# Scrambling the spectral form factor: unitarity constraints and exact results

## TL;DR

This paper investigates the spectral form factor and quantum speed limits to understand information scrambling, providing exact results for certain models and revealing universal features of quantum evolution related to unitarity constraints.

## Contribution

It derives an exponential lower bound on survival probability, analyzes universal spectral features, and provides exact spectral form factor results in the Gaussian unitary ensemble.

## Key findings

- Exponential lower bound on survival probability governed by energy fluctuations
- Universal non-exponential behavior at short and long times
- Exact spectral form factor in Gaussian unitary ensemble with agreement to numerical studies

## Abstract

Quantum speed limits set an upper bound to the rate at which a quantum system can evolve and as such can be used to analyze the scrambling of information. To this end, we consider the survival probability of a thermofield double state under unitary time-evolution which is related to the analytic continuation of the partition function. We provide an exponential lower bound to the survival probability with a rate governed by the inverse of the energy fluctuations of the initial state. Further, we elucidate universal features of the non-exponential behavior at short and long times of evolution that follow from the analytic properties of the survival probability and its Fourier transform, both for systems with a continuous and a discrete energy spectrum. We find the spectral form factor in a number of illustrative models, notably we obtain the exact answer in the Gaussian unitary ensemble for any $N$ with excellent agreement with recent numerical studies. We also discuss the relationship of our findings to models of black hole information loss, such as the Sachdev-Ye-Kitaev model dual to AdS$_2$ as well as higher-dimensional versions of AdS/CFT.