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

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.