Towards the fast scrambling conjecture

TL;DR

This work probes the fast scrambling conjecture by constructing two clear examples—Brownian quantum circuits and antiferromagnetic Ising on sparse random graphs—that scramble information in time scaling as log n, illustrating that logarithmic scrambling is achievable even with simple, nonlocal interactions, though neither example fully satisfies all criteria for an ideal fast scrambler. It then establishes rigorous lower bounds using generalized Lieb-Robinson techniques, showing that dense two-body Hamiltonians with finite-norm terms cannot scramble faster than O(log n), and extends some arguments to four-body Hamiltonians akin to BFSS matrix models; for sparse graphs, the bounds weaken to about O(√log n). The paper also discusses the role of initial state purity and mean-field approximations, clarifying where scrambling-induced signalling must occur and how locality-like constraints emerge from nonlocal interactions. Together, these results advance the understanding of scrambling dynamics in many-body quantum systems and illuminate the challenges in modeling black-hole information leakage with time-independent Hamiltonians.

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.

Figures (7)

• Figure 1: Schematic plot of the decay of the average purity $\overline{h}_k(t)$ of a subsystem $S$ of size $k$. When the initial state is a pure product state all purities begin equal to one. The scrambling time for a system of size $k$ is defined as the amount of time required before purity of subsystems of size $k$ becomes less than $(1+\delta)2^{-k}$; a purity of exactly $2^{-k}$ corresponds to the maximally mixed state. For subsystems of size smaller than $n/2$, the dynamics ensures that larger systems have smaller purities, a property not necessarily true of general entangled states.
• Figure 2: Antiferromagnetic Ising interaction on an undirected graph $G=(V,E)$. There is term $H_{\langle u,v \rangle}$ in the Hamiltonian for each edge $\langle u,v \rangle \in E$ of the graph. Generic sparse graphs with average vertex degree roughly $\log |V|$ will quickly scramble information stored in the simultaneous $\{ \sigma^x_v : v \in V \}$ eigenbasis.
• Figure 3: Scrambling implies signalling. Site $1$ is prepared in one of two orthogonal states $|j\rangle$ for $j$ either $0$ or $1$. All other sites are prepared in states that are independent of $j$. After the scrambling time $t_*$ any subsystem $S$ of size at most $\kappa n$ will be essentially independent of $j$, but the reduced states $\Psi_{S^c}^{(j)}(t_*)$ on the complementary subsystem $S^c$ will be nearly orthogonal. Scrambling therefore implies signalling from the first site to the complementary system $S^c$.
• Figure 4: Proving the Lieb-Robinson bound on a graph involves a sum over pairs of vertices that contain paths between $x$ and $y$. Starting with a set of edges, paths can be visualized for the purpose of counting as different ways of identifying vertices in successive edges of a sequence. For example, in the figure, each bubble represents an edge and the blue lines indicate the identified vertices: $z = z_1 = z_2'$, $z_2=z_3'$ and $z_3 = y$. There is therefore a path with the following edges: ${\langle x, z \rangle}$, ${\langle z, z'_1 \rangle}$, ${\langle z, z_2 \rangle}$, ${\langle z_2, y \rangle}$ and ${\langle y, z_4 \rangle}$.
• Figure 5: Scrambling implies signalling for mixed initial states. Site $1$ is prepared in one of two orthogonal states $|j\rangle$ for $j$ either $0$ or $1$, and all other states are prepared in states that are independent of $j$ and highly mixed. These mixed states can be viewed as parts of pure states that are entangled with environmental degrees of freedom $E_2$ through $E_n$. When the initial states are maximally mixed, it is possible to scramble subsystems $S$ of size $n - O(1)$. This leads to signalling to the complementary degrees of freedom $S^c$, adjoined with the environmental degrees of freedom $E = E_2 \cdots E_n$. That is, the states $\Psi_{S^c E}^{(j)}(t_*)$ are nearly orthogonal to each other. Because $S^c$ can be taken to be constant-sized, the Lieb-Robinson bound provides nontrivial lower bounds on the signalling, and hence scrambling, time in this setting without the need for additional argument.