Separation of Out-Of-Time-Ordered Correlation and Entanglement

Abstract

The out-of-time-ordered correlation (OTOC) and entanglement are two physically motivated and widely used probes of the “scrambling” of quantum information, a phenomenon that has drawn great interest recently in quantum gravity and many-body physics. We argue that the corresponding notions of scrambling can be fundamentally different, by proving an asymptotic separation between the time scales of the saturation of OTOC and that of entanglement entropy in a random quantum-circuit model defined on graphs with a tight bottleneck, such as tree graphs. Our result counters the intuition that a random quantum circuit mixes in time proportional to the diameter of the underlying graph of interactions. It also provides a more rigorous justification for an argument in our previous work [Shor P.W., Scrambling time and causal structure of the photon sphere of a Schwarzschild black hole, arXiv:1807.04363 (2018)], that black holes may be slow information scramblers, which in turn relates to the black-hole information problem. The bounds we obtain for OTOC are interesting in their own right in that they generalize previous studies of OTOC on lattices to the geometries on graphs in a rigorous and general fashion.