Chaos in quantum channels

  • Pavan Hosur
  • Xiao-Liang Qi
  • Daniel A. RobertsEmail author
  • Beni Yoshida
Open Access
Regular Article - Theoretical Physics


We study chaos and scrambling in unitary channels by considering their entanglement properties as states. Using out-of-time-order correlation functions to diagnose chaos, we characterize the ability of a channel to process quantum information. We show that the generic decay of such correlators implies that any input subsystem must have near vanishing mutual information with almost all partitions of the output. Additionally, we propose the negativity of the tripartite information of the channel as a general diagnostic of scrambling. This measures the delocalization of information and is closely related to the decay of out-of-time-order correlators. We back up our results with numerics in two non-integrable models and analytic results in a perfect tensor network model of chaotic time evolution. These results show that the butterfly effect in quantum systems implies the information-theoretic definition of scrambling.


AdS-CFT Correspondence Holography and condensed matter physics (AdS/CMT) Gauge-gravity correspondence Random Systems 


Open Access

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  1. [1]
    P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  2. [2]
    Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].CrossRefADSGoogle Scholar
  3. [3]
    N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the Fast Scrambling Conjecture, JHEP 04 (2013) 022 [arXiv:1111.6580] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  4. [4]
    D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
  5. [5]
    D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  6. [6]
    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  7. [7]
    S.H. Shenker and D. Stanford, Multiple Shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].CrossRefADSGoogle Scholar
  8. [8]
    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].CrossRefADSGoogle Scholar
  9. [9]
    S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  10. [10]
    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, arXiv:1503.01409 [INSPIRE].
  11. [11]
    A. Kitaev, A simple model of quantum holography, talks given at The Kavli Institute for Theoretical Physics (KITP), University of California, Santa Barbara, U.S.A., 7 April 2015 and 27 May 2015.Google Scholar
  12. [12]
    T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  13. [13]
    H. Liu and S.J. Suh, Entanglement Tsunami: Universal Scaling in Holographic Thermalization, Phys. Rev. Lett. 112 (2014) 011601 [arXiv:1305.7244] [INSPIRE].CrossRefADSGoogle Scholar
  14. [14]
    H. Liu and S.J. Suh, Entanglement growth during thermalization in holographic systems, Phys. Rev. D 89 (2014) 066012 [arXiv:1311.1200] [INSPIRE].ADSGoogle Scholar
  15. [15]
    F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, JHEP 06 (2015) 149 [arXiv:1503.06237] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  16. [16]
    M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (2000).zbMATHGoogle Scholar
  17. [17]
    A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  18. [18]
    M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96 (2006) 110405 [INSPIRE].CrossRefADSGoogle Scholar
  19. [19]
    P. Hayden, M. Headrick and A. Maloney, Holographic Mutual Information is Monogamous, Phys. Rev. D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].ADSGoogle Scholar
  20. [20]
    H. Gharibyan and R.F. Penna, Are entangled particles connected by wormholes? Evidence for the ER=EPR conjecture from entropy inequalities, Phys. Rev. D 89 (2014) 066001 [arXiv:1308.0289] [INSPIRE].ADSGoogle Scholar
  21. [21]
    M. Rangamani and M. Rota, Entanglement structures in qubit systems, J. Phys. A 48 (2015) 385301 [arXiv:1505.03696] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  22. [22]
    A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  23. [23]
    S.L. Braunstein, S. Pirandola and K. Życzkowski, Better Late than Never: Information Retrieval from Black Holes, Phys. Rev. Lett. 110 (2013) 101301 [arXiv:0907.1190] [INSPIRE].CrossRefADSGoogle Scholar
  24. [24]
    A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An Apologia for Firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].CrossRefADSGoogle Scholar
  25. [25]
    J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  26. [26]
    A. Kitaev, Hidden correlations in the hawking radiation and thermal noise, talk given at The Fundamental Physics Prize Symposium, Stanford University, Standford, U.S.A., 10 November 2014.Google Scholar
  27. [27]
    S. Sachdev and J.-w. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
  28. [28]
    S. Sachdev, Bekenstein-Hawking Entropy and Strange Metals, Phys. Rev. X 5 (2015) 041025 [arXiv:1506.05111] [INSPIRE].Google Scholar
  29. [29]
    D.A. Roberts and B. Swingle, to appear.Google Scholar
  30. [30]
    E.H. Lieb and D.W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28 (1972) 251 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  31. [31]
    M.B. Hastings and T. Koma, Spectral gap and exponential decay of correlations, Commun. Math. Phys. 265 (2006) 781 [math-ph/0507008] [INSPIRE].
  32. [32]
    M.B. Hastings, Locality in quantum systems, arXiv:1008.5137.
  33. [33]
    S. Leichenauer and M. Moosa, Entanglement Tsunami in (1+1)-Dimensions, Phys. Rev. D 92 (2015) 126004 [arXiv:1505.04225] [INSPIRE].ADSGoogle Scholar
  34. [34]
    H. Casini, H. Liu and M. Mezei, Spread of entanglement and causality, arXiv:1509.05044 [INSPIRE].
  35. [35]
    D. Stanford, Scrambling and the entanglement wedge, unpublished.Google Scholar
  36. [36]
    M.C. Bañuls, J.I. Cirac and M.B. Hastings, Strong and weak thermalization of infinite nonintegrable quantum systems, Phys. Rev. Lett. 106 (2011) 050405 [arXiv:1007.3957].CrossRefADSGoogle Scholar
  37. [37]
    F.G.S.L. Brandao, A.W. Harrow and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, arXiv:1208.0692.
  38. [38]
    D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].ADSGoogle Scholar
  39. [39]
    B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].ADSGoogle Scholar
  40. [40]
    G. Vidal, Entanglement Renormalization, Phys. Rev. Lett. 99 (2007) 220405 [cond-mat/0512165] [INSPIRE].
  41. [41]
    Z. Yang, P. Hayden and X.-L. Qi, Bidirectional holographic codes and sub-AdS locality, arXiv:1510.03784 [INSPIRE].
  42. [42]
    A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  43. [43]
    H. Kim and D.A. Huse, Ballistic spreading of entanglement in a diffusive nonintegrable system, Phys. Rev. Lett. 111 (2013) 127205 [arXiv:1306.4306].CrossRefADSGoogle Scholar
  44. [44]
    S.G. Nezami, P. Hayden, X.L. Qi, N. Thomas, M. Walters and Z. Yang, Random tensor networks as models of holography, to appear.Google Scholar
  45. [45]
    M. Hastings, Random mera states and the tightness of the brandao-horodecki entropy bound, arXiv:1505.06468.
  46. [46]
    D. Marolf, H. Maxfield, A. Peach and S.F. Ross, Hot multiboundary wormholes from bipartite entanglement, Class. Quant. Grav. 32 (2015) 215006 [arXiv:1506.04128] [INSPIRE].CrossRefADSzbMATHMathSciNetGoogle Scholar
  47. [47]
    K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Almqvist & Wiksell (1930).Google Scholar
  48. [48]
    L. Susskind, The Typical-State Paradox: Diagnosing Horizons with Complexity, Fortsch. Phys. 64 (2016) 84 [arXiv:1507.02287] [INSPIRE].CrossRefGoogle Scholar
  49. [49]
    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity Equals Action, arXiv:1509.07876 [INSPIRE].
  50. [50]
    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, Action and Black Holes, arXiv:1512.04993 [INSPIRE].
  51. [51]
    G. ’t Hooft, Dimensional reduction in quantum gravity, in Salamfest (1993) 0284 [gr-qc/9310026] [INSPIRE].
  52. [52]
    L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].CrossRefADSMathSciNetzbMATHGoogle Scholar
  53. [53]
    L. Susskind, Entanglement is not Enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].CrossRefGoogle Scholar
  54. [54]
    M. Fannes, A continuity property of the entropy density for spin lattice systems, Commun. Math. Phys. 31 (1973) 291.CrossRefADSMathSciNetzbMATHGoogle Scholar
  55. [55]
    K.M.R. Audenaert, A sharp fannes-type inequality for the von neumann entropy, J. Phys. A 40 (2007) 8127 [quant-ph/0610146].
  56. [56]
    P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].
  57. [57]
    P. Calabrese and J. Cardy, Quantum Quenches in Extended Systems, J. Stat. Mech. 0706 (2007) P06008 [arXiv:0704.1880] [INSPIRE].
  58. [58]
    P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  59. [59]
    C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches, JHEP 02 (2015) 171 [arXiv:1410.1392] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  60. [60]
    V. Balasubramanian, A. Bernamonti, N. Copland, B. Craps and F. Galli, Thermalization of mutual and tripartite information in strongly coupled two dimensional conformal field theories, Phys. Rev. D 84 (2011) 105017 [arXiv:1110.0488] [INSPIRE].ADSGoogle Scholar
  61. [61]
    A. Allais and E. Tonni, Holographic evolution of the mutual information, JHEP 01 (2012) 102 [arXiv:1110.1607] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  62. [62]
    C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Entanglement Scrambling in 2d Conformal Field Theory, JHEP 09 (2015) 110 [arXiv:1506.03772] [INSPIRE].CrossRefADSGoogle Scholar
  63. [63]
    D. Fattal, T.S. Cubitt, Y. Yamamoto, S. Bravyi and I.L. Chuang, Entanglement in the stabilizer formalism, quant-ph/0406168.
  64. [64]
    B. Yoshida and I.L. Chuang, Framework for classifying logical operators in stabilizer codes, Phys. Rev. A 81 (2010) 052302 [arXiv:1002.0085].CrossRefADSGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Pavan Hosur
    • 1
  • Xiao-Liang Qi
    • 1
  • Daniel A. Roberts
    • 2
    Email author
  • Beni Yoshida
    • 3
    • 4
  1. 1.Department of PhysicsStanford UniversityStanfordU.S.A.
  2. 2.Center for Theoretical Physics and Department of PhysicsMassachusetts Institute of TechnologyCambridgeU.S.A.
  3. 3.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  4. 4.Walter Burke Institute for Theoretical PhysicsCalifornia Institute of TechnologyPasadenaU.S.A.

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