D 77 (2008) 126006 [arXiv:0712.0805] [INSPIRE].
M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, The Viscosity Bound and Causality Violation, Phys. Rev. Lett. 100 (2008) 191601 [arXiv:0802.3318] [INSPIRE].
T. Banks and G. Festuccia, The Regge Limit for Green Functions in Conformal Field Theory, JHEP 06 (2010) 105 [arXiv:0910.2746] [INSPIRE].
R.C. Brower, J. Polchinski, M.J. Strassler and C.-I. Tan, The pomeron and gauge/string duality, JHEP 12 (2007) 005 [hep-th/0603115] [INSPIRE].
T. Dray and G. ’t Hooft, The Gravitational Shock Wave of a Massless Particle, Nucl. Phys. B 253 (1985) 173 [INSPIRE].
K. Sfetsos, On gravitational shock waves in curved space-times, Nucl. Phys. B 436 (1995) 721 [hep-th/9408169] [INSPIRE].
X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, Causality Constraints on Corrections to the Graviton Three-Point Coupling, JHEP 02 (2016) 020 [arXiv:1407.5597] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal Approximation in AdS/CFT: From Shock Waves to Four-Point Functions, JHEP 08 (2007) 019 [hep-th/0611122] [INSPIRE].
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
Additional information
ArXiv ePrint: 1503.01409
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Maldacena, J., Shenker, S.H. & Stanford, D. A bound on chaos. J. High Energ. Phys. 2016, 106 (2016). https://doi.org/10.1007/JHEP08(2016)106
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2016)106
Keywords
- 1/N Expansion
- Black Holes
- AdS-CFT Correspondence