Robust Randomness Amplifiers: Upper and Lower Bounds

TL;DR

This paper studies the fundamental limits of randomness amplification protocols, providing new bounds on their maximum expansion and analyzing the robustness of various classes of protocols.

Contribution

It introduces a systematic framework for analyzing randomness amplifiers and establishes upper bounds on their expansion capabilities, especially for non-adaptive and CHSH-based protocols.

Findings

Non-adaptive, noise-robust amplifiers cannot exceed doubly exponential expansion.

Protocols based on the CHSH game are limited to exponential expansion with non-signaling adversaries.

The paper provides improved analysis of a robust exponential expansion randomness amplifier.

Abstract

A recent sequence of works, initially motivated by the study of the nonlocal properties of entanglement, demonstrate that a source of information-theoretically certified randomness can be constructed based only on two simple assumptions: the prior existence of a short random seed and the ability to ensure that two black-box devices do not communicate (i.e. are non-signaling). We call protocols achieving such certified amplification of a short random seed randomness amplifiers. We introduce a simple framework in which we initiate the systematic study of the possibilities and limitations of randomness amplifiers. Our main results include a new, improved analysis of a robust randomness amplifier with exponential expansion, as well as the first upper bounds on the maximum expansion achievable by a broad class of randomness amplifiers. In particular, we show that non-adaptive randomness amplifiers that are robust to noise cannot achieve more than doubly exponential expansion. Finally, we show that a wide class of protocols based on the use of the CHSH game can only lead to (singly) exponential expansion if adversarial devices are allowed the full power of non-signaling strategies. Our upper bound results apply to all known non-adaptive randomness amplifier constructions to date.