Randomness can be certified in energy-constrained semi-device-independent scenarios

In semi-device-independent (SDI) quantum cryptography, researchers aim to certify the randomness of a prepare-and-measure (PM) system by making only minimal assumptions about the hardware. A popular approach is to constrain the energy of the prepared states, which is experimentally practical in photonic systems. While previous studies established randomness certification under the assumption that the preparation and measurement devices share only classical correlations, recent work suggested that an adversary with access to pre-distributed entanglement could potentially undermine these security guarantees. This paper addresses the fundamental question of whether randomness can still be certified when such quantum adversaries are present.

To resolve this, the authors adapt the Navascués-Pironio-Acín (NPA) hierarchy—a standard tool for bounding quantum correlations—to the energy-constrained PM framework. By decomposing the measurement operator into blocks that act on the message and measurement subsystems, they construct a hierarchy of semidefinite programs (SDPs). Crucially, this construction allows them to derive certified lower bounds on the guessing probability (and thus the extractable randomness) without imposing any restrictions on the Hilbert-space dimensions of the preparation, measurement, or adversarial systems.

The authors show that certified randomness generation is theoretically possible even in the presence of a fully quantum adversary. Their dimension-independent lower bounds on min-entropy remain strictly positive for energy parameters up to approximately 0.18. While these rates are lower than those achievable in scenarios without shared entanglement, they provide a rigorous security guarantee that holds regardless of the adversary's strategy or the dimension of the underlying quantum systems. Furthermore, the authors demonstrate that their technique can be adapted to provide tighter bounds if one is willing to assume a specific source dimension, and they show that their approach is robust to experimental imperfections in the observed correlator statistics.