The non-locality of n noisy Popescu-Rohrlich boxes

TL;DR

This paper analyzes how the non-locality of noisy Popescu-Rohrlich boxes scales with the number of systems, providing quantitative bounds on their local parts based on noise levels.

Contribution

It introduces a method to quantify the non-locality in multiple noisy PRBs and derives bounds on their local components depending on noise and bias.

Findings

Local part scales as Θ(ε^{ceil(n/2)}) in isotropic case

Local part is (3ε)^n in maximally biased case

Provides quantitative bounds on non-locality in noisy PRBs

Abstract

We quantify the amount of non-locality contained in n noisy versions of so-called Popescu-Rohrlich boxes (PRBs), i.e., bipartite systems violating the CHSH Bell inequality maximally. Following the approach by Elitzur, Popescu, and Rohrlich, we measure the amount of non-locality of a system by representing it as a convex combination of a local behaviour, with maximal possible weight, and a non-signalling system. We show that the local part of n systems, each of which approximates a PRB with probability $1-\epsilon$, is of order $\Theta(\epsilon^{\lceil n/2\rceil})$ in the isotropic, and equal to $(3\epsilon)^n$ in the maximally biased case.