Towards the fast scrambling conjecture (1111.6580v2)

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.

• The paper demonstrates that two quantum systems achieve logarithmic scrambling times, reinforcing the notion that black holes are nature’s fastest scramblers.
• The authors use stochastic calculus and graph-theoretic arguments in Brownian circuits and sparse Ising models to validate rapid information dispersion.
• The study extends Lieb-Robinson bounds to nonlocal interactions, establishing fundamental limits that preclude sub-logarithmic scrambling times.

Overview of "Towards the Fast Scrambling Conjecture"

The paper "Towards the Fast Scrambling Conjecture" by Lashkari et al. addresses the conjecture that black holes are the fastest scramblers in nature, capable of scrambling information in a time logarithmic in their entropy. This conjecture, as posited by Sekino and Susskind, connects the rapid thermalization characteristics of black holes with quantum information dynamics, suggesting that no physical system can scramble information faster than black holes. The paper explores this conjecture from both constructive and theoretical perspectives.

Constructive Approach

The authors present two quantum systems that exhibit the capacity for fast scrambling in logarithmic time:

1. Brownian Quantum Circuits: These circuits are modeled with time-evolving Hamiltonians characterized by stochastic variations, specifically in two-body interactions. The authors prove via stochastic calculus that such Brownian quantum circuits achieve scrambling in $O(\log n)$ time, where $n$ is the number of qubits. This is demonstrated by tracking the evolution of purity in subsystems, providing evidence that mean-field approximations hold for short times, thereby scrambling information rapidly across the entire system.
2. Antiferromagnetic Ising Model on Sparse Random Graphs: This model, utilizing an antiferromagnetic Ising interaction on a graph where each node represents a spin, scrambles a specific basis efficiently. In these systems, information initially stored in the $\sigma^x$ basis becomes indistinguishable in expected logarithmic time $\pi(\log n + c)$, where $c$ is a constant. This is achieved using graph theoretic arguments, specifically examining the rank of adjacency matrices on these graphs.

Neither of these examples precisely models a natural fast scrambler as required by the conjecture; Brownian quantum circuits rely on explicit time-dependent interaction tuning, and the Ising models do not universally scramble all initial configurations. Nonetheless, they illustrate the plausibility of constructing quantum systems that approach the fast scrambling time scale theoretically associated with black holes.

Theoretical Approach

The paper further explores limits on scrambling times using Lieb-Robinson bounds, traditionally applied to establish locality in quantum lattice systems. The authors adapt these techniques to the more abstract setting of systems with nonlocal Hamiltonians:

• Lieb-Robinson Techniques: By analyzing information propagation, the paper establishes that for two-body systems with bounded operator norms and interaction graphs of maximum degree $D$, scrambling cannot occur in less time than $O(\log D)$. This bound holds regardless of interactions allowing any degree of freedom to access all others directly, hence supporting the impossibility of sub-logarithmic scrambling times without contravening physical laws.
• Mean-Field Approximation: Extended arguments demonstrate that mean-field approximations remain valid up to logarithmic times in systems with dense interactions. Here, it is proven that such systems cannot produce significant entanglement before $O(\log n)$ time scales, underpinning the conjecture’s claim of a lower bound on scrambling speed.

Implications and Future Directions

The implications of this research are profound for quantum gravity, particularly in understanding the microscopic dynamics of black holes and their role in information theory. The constructive models and theoretical analyses provide both a confirmation and a cross-examination of the conjecture, suggesting pathways for further exploring nonlocal interactions in high-energy physics and condensed matter systems.

Future work could extend these models to more physically realizable systems, potentially incorporating additional constraints such as energy conservation to produce more realistic scramblers. Furthermore, the unbounded operators in theories approximating black hole dynamics, such as in matrix models, present additional challenges, potentially bridging the gap between quantum computation and gravitational research.

In conclusion, Lashkari et al.'s paper advances the understanding of scrambling dynamics, contributing significantly to the discourse surrounding one of the most fascinating conjectures in theoretical physics, while setting a foundation for future theoretical and experimental investigations. The confirmation of logarithmic scrambling as a potential universal bound remains a tantalizing task, promising deeper insights into the nature of information dynamics in black holes and beyond.