Maximum Quantum Non-Locality is not always Sufficient for Device-Independent Randomness Generation

Abstract: The outcomes of local measurements on entangled quantum systems can be certified to be genuinely random through the violation of a Bell Inequality. The randomness of the outcomes with respect to an adversary is quantified by the guessing probability, conditioned upon the observation of a specific amount of Bell violation or upon the observation of the entire input-output behavior. It has been an open question whether standard device-independent randomness generation protocols against classical or quantum adversaries can be constructed on the basis of any arbitrary Bell inequality, i.e., does there exist a Bell inequality for which the guessing probability is one for any chosen input even upon observing the maximal violation of the inequality? A strengthened version of the question asks whether there exists a quantum behavior that exhibits maximum non-locality but zero certifiable randomness for any arbitrary input. In this paper, we present an affirmative answer to both questions by constructing families of $n$-player non-local games for $n \geq 2$ and families of non-local behaviors on the quantum boundary that do not allow to certify any randomness against a classical adversary. Our results show the existence of a form of bound randomness against classical adversaries, highlighting that device-independent randomness and quantum non-locality can be maximally inequivalent resources.