Revealing Incommensurability between Device-Independent Randomness, Nonlocality, and Entanglement using Hardy and Hardy-type Relations

TL;DR

This paper investigates the complex relationships between device-independent randomness, nonlocality, and entanglement using Hardy and CLL relations, revealing non-uniqueness in Hardy-based randomness and demonstrating bounds that are achievable with non-maximal entanglement.

Contribution

It provides a detailed analytical and numerical analysis of Hardy and CLL nonlocality relations, highlighting the incommensurability between randomness, nonlocality, and entanglement in quantum systems.

Findings

Hardy nonlocality allows a range of certifiable randomness for a given nonlocality value.

For CLL nonlocality, certifiable randomness is uniquely determined by the nonlocality measure.

Maximum certifiable randomness can be achieved with non-maximally entangled states.

Abstract

A comprehensive treatment of the quantification of randomness certified device-independently by using the Hardy and Cabello-Liang-Li (CLL) nonlocality relations is provided in the two parties - two measurements per party - two outcomes per measurement (2-2-2) scenario. For the Hardy nonlocality, it is revealed that for a given amount of nonlocality signified by a particular non-zero value of the Hardy parameter, the amount of Hardy-certifiable randomness is not unique, unlike the way the amount of certifiable randomness is related to the CHSH nonlocality. This is because any specified non-maximal value of Hardy nonlocality parameter characterises a set of quantum extremal distributions. Then this leads to a range of certifiable amounts of randomness corresponding to a given Hardy parameter. On the other hand, for a given amount of CLL-nonlocality, the certifiable randomness is unique, similar to that for the CHSH nonlocality. Furthermore, the tightness of our analytical treatment evaluating the respective guaranteed bounds for the Hardy and CLL relations is demonstrated by their exact agreement with the Semi-Definite-Programming based computed bounds. Interestingly, the analytically evaluated maximum achievable bounds of both Hardy and CLL-certified randomness have been found to be realisable for non-maximal values of the Hardy and CLL nonlocality parameters. In particular, we have shown that even close to the maximum 2 bits of CLL-certified randomness can be realised from non-maximally entangled pure two-qubit states corresponding to small values of the CLL nonlocal parameter. This, therefore, clearly illustrates the quantitative incommensurability between randomness, nonlocality and entanglement.