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As an analogy of best separable approximation (BSA) in the framework ofentanglement theory, here we concentrate on the notion of best incoherentapproximation, with application to characterizing and quantifying quantumcoherence. From both analytical and numerical perspectives, we havedemonstrated that the weight-based coherence measure displays some unusualproperties, in sharp contrast to other popular coherence quantifiers. First, byderiving a closed formula for qubit states, we have showed the weight-basedcoherence measure exhibits a rich (geometrical) structure even in this simplestcase. Second, we have identified the existence of mixed maximally coherentstates (MMCS) with respect to this coherence measure and discussed thecharacteristic feature of MMCS in high-dimensional Hilbert spaces. Especially,we present several important families of MMCS by gaining insights from thenumerical simulations. Moreover, it is pointed out that some considerations inthis work can be generalized to general convex resource theories and anumerical method of improving the computational efficiency for finding the BSAis also discussed.

Anomalies of weight-based coherence measure and mixed maximally coherent states