A quantum measure of coherence and incompatibility

The well-known two-slit interference is understood as a special relation between an observable (localization at the slits) and a state (being on both slits). The relation between an observable and a quantum state is investigated in the general case. It is assumed that the amount of coherence equals that of the incompatibility between an observable and a state. On these grounds, an argument is presented 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 paper. A precise relation between the measure of coherence of an observable and that of its coarsening is obtained and discussed from the intuitive 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 the coherence information are derived with a view to investigating the final-state entanglement in general repeatable measurement, and, more importantly, general bipartite entanglement in follow-ups of this study