A quantum measure of coherence and incompatibility

The well-known two-slit interference is understood as a special relation between observable (localization at the slits) and state (being on both slits). Relation between an observable and a quantum state is investigated in the general case. It is assumed that the amount of ceherence equals that of incompatibility between observable and state. On ground of this, an argument is peresented that leads to a natural quantum measure of coherence, called "coherence or incompatibility information". Its properties are studied in detail making use of 'the mixing property of relative entropy' derived in this article. A precise relation between the measure of coherence of an observable and that of its coarsening is obtained and discussed from the intutitive point of view. Convexity of the measure is proved, and thus the fact that it is an information entity is established. A few more detailed properties of coherence information are derived with a view to investigate final-state entanglement in general repeatable measurement, and, more importantly, general bipartite entanglement in follow ups of this study.