Towards the fast scrambling conjecture

TL;DR

This paper investigates the fast scrambling conjecture in quantum systems related to black holes, providing examples that scramble in logarithmic time and proving a lower bound on scrambling time using Lieb-Robinson techniques.

Contribution

It presents examples of systems that scramble in logarithmic time and establishes a logarithmic lower bound on scrambling time for systems with finite norm Hamiltonians.

Findings

Brownian quantum circuits scramble in logarithmic time

Antiferromagnetic Ising model on sparse graphs scrambles in logarithmic time

A logarithmic lower bound on scrambling time is proven for finite norm Hamiltonian systems

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.