# A Separation of Out-of-time-ordered Correlation and Entanglement

## TL;DR

This paper demonstrates that out-of-time-ordered correlations and entanglement entropy can saturate at different times in quantum systems, challenging the assumption they always scramble information simultaneously, especially in graph-structured circuits.

## Contribution

It provides a rigorous proof of the fundamental difference in scrambling times between OTOC and entanglement in quantum circuits on graphs with bottlenecks.

## Key findings

- OTOC and entanglement saturate at different times in certain graph-based circuits.
- The results generalize previous OTOC studies from lattices to complex graph geometries.
- Supports the idea that black holes are slow scramblers of quantum information.

## 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 arXiv:1807.04363, that black holes may be slow information scramblers, which in turn relates to the black hole information problem. The bounds we obtained 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.