Anomalies of the weight-based coherence measure and mixed maximally coherent states
Yao Yao, Dong Li, C. P. Sun · Physical Review A · 2020
As an analogy of best separable approximation (BSA) in the framework of entanglement theory, here we concentrate on the notion of best incoherent approximation, with application to characterizing and quantifying quantum coherence. From both analytical and numerical perspectives, we have demonstrated that the weight-based coherence measure displays some unusual properties, in sharp contrast to other popular coherence quantifiers. First, by deriving a closed formula for single-qubit states, we have shown the exact value of weight-based coherence measure does not depend solely on the off-diagonal element but relies on the geometry of a given qubit state, thus exhibiting a rich structure even in this simplest case. Second, albeit the fact that almost all popular coherence measures only assign a maximal value to pure maximally coherent states, we have demonstrated the existence of mixed maximally coherent states (MMCS) with respect to this coherence measure and discussed the characteristic feature of MMCS in high-dimensional Hilbert spaces. Especially, we present several important families of MMCS by gaining insights from the numerical simulations. Moreover, it is pointed out that some considerations in this work can be generalized to general convex resource theories and a numerical method of improving the computational efficiency for finding the BSA is also discussed.