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 Mso provide the relationship between the dimension witness value,the error rate and the security key rate by the numerical simulation.