Semi-device-independent randomness certification with partially free random sources using 4→1 quantum random access code

The semi-device-independent (SDI) quantum random number expansion protocols are based on n → 1 quantum random access codes (QRACs). These protocols do not require the use of entanglement and are easier to implement than ones which are in device-independent scenario. Since there are only three mutually orthogonal directions in R 3 , it is difficult to design an SDI quantum random number expansion protocol for n > 3 , especially when partially free bits are used as seed. In this paper, we propose the SDI randomness expansion using 4 → 1 QRAC and obtain the corresponding classical bound and quantum bound of the two-dimensional quantum witness. Subsequently, we get the condition to be satisfied by the partially free random sources to generate new randomness. Moreover, we prove the analytic relationship between the certified randomness and the two-dimensional quantum witness violation. Furthermore, the other SDI randomness expansions using n → 1 QRACs for n > 4 can be studied further by drawing on our methods.