Proof of Security of a Semi-Device-Independent Quantum Key Distribution Protocol

Semi-device-independent quantum key distribution (SDI-QKD) has been proposed by applying the quantum dimension correlation, and the security relies on the violation of quantum dimension witness inequalities. We prove the security of the SDI-QKD protocol under the depolarization channel by considering the quantum dimension witness inequalities and minimum entropy and the specific process of the QKD protocol, combining with a four-quantum-state preparation and three measurement bases. We also provide the relationship between the dimension witness value, the error rate and the security key rate by the numerical simulation.