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.
- Received 11 October 2019
- Revised 22 December 2020
- Accepted 4 May 2021
DOI:https://doi.org/10.1103/PRXQuantum.2.020339
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
“Scrambling” refers to the process in which initial information stored locally in a system spread out to the entire system as it evolves quantum mechanically, and becomes inaccessible when only looking at a part of the system. It plays central roles in the recent study of many key areas in physics, including quantum information, quantum gravity, and quantum many-body physics. Researchers have proposed several ways to characterize how scrambled the system becomes, but their relations are not clear. In this work we show that two of the most prominent types of measures are fundamentally inequivalent. We construct meaningful models motivated by black holes, and show that the scrambling times as indicated by these two measures exhibit vastly different scalings.
The first type we study is based on the so-called out-of-time-ordered correlation (OTOC), which describes the spreading of the perturbation happening on one site towards another site. We prove that such spreading is linear, i.e., the time needed for the spreading is proportional to the length of the shortest path between the sites, even if these path lengths are defined in terms of exotic geometries. Therefore, the scrambling time measured using OTOC is proportional to the distance between farthest sites of the system. The other important type is based on quantum entanglement, and it relies differently on the geometry in which the sites interact. Our observation is that the scrambling time measured by entanglement could be large when there is a tight bottleneck in the geometry. In such cases the two measures give significantly different time scales for scrambling.
Our results on the separation of these two important measures advance the understanding on the essence of scrambling, and are furthermore expected to have important implications for the black-hole information problem as well as many-body physics and quantum computation.
