Quantum chaos can be characterized by an exponential growth of the thermal out-of-time-order four-point function up to a scrambling time ${\widehat{u}}_{*}$. We discuss generalizations of this statement for certain higher-point correlation functions. For concreteness, we study the Schwarzian theory of a one-dimensional time reparametrization mode, which describes two-dimensional anti–de Sitter space (${\mathrm{AdS}}_2$) gravity and the low-energy dynamics of the Sachdev-Ye-Kitaev model. We identify a particular set of $2k$-point functions, characterized as being both “maximally braided” and “$k$-out of time order,” which exhibit exponential growth until progressively longer time scales ${\widehat{u}}_{*}^{(k)}\sim (k-1){\widehat{u}}_{*}$. We suggest an interpretation as scrambling of increasingly fine grained measures of quantum information, which correspondingly take progressively longer time to reach their thermal values.

• Received 21 December 2017

DOI:https://doi.org/10.1103/PhysRevLett.120.121601

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Published by the American Physical Society