Quantum Nonlocality and Device-Independent Randomness Robust to Relaxations of Bell Assumptions

The question of certifying quantum nonlocality under a relaxation of the assumptions in the Bell theorem has gained traction, with potential for device-independent applications under weak seeds and cross-talk. Recently, it was shown that quantum nonlocality can be certified even under a simultaneous arbitrary (but not full) relaxation of the assumptions of Measurement Independence (MI) and Parameter Independence (PI), using states of local dimension $d = poly((1-\epsilon)^{-1})$ for an $\epsilon \in [0,1)$-relaxation. Here, we derive three results strengthening the state-of-art. Firstly, we show that states of constant local dimension $d$ are already sufficient to certify quantum nonlocality under arbitrary MI and PI relaxation, albeit in a non-robust manner. Secondly, and as a theoretical paradigm to derive the above, we introduce the notion of \textit{measurement-dependent parameter-dependent locality} as the set of input-output behaviors under simultaneous relaxations of measurement and parameter independence. We provide a rigorous characterisation of the vertices of the polytope of joint input-output behaviors that obey a $\mu$-relaxation of MI and $\epsilon$-relaxation of PI. We highlight a relation between nonlocality certification under PI relaxation and that under detection inefficiencies by pointing out alternative extremal correlations to the Eberhard correlations that also allow to achieve detection efficiency of $\eta = 2/3$ in the two-input scenario. Finally, we study the implication of the relaxed assumptions for device-independent randomness certification. We analytically derive the quantum guessing probability for one player's outcomes in the CHSH Bell test, as a function of the noise in the test as well as of a leakage of an average amount of $I(X:B) < 1$ bits of input information per measurement round.