Entropic diagram characterization of quantum coherence: Degenerate distillation and the maximum eigenvalue uncertainty bound

We present a geometric approach to characterize quantum coherence through entropic diagrams that map the eigenvalue spectra of density matrices into a two-dimensional space defined by von Neumann and Tsallis-2 entropies. We derive the relative entropy of coherence using the Schur-Horn majorization theorem and introduce a new coherence monotone—the relative cross-entropy of coherence—which is valid only for mixed states and whose behavior under various classes of quantum operations is examined in detail. Interestingly, our approach reveals distinct entropy-coherence distillation curves and introduces the notion of degenerate coherence distillation, where coherence is concentrated into higher-dimensional, partially uniform spectral configurations. Although the degenerate coherence distillation does not always surpass standard qubit coherence distillation in asymptotic rate, it may offer an operationally meaningful alternative for protocols that require high-dimensional or symmetrically structured coherence resources. In addition, we derive a refined entropic uncertainty relation that incorporates the maximum eigenvalue of post-measurement states, which leads to state-dependent bounds that improve upon the conventional Maassen-Uffink bound. These findings suggest that entropic diagrams may serve not only as a useful visual and analytical tool but also as a guide for developing resource-efficient strategies in quantum information processing.