Bell scenarios in which nonlocality and entanglement are inversely related

TL;DR

This paper analytically demonstrates that in two-qubit chained Bell inequalities, nonlocality and entanglement are inversely related, with the relationship holding in both noiseless and noisy scenarios, across various Bell inequalities.

Contribution

It provides the first analytical proof of the inverse relationship between nonlocality and entanglement in chained Bell inequalities and extends this to other Bell scenarios.

Findings

Nonlocality and entanglement are inversely related in chained Bell inequalities.

Robustness of nonlocality and concurrence are inversely related in noisy conditions.

The inverse relationship also applies to specific two-qubit and two-qutrit Bell inequalities.

Abstract

We show that for two-qubit chained Bell inequalities with an arbitrary number of measurement settings, nonlocality and entanglement are not only different properties but are inversely related. Specifically, we analytically prove that in absence of noise, robustness of nonlocality, defined as the maximum fraction of detection events that can be lost such that the remaining ones still do not admit a local model, and concurrence are inversely related for any chained Bell inequality with an arbitrary number of settings. The closer quantum states are to product states, the harder it is to reproduce quantum correlations with local models. We also show that, in presence of noise, nonlocality and entanglement are simultaneously maximized only when the noise level is equal to the maximum level tolerated by the inequality; in any other case, a more nonlocal state is always obtained by reducing the entanglement. In addition, we observed that robustness of nonlocality and concurrence are also inversely related for the Bell scenarios defined by the tight two-qubit three-setting $I_{3322}$ inequality, and the tight two-qutrit inequality $I_3$.