Trade-off relations of geometric coherence

Abstract: Quantum coherence is an important quantum resource and it is intimately related to various research fields. The geometric coherence is a coherence measure both operationally and geometrically. We study the trade-off relation of geometric coherence in qubit systems. We first derive an upper bound for the geometric coherence by the purity of quantum states. Based on this, a complementarity relation between the quantum coherence and the mixedness is established. We then derive the quantum uncertainty relations of the geometric coherence on two and three general measurement bases in terms of the incompatibility respectively, which turn out to be state-independent for pure states. These trade-off relations provide the limit to the amount of quantum coherence. As a byproduct,the complementarity relation between the minimum error probability for discriminating a pure-states ensemble and the mixedness of quantum states is established.