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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 randomsources or partially free ones as seed. Subsequently, Li et al. [Phys. Rev. A84, 034301 (2011)] studied this topic in the framework ofsemi-device-independent and proved that new randomness can be obtained fromperfectly free random sources. Here we discuss whether and how partially freerandom sources bring us new randomness in semi-device-independent scenario. Wepropose a semi-device-independent randomness expansion protocol with partiallyfree random sources, and obtain the condition that the partially free randomsources should satisfy to generate new randomness. In the process of analysis,we acquire a new 2-dimensional quantum witness. Furthermore, we get theanalytic relationship between the generated randomness and the 2-dimensionalquantum witness violation.

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