Relative quantum coherence, incompatibility, and quantum correlations of states

Quantum coherence, incompatibility, and quantum correlations are fundamental features of quantum physics. A unified view of those features is crucial for revealing quantitatively their intrinsic connections. We define the relative quantum coherence of two states as the coherence of one state in the reference basis spanned by the eigenvectors of another one and establish its quantitative connections with the extent of mutual incompatibility of two states. We also show that the proposed relative quantum coherence, which can take any form of measures such as $l_1$ norm and relative entropy, can be interpreted as or connected to various quantum correlations such as quantum discord, symmetric discord, entanglement of formation, and quantum deficits. Our results reveal conceptual implications and basic connections of quantum coherence, mutual incompatibility, and quantum correlations.