Close the gap between certified lower and adversarial upper bounds on randomness

Determine the true randomness-generation rate in the energy-constrained semi-device-independent prepare-and-measure scenario with arbitrary shared entanglement by closing the gap between the certified lower bounds obtained from semidefinite-programming relaxations and the seesaw upper bounds, using tighter relaxations or stronger explicit attacks.

Background

The paper establishes dimension-independent certified lower bounds on the extractable min-entropy when the preparation and measurement devices may share arbitrary entanglement. These lower bounds vanish for energy parameters above approximately 0.18, whereas existing seesaw-based upper bounds remain positive up to approximately 0.32. Consequently, the exact randomness-generation rate is not characterized, and the authors explicitly identify closing this interval as an unresolved problem.