Chaos in quantum channels


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.

A preprint version of the article is available at ArXiv.


  1. [1]

    P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  2. [2]

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

    Article  ADS  Google 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].

    Article  ADS  MathSciNet  Google 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].

    Article  ADS  MathSciNet  Google Scholar 

  6. [6]

    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  7. [7]

    S.H. Shenker and D. Stanford, Multiple Shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].

    Article  ADS  Google 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].

    Article  ADS  Google Scholar 

  9. [9]

    S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].

    Article  ADS  MathSciNet  Google 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.

  12. [12]

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

    Article  ADS  MathSciNet  Google 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].

    Article  ADS  Google 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].

    ADS  Google 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].

    Article  ADS  MathSciNet  Google Scholar 

  16. [16]

    M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (2000).

    MATH  Google Scholar 

  17. [17]

    A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].

    Article  ADS  MathSciNet  Google 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].

    Article  ADS  Google 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].

    ADS  Google 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].

    ADS  Google Scholar 

  21. [21]

    M. Rangamani and M. Rota, Entanglement structures in qubit systems, J. Phys. A 48 (2015) 385301 [arXiv:1505.03696] [INSPIRE].

    ADS  MathSciNet  MATH  Google 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].

    Article  ADS  MathSciNet  Google 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].

    Article  ADS  Google 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].

    Article  ADS  Google Scholar 

  25. [25]

    J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].

    Article  ADS  MathSciNet  Google 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.

  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.

  30. [30]

    E.H. Lieb and D.W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28 (1972) 251 [INSPIRE].

    Article  ADS  MathSciNet  Google 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].

    ADS  Google 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.

  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].

    Article  ADS  Google 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].

    ADS  Google Scholar 

  39. [39]

    B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].

    ADS  Google 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].

    Article  ADS  MathSciNet  Google 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].

    Article  ADS  Google 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.

  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].

    Article  ADS  MATH  MathSciNet  Google Scholar 

  47. [47]

    K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Almqvist & Wiksell (1930).

  48. [48]

    L. Susskind, The Typical-State Paradox: Diagnosing Horizons with Complexity, Fortsch. Phys. 64 (2016) 84 [arXiv:1507.02287] [INSPIRE].

    Article  Google 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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. [53]

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

    Article  Google Scholar 

  54. [54]

    M. Fannes, A continuity property of the entropy density for spin lattice systems, Commun. Math. Phys. 31 (1973) 291.

    Article  ADS  MathSciNet  MATH  Google 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].

    MathSciNet  MATH  Google 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].

    Article  ADS  MathSciNet  Google 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].

    ADS  Google Scholar 

  61. [61]

    A. Allais and E. Tonni, Holographic evolution of the mutual information, JHEP 01 (2012) 102 [arXiv:1110.1607] [INSPIRE].

    Article  ADS  MATH  Google 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].

    Article  ADS  Google 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].

    Article  ADS  Google Scholar 

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



Corresponding author

Correspondence to Daniel A. Roberts.

Additional information

ArXiv ePrint: 1511.04021

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hosur, P., Qi, XL., Roberts, D.A. et al. Chaos in quantum channels. J. High Energ. Phys. 2016, 4 (2016).

Download citation


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