  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.

Quantum Physics