Ravishankar Ramanathan

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.

We consider quantum statistics from the perspective of post-quantum no-signaling theories in which either none or only a certain number of systems are trusted. These scenarios can be fully described by so-called no-signaling boxes or no-signaling assemblages respectively. It has been shown so far that in the usual Bell non-locality scenario with a single measurement run, quantum correlations can never reproduce an extremal non-local point within the set of no-signaling boxes. We provide here a general no-go rule showing that the latter stays true even if arbitrary sequential measurements are allowed. On the other hand, we prove a positive result showing that already a single trusted qubit is enough for quantum theory to produce a self-testable extremal point within the corresponding set of no-signaling assemblages. This result provides a tool that opens up possibilities for security proofs of cryptographic protocols against general no-signaling adversaries in semi-device-independent scenarios.

A direct analysis of the protocol of randomness amplification using Bell inequality violation is performed in terms of the convex combination of no-signaling boxes required to simulate quantum violation of the inequality. The probability distributions of bits generated by a Santha-Vazirani source are shown to be mixtures of permutations of Bernoulli distributions with parameter defined by the source. An intuitive proof is provided for the range of partial randomness from which perfect randomness can be extracted using quantum correlations violating the chain inequalities. Exact values are derived in the asymptotic limit of a large number of measurement settings.