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.
We characterize the operational task of environment-assisted distillation of quantum coherence under different sets of free operations when only a finite supply of copies of a given state is available. We first evaluate the one-shot assisted distillable coherence exactly, and introduce a semidefinite programming bound on it in terms of a smooth entropic quantity. We prove the bound to be tight for all systems in dimensions 2 and 3, which allows us to obtain computable expressions for the one-shot rate of distillation, establish an analytical expression for the best achievable fidelity of assisted distillation for any finite number of copies, and fully solve the problem of asymptotic zero-error assisted distillation for qubit and qutrit systems. Our characterization shows that all relevant sets of free operations in the resource theory of coherence have exactly the same power in the task of one-shot assisted coherence distillation, and furthermore resolves a conjecture regarding the additivity of coherence of assistance in dimension 3.
The uncertainty principle constitutes a fundamental pillar of quantum theory, representing one of the most distinctive features that differentiates quantum mechanics from classical physics. In this study, we firstly propose a novel generalized entropic uncertainty relation (EUR) for arbitrary multi-measurement in the many-body systems, and rigorously derive a significantly tighter bound compared to existing formulations. Specifically, we discuss the proposed EUR in the context of Schwarzschild black hole, where we demonstrate the superior tightness of our derived bound. The study further elucidates the dynamical evolution of multipartite quantum coherence and entanglement in the curved spacetime. A particularly noteworthy finding reveals the exact equivalence between entanglement and
l1-norm coherence for arbitrary
N-partite Greenberger-Horne-Zeilinger-type (GHZ-type) states. Moreover, we find that quantum coherence is significantly diminished and the measurement uncertainty increases to a stable maximum with increasing Hawking temperature. Thus, the findings of this study contribute to a deeper understanding of non-classicality and quantum resources in black holes.