Quantum-proof randomness extractors via operator space theory

TL;DR

This paper introduces a novel operator space framework to analyze quantum-proof randomness extractors and condensers, establishing their security against quantum adversaries through bounded and completely bounded norm conditions.

Contribution

It applies operator space theory to characterize quantum-proof extractors and condensers, linking their security to well-studied mathematical norms and bipartite graph problems.

Findings

High min-entropy extractors are approximately quantum-proof.

Extractors with small output are approximately quantum-proof.

Quantum-proof condensers correspond to Bell inequalities that quantum mechanics cannot violate.

Abstract

Quantum-proof randomness extractors are an important building block for classical and quantum cryptography as well as device independent randomness amplification and expansion. Furthermore they are also a useful tool in quantum Shannon theory. It is known that some extractor constructions are quantum-proof whereas others are provably not [Gavinsky et al., STOC'07]. We argue that the theory of operator spaces offers a natural framework for studying to what extent extractors are secure against quantum adversaries: we first phrase the definition of extractors as a bounded norm condition between normed spaces, and then show that the presence of quantum adversaries corresponds to a completely bounded norm condition between operator spaces. From this we show that very high min-entropy extractors as well as extractors with small output are always (approximately) quantum-proof. We also study a generalization of extractors called randomness condensers. We phrase the definition of condensers as a bounded norm condition and the definition of quantum-proof condensers as a completely bounded norm condition. Seeing condensers as bipartite graphs, we then find that the bounded norm condition corresponds to an instance of a well studied combinatorial problem, called bipartite densest subgraph. Furthermore, using the characterization in terms of operator spaces, we can associate to any condenser a Bell inequality (two-player game) such that classical and quantum strategies are in one-to-one correspondence with classical and quantum attacks on the condenser. Hence, we get for every quantum-proof condenser (which includes in particular quantum-proof extractors) a Bell inequality that can not be violated by quantum mechanics.