Yuan Liu

Violations of Bell inequalities are often regarded as evidence that quantum nonlocality guarantees intrinsic randomness, effectively playing the role of a “dice” at the heart of device-independent (DI) cryptographic protocols. Yet the precise connection between nonlocality and randomness is more nuanced. We first show that there exist nontrivial Bell inequalities that are maximally violated by quantum correlations while certifying no randomness for any fixed input pair, rendering them ineffective for a large class of standard DI schemes. Moreover, we construct maximally nonlocal quantum correlations that remain deterministic for every fixed input pair, in the sense that for any chosen inputs they can be decomposed into strategies with fixed outputs. Conversely, we show when all input pairs are used for randomness generation, any amount of quantum nonlocality suffices to certify randomness, implying that no form of bound randomness exists in quantum nonlocality: every nonlocal behavior can be useful for DI randomness generation under an appropriately designed protocol. Building on this, we introduce the average guessing probability over all inputs, in contrast to the hitherto considered fixed-input guessing probability, as a faithful and monotonic quantifier of nonlocality. Using this measure, we prove that, contrary to recent findings in PRL 134, 090201, the detection efficiency threshold for certifying randomness is never lower than that required for detecting nonlocality. Finally, we analytically compute the average guessing probability by a quantum adversary in the standard CHSH test and show how this leads to improved generation rates in state-of-the-art amplification protocols. Together, our results precisely delineate the limits of determinism compatible with quantum nonlocality and establish average guessing probability as the correct operational bridge between nonlocality and DI randomness.