Uncertainty relation based on metric-adjusted skew information with quantum memory

The uncertainty principle is one of the most essential features of quantum mechanics. Recently, uncertainty relations of skew information have been widely studied. In this paper, general and conditional uncertainty relations based on metric-adjusted skew information are put forward to study the case of the uncertainty relation with the existence of a quantum memory for the bipartite quantum system. These uncertainty relations include the product form and the sum form. The results show that the lower bounds contain two parts: one is characterizing the degree of compatibility of two measurements, and the other one is the quantum correlation between the measured system and the quantum memory. These results may play an important role in witnessing entanglement and quantum key distribution.