Randomness versus Nonlocality in Multiple-Input and Multiple-Output Quantum Scenario

Device-independent randomness certification based on Bell nonlocality does not require any assumptions about the devices, thus provides adequate security. Great effort has been made to demonstrate that nonlocality is necessary for generating quantum randomness, but the minimal resource required has not been clarified. Here we prove and experimentally demonstrate that violating any two-input Bell inequality is both necessary and sufficient for certifying randomness, however, for the multiple-input cases, this sufficiency ceases to apply, leading to certain states exhibiting Bell nonlocality without the capability to certify randomness. We examine two typical classes of Bell inequalities with multiple input and multiple output, the facet inequalities and Salavrakos-Augusiak-Tura-Wittek-Acín-Pironio inequalities, in the high-dimensional photonic system, and observe that the violation of the latter one can always certify randomness which is not true for the former. The private randomness with a generation rate of 1.867±0.018  bits per photon pair is obtained in the scenario of Salavrakos-Augusiak-Tura-Wittek-Acín-Pironio inequalities with 3-input and 4-output. Our Letter unravels the internal connection between randomness and nonlocality, effectively enhancing the performance of tasks such as device-independent random number generation.