Semi-device-independent randomness expansion with partially free random sources

By proposing device-independent protocols, S. Pironio et al. [Nature 464, 1021-1024 (2010)] and R. Colbeck et al. [Nature Physics 8, 450-453 (2012)] proved that new randomness can be generated by using perfectly free random sources or partially free ones as seed. Subsequently, Li et al. [Phys. Rev. A 84, 034301 (2011)] studied this topic in the framework of semi-device-independent and proved that new randomness can be obtained from perfectly free random sources. Here we discuss whether and how partially free random sources bring us new randomness in semi-device-independent scenario. We propose a semi-device-independent randomness expansion protocol with partially free random sources, and obtain the condition that the partially free random sources should satisfy to generate new randomness. In the process of analysis, we acquire a new 2-dimensional quantum witness. Furthermore, we get the analytic relationship between the generated randomness and the 2-dimensional quantum witness violation.