Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Unscrambling the physics of out-of-time-order correlators

Quantitative tools for measuring the propagation of information through quantum many-body systems, originally developed to study quantum chaos, have recently found many new applications from black holes to disordered spin systems.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Schematic of the growth of operators with time.


  1. 1.

    Larkin, A. & Ovchinnikov, Y. N. Quasiclassical method in the theory of superconductivity. J. Exp. Theor. Phys. 28, 1200–1205 (1969).

    ADS  Google Scholar 

  2. 2.

    Hahn, E. L. Spin echoes. Phys. Rev. 80, 580–594 (1950).

    ADS  Article  Google Scholar 

  3. 3.

    Peres, A. Stability of quantum motion in chaotic and regular systems. Phys. Rev. A 30, 1610–1615 (1984).

    ADS  MathSciNet  Article  Google Scholar 

  4. 4.

    Jalabert, R. A. & Pastawski, H. M. Environment-independent decoherence rate in classically chaotic systems. Phys. Rev. Lett. 86, 2490–2493 (2001).

    ADS  Article  Google Scholar 

  5. 5.

    Gorin, T., Prosen, T., Seligman, T. H. & Žnidarič, M. Dynamics of Loschmidt echoes and delity decay. Phys. Rep. 435, 33–156 (2006).

    ADS  Article  Google Scholar 

  6. 6.

    Rhim, W.-K., Pines, A. & Waugh, J. S. Time-reversal experiments in dipolar-coupled spin systems. Phys. Rev. B 3, 684–696 (1971).

    ADS  Article  Google Scholar 

  7. 7.

    Zhang, S., Meier, B. H. & Ernst, R. R. Polarization echoes in NMR. Phys. Rev. Lett. 69, 2149–2151 (1992).

    ADS  Article  Google Scholar 

  8. 8.

    Levstein, P. R., Usaj, G. & Pastawski, H. M. Attenuation of polarization echoes in nuclear magnetic resonance: A study of the emergence of dynamical irreversibility in many-body quantum systems. J. Chem. Phys. 108, 2718–2724 (1998).

    ADS  Article  Google Scholar 

  9. 9.

    Swingle, B., Bentsen, G., Schleier-Smith, M. & Hayden, P. Measuring the scrambling of quantum information. Phys. Rev. A 94, 040302 (2016).

    ADS  MathSciNet  Article  Google Scholar 

  10. 10.

    Yao, N. Y. et al. Interferometric approach to probing fast scrambling. Preprint at (2016).

  11. 11.

    Zhu, G., Hafezi, M. & Grover, T. Measurement of many-body chaos using a quantum clock. Phys. Rev. A 94, 062329 (2016).

    ADS  Article  Google Scholar 

  12. 12.

    Campisi, M. & Goold, J. Thermodynamics of quantum information scrambling. Phys. Rev. E 95, 062127 (2017).

    ADS  Article  Google Scholar 

  13. 13.

    Yunger Halpern, N. Jarzynski-like equality for the out-of-time-ordered correlator. Phys. Rev. A 95, 012120 (2017).

    ADS  Article  Google Scholar 

  14. 14.

    Gärttner, M. et al. Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet. Nat. Phys. 13, 781–786 (2017).

    Article  Google Scholar 

  15. 15.

    Li, J. et al. Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator. Phys. Rev. X 7, 031011 (2017).

    Google Scholar 

  16. 16.

    Wei, K. X., Ramanathan, C. & Cappellaro, P. Exploring localization in nuclear spin chains. Phys. Rev. Lett. 120, 070501 (2018).

    ADS  Article  Google Scholar 

  17. 17.

    Meier, E. J., Ang’ong’a, J., An, F. A. & Gadway, B. Exploring quantum signatures of chaos on a Floquet synthetic lattice. Preprint at (2017).

  18. 18.

    Roberts, D. A. & Swingle, B. Lieb-Robinson bound and the butterfly effect in quantum field theories. Phys. Rev. Lett. 117, 091602 (2016).

    ADS  Article  Google Scholar 

  19. 19.

    Shenker, S. H. & Stanford, D. Black holes and the butterfly effect. J. High Energy Phys. 2014, 67 (2014).

    MathSciNet  Article  Google Scholar 

  20. 20.

    Kitaev, A. A simple model of quantum holography. KITP (2015).

  21. 21.

    Maldacena, J., Shenker, S. H. & Stanford, D. A bound on chaos. J. High Energy Phys. 2016, 106 (2016).

    MathSciNet  Article  Google Scholar 

  22. 22.

    Sachdev, S. & Ye, J. Gapless spin-liquid ground state in a random quantum Heisenberg magnet. Phys. Rev. Lett. 70, 3339–3342 (1993).

    ADS  Article  Google Scholar 

  23. 23.

    Polchinski, J. & Rosenhaus, V. The spectrum in the Sachdev-Ye-Kitaev model. J. High Energy Phys. 2016, 1 (2016).

    MathSciNet  Article  Google Scholar 

  24. 24.

    Maldacena, J. & Stanford, D. Remarks on the Sachdev-Ye-Kitaev model. Phys. Rev. D 94, 106002 (2016).

    ADS  MathSciNet  Article  Google Scholar 

  25. 25.

    Kukuljan, I., Grozdanov, S. & Prosen, T. Weak quantum chaos. Phys. Rev. B 96, 060301 (2017).

    ADS  Article  Google Scholar 

  26. 26.

    Hayden, P. & Preskill, J. Black holes as mirrors: quantum information in random subsystems. J. High Energy Phys. 2007, 120 (2007).

    MathSciNet  Article  Google Scholar 

  27. 27.

    Hosur, P., Qi, X.-L., Roberts, D. A. & Yoshida, B. Chaos in quantum channels. J. High Energy Phys. 2016, 4 (2016).

    MathSciNet  Article  Google Scholar 

  28. 28.

    Maldacena, J., Stanford, D. & Yang, Z. Diving into traversable wormholes. Fortschr. Phys. 65, 1700034 (2017).

    MathSciNet  Article  Google Scholar 

  29. 29.

    Yoshida, B. & Kitaev, A. Efficient decoding for the Hayden-Preskill protocol. Preprint at (2017).

  30. 30.

    Gao, P., Jafferis, D. L. & Wall, A. C. Traversable wormholes via a double trace deformation. J. High Energy Phys. 2017, 151 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  31. 31.

    Huse, D. A., Nandkishore, R. & Oganesyan, V. Phenomenology of fully many-body-localized systems. Phys. Rev. B 90, 174202 (2014).

    ADS  Article  Google Scholar 

  32. 32.

    Serbyn, M., Papic, Z. & Abanin, D. A. Local conservation laws and the structure of the many-body localized states. Phys. Rev. Lett. 111, 127201 (2013).

    ADS  Article  Google Scholar 

  33. 33.

    Huang, Y., Zhang, Y.-L. & Chen, X. Out-of-time-ordered correlator in many-body localized systems. Ann. Phys. 529, 1600318 (2017).

    Article  Google Scholar 

  34. 34.

    Fan, R., Zhang, P., Shen, H. & Zhai, H. Out-of-time-order correlation for many-body localization. Sci. Bull. 62, 707–711 (2017).

    Article  Google Scholar 

  35. 35.

    Swingle, B. & Chowdhury, D. Slow scrambling in disordered quantum systems. Phys. Rev. B 95, 060201 (2017).

    ADS  Article  Google Scholar 

  36. 36.

    Chen, Y. Quantum logarithmic butterfly in many body localization. Preprint at (2016).

  37. 37.

    Slagle, K., Bi, Z., You, Y.-Z. & Xu, C. Out-of-time-order correlation in marginal many-body localized systems. Phys. Rev. B 95, 165136 (2017).

    ADS  Article  Google Scholar 

  38. 38.

    Yunger Halpern, N., Swingle, B. & Dressel, J. The quasiprobability behind the out-of-time-ordered correlator. Phys. Rev. A 97, 042105 (2018).

    ADS  Article  Google Scholar 

  39. 39.

    Davis, E., Bentsen, G. & Schleier-Smith, M. Approaching the Heisenberg limit without single-particle detection. Phys. Rev. Lett. 116, 053601 (2016).

    ADS  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Brian Swingle.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Swingle, B. Unscrambling the physics of out-of-time-order correlators. Nature Phys 14, 988–990 (2018).

Download citation

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing