Generalized Entanglement Entropies of Quantum Designs

TL;DR

This paper explores how quantum designs influence entanglement entropy, revealing that certain designs nearly maximize entanglement and relate to quantum chaos, thus extending understanding of randomness in quantum systems.

Contribution

It establishes a connection between quantum designs and generalized entropies, showing designs of specific orders nearly maximize entanglement, and generalizes the fast scrambling conjecture.

Findings

Renyí entropies averaged over designs are nearly maximal.

Designs of logarithmic order maximize all Renyi entropies.

Results relate entanglement behavior to quantum chaos and complexity.

Abstract

The entanglement properties of random quantum states or dynamics are important to the study of a broad spectrum of disciplines of physics, ranging from quantum information to high energy and many-body physics. This work investigates the interplay between the degrees of entanglement and randomness in pure states and unitary channels. We reveal strong connections between designs (distributions of states or unitaries that match certain moments of the uniform Haar measure) and generalized entropies (entropic functions that depend on certain powers of the density operator), by showing that R\'enyi entanglement entropies averaged over designs of the same order are almost maximal. This strengthens the celebrated Page's theorem. Moreover, we find that designs of an order that is logarithmic in the dimension maximize all R\'enyi entanglement entropies, and so are completely random in terms of the entanglement spectrum. Our results relate the behaviors of R\'enyi entanglement entropies to the complexity of scrambling and quantum chaos in terms of the degree of randomness, and suggest a generalization of the fast scrambling conjecture.