Gluing Randomness via Entanglement: Tight Bound from Second R\'enyi Entropy

TL;DR

This paper establishes a tight bound linking the second R'enyi entanglement entropy of an initial state to the quality of approximate random states generated via local unitaries, highlighting entanglement as a key resource.

Contribution

It introduces a precise relationship between second R'enyi entropy and the ability to generate approximate random states using local unitaries, and extends this to resource-free operations and pseudorandom state generation.

Findings

Approximate random states form a design with error $ heta(e^{- ext{second R'enyi entropy})}$.

Second R'enyi entropy provides the tightest bounds among all R'enyi entropies.

Entanglement acts as a resource that determines the quality of randomness generation.

Abstract

The efficient generation of random quantum states is a long-standing challenge, motivated by their diverse applications in quantum information processing tasks. In this work, we identify entanglement as the key resource that enables local random unitaries to generate global random states by effectively gluing randomness across the system. Specifically, we demonstrate that approximate random states can be produced from an entangled state $|\psi\rangle$ through the application of local random unitaries. We show that the resulting ensemble forms an approximate state design with an error saturating as $\Theta(e^{-\mathcal{N}_2(\psi)})$, where $\mathcal{N}_2(\psi)$ is the second R\'enyi entanglement entropy of $|\psi\rangle$. Furthermore, we prove that this tight bound also applies to the second R\'enyi entropy of coherence when the ensemble is constructed using coherence-free operations. These results imply that, when restricted to resource-free gates, the quality of the generated random states is determined entirely by the resource content of the initial state. Notably, we find that among all $\alpha$-R\'enyi entropeis, the second R\'enyi entropy yields the tightest bounds. Consequently, these second R\'enyi entropies can be interpreted as the maximal capacities for generating randomness using resource-free operations. Finally, moving beyond approximate state designs, we utilize this entanglement-assisted gluing mechanism to present a novel method for generating pseudorandom states in multipartite systems from a locally entangled state via pseudorandom unitaries in each of parties.