Uncertainty Relation of Quantum Coherence Measure Based on Hellinger Distance

In this paper, we focus on the uncertainty relations based on the quantum coherence measure of Hellinger distance. First, we discuss two kinds of quantum uncertainty relations for single qubit state based on two orthogonal bases. The lower bound of the coherence sum under two sets of orthogonal bases is related to the overlap of the two sets of bases and the mixedness of single-qubit state. We find that the lower bound \(B_1\) $B_1$ decrease with the growing of the mixedness, and increase with the growing of the overlap of the two sets of bases. Second, we consider uncertainty relation in a subsystem for a two-qubit quantum state. The result shows that the lower bound increases with the growing of entanglement. In addition, we find that the sum of coherence measure will be affected by the existence of entanglement by numerical analysis. Finally, the uncertainty relations of d-dimensional quantum systems are studied.