Towards the fast scrambling conjecture

Abstract

Many proposed quantum mechanical models of black holes include highly nonlocal interactions. The time required for thermalization to occur in such models should reflect the relaxation times associated with classical black holes in general relativity. Moreover, the time required for a particularly strong form of thermalization to occur, sometimes known as scrambling, determines the time scale on which black holes should start to release information. It has been conjectured that black holes scramble in a time logarithmic in their entropy, and that no system in nature can scramble faster. In this article, we address the conjecture from two directions. First, we exhibit two examples of systems that do indeed scramble in logarithmic time: Brownian quantum circuits and the antiferromagnetic Ising model on a sparse random graph. Unfortunately, both fail to be truly ideal fast scramblers for reasons we discuss. Second, we use Lieb-Robinson techniques to prove a logarithmic lower bound on the scrambling time of systems with finite norm terms in their Hamiltonian. The bound holds in spite of any nonlocal structure in the Hamiltonian, which might permit every degree of freedom to interact directly with every other one.

Black holes as random particles: entanglement dynamics in infinite range and matrix models

“…Also, as shown in the previous section, notice that any other entanglement entropy associated to other subsystems reach the entanglement plateaux at this time scale as well. At this point, we want to remark that this result, if correct, furnish a counterexample of the mean field bound presented in [36]. In ref.…”

“…We want to stress again that the lower bounds presented in ref. [36] seem not to be valid generically. The breakdown of the mean field approximation, or analogously the breakdown of the factorization of the initial state (3.5), occurs on a time scale given by t = 1/R, independent of the system size, see (3.6).…”

“…We believe that our results, formulas (2.23), (2.28) and (2.37), which follow from relatively simple computations, show that production and stationarity of entanglement entropy are attained in a time scale independent of the system size. If correct, they furnish a counterexample to the arguments presented in [36], since this early time entanglement cannot be explained with a mean field approximation, such as (3.9). It is obvious that this issue deserves further consideration.…”

“…In local theories the intuition leads naturally to consider diffusion processes. It was then more rigorously (and abstractly) defined by the 'Page's test' [35], as the time by which subsystems are maximally entangled with the environment, see [5,28,36]. The logic behind this last definition is well known in the context of quantum thermalization, see the recent review [17].…”

“…In ref. [36] it was claimed that there is a lower bound for the time scale of entanglement production in gaussian systems with fully non-local interactions. This lower bound was claimed to be proportional to log N .…”