Violation of Bell inequalities in larger Hilbert spaces: robustness and challenges

TL;DR

This paper investigates the practical challenges of detecting Bell inequality violations in high-dimensional quantum systems, highlighting increasing difficulties due to imperfections and perturbations.

Contribution

It analyzes the robustness of Bell inequality violations in large Hilbert spaces under realistic imperfections and quantifies the effort needed for certification.

Findings

Bell violations are theoretically possible at any dimension.

Detection becomes increasingly difficult with system size and imperfections.

Random perturbations significantly hinder violation certification.

Abstract

We explore the challenges posed by the violation of Bell-like inequalities by $d$-dimensional systems exposed to imperfect state-preparation and measurement settings. We address, in particular, the limit of high-dimensional systems, naturally arising when exploring the quantum-to-classical transition. We show that, although suitable Bell inequalities can be violated, in principle, for any dimension of given subsystems, it is in practice increasingly challenging to detect such violations, even if the system is prepared in a maximally entangled state. We characterize the effects of random perturbations on the state or on the measurement settings, also quantifying the efforts needed to certify the possible violations in case of complete ignorance on the system state at hand.