Certified quantum randomness within purity constraints

Quantum random number generators (QRNGs) exploit quantum mechanics to produce true randomness, but practical implementations suffer from device imperfections that can compromise security. We propose a semi–device-independent QRNG based on measurement uncertainty under purity constraints for conditional min-entropy estimation. Our protocol requires no detailed modeling of the two-dimensional source and the countable-dimensional measurement and remains secure against quantum attacks on measurement devices. We uncover an implicit uncertainty relation arising from naturally satisfied purity conditions, define a randomness parameter C using test states, and connect it to an analytical bound on extractable randomness via unitary rotation symmetries of states and measurements. Using the generalized entropy accumulation theorem, we extend the single-round bound to multirounds and obtain composable security against coherent attacks. Implemented on a standard commercial fiber system, our high-tolerance protocol achieves exponential secure randomness expansion and is suitable for integrated-photonic platforms.