# Entanglement, quantum randomness, and complexity beyond scrambling

## TL;DR

This paper develops a mathematical framework to analyze quantum randomness and entanglement beyond traditional scrambling, revealing a hierarchy of complexity levels and conditions under which maximal entanglement can be achieved.

## Contribution

It introduces a hierarchy of entanglement complexities based on Rényi entropies and establishes bounds for designs of different orders, extending the fast scrambling conjecture.

## Key findings

- Rényi entanglement entropies are almost maximal for designs of the same order.
- Existence of 2-designs with higher-order entropies bounded away from maximum.
- Logarithmic designs achieve maximal entanglement, indicating max-scrambling.

## Abstract

Scrambling is a process by which the state of a quantum system is effectively randomized due to the global entanglement that "hides" initially localized quantum information. In this work, we lay the mathematical foundations of studying randomness complexities beyond scrambling by entanglement properties. We do so by analyzing the generalized (in particular R\'enyi) entanglement entropies of designs, i.e. ensembles of unitary channels or pure states that mimic the uniformly random distribution (given by the Haar measure) up to certain moments. A main collective conclusion is that the R\'enyi entanglement entropies averaged over designs of the same order are almost maximal. This links the orders of entropy and design, and therefore suggests R\'enyi entanglement entropies as diagnostics of the randomness complexity of corresponding designs. Such complexities form a hierarchy between information scrambling and Haar randomness. As a strong separation result, we prove the existence of (state) 2-designs such that the R\'enyi entanglement entropies of higher orders can be bounded away from the maximum. However, we also show that the min entanglement entropy is maximized by designs of order only logarithmic in the dimension of the system. In other words, logarithmic-designs already achieve the complexity of Haar in terms of entanglement, which we also call max-scrambling. This result leads to a generalization of the fast scrambling conjecture, that max-scrambling can be achieved by physical dynamics in time roughly linear in the number of degrees of freedom.