Necessary and Sufficient Condition for Randomness Certification from Incompatibility

TL;DR

This paper establishes the exact conditions under which quantum measurements can certify randomness, linking measurement incompatibility structures to the ability to generate certifiable quantum randomness.

Contribution

It provides a necessary and sufficient condition for randomness certification based on measurement incompatibility, extending to Bell scenarios and chain inequalities.

Findings

Certifiable randomness requires non-isomorphic measurement compatibility structures.

Violation of chain inequalities confirms the absence of certain compatibility structures.

Results identify minimal quantum resources needed for randomness certification.

Abstract

Quantum randomness can be certified from probabilistic behaviors demonstrating Bell nonlocality or Einstein-Podolsky-Rosen steering, leveraging outcomes from uncharacterized devices. However, such nonlocal correlations are not always sufficient for this task, necessitating the identification of required minimum quantum resources. In this work, we provide the necessary and sufficient condition for nonzero certifiable randomness in terms of measurement incompatibility and develop approaches to detect them. Firstly, we show that the steering-based randomness can be certified if and only if the correlations arise from a measurement compatibility structure that is not isomorphic to a hypergraph containing a star subgraph. In such a structure, the central measurement is individually compatible with the measurements at branch sites, precluding certifiable randomness in the central measurement outcomes. Subsequently, we generalize this result to the Bell scenario, proving that the violation of any chain inequality involving $m$ inputs and $d$ outputs rules out such a compatibility structure, thereby validating all chain inequalities as credible witnesses for randomness certification. Our results point out the role of incompatibility structure in generating random numbers, offering a way to identify minimum quantum resources for the task.