Entanglement in non-local games and the hyperlinear profile of groups

TL;DR

This paper connects the entanglement needed for near-optimal play in linear-system non-local games to the hyperlinear profile of groups, providing a quantitative measure of entanglement requirements that grow as the game becomes more precise.

Contribution

It introduces a novel link between group hyperlinear profiles and quantum entanglement in non-local games, offering explicit bounds on entanglement as a function of game accuracy.

Findings

Entanglement required grows at least as 1/ε^k for some k>0

Provides a concrete example of a non-local game with unbounded entanglement requirements

Quantifies the relationship between group properties and quantum resources

Abstract

We relate the amount of entanglement required to play linear-system non-local games near-optimally to the hyperlinear profile of finitely-presented groups. By calculating the hyperlinear profile of a certain group, we give an example of a finite non-local game for which the amount of entanglement required to play $\varepsilon$-optimally is at least $\Omega(1/\varepsilon^k)$, for some $k>0$. Since this function approaches infinity as $\varepsilon$ approaches zero, this provides a quantitative version of a theorem of the first author.