The uncertainty relation reveals the intrinsic difference that distinguishes the quantum world from the classical world. In this paper, the quantum uncertainty relations of the sandwiched R\'enyi relative entropy of coherence (SRREOC) and unified $(\ensuremath{\alpha},\ensuremath{\beta})$-relative entropy of coherence (UREOC) are studied. First, the uncertainty relation of the SRREOC for pure states, with extension to mixed states by convex-roof construction, is given in a
d-dimensional quantum system. Second, the analytical formula of the UREOC is presented; it is shown that the UREOC is a good candidate for a quantum coherence measure. Furthermore, the quantum uncertainty relation of the UREOC for pure states is obtained in a
d-dimensional quantum system, then it is generalized to mixed states. In addition, our results cover the quantum uncertainty relations of the other three coherence measures. Third, two methods are given to compare the lower bounds of the SRREOC and UREOC for single-qubit pure states. By comparison, we find that the results in this paper not only can be applied to single-qubit states but also can be extended to a
d-dimensional quantum system.
Abstract We establish a rigorous equivalence between quantum coherence measures defined by the sandwiched Rényi relative entropy and generalized conditional entropies associated with auxiliary quantum systems, providing a direct operational interpretation of sandwiched Rényi relative entropy of coherence as the conditional uncertainty of projective measurement outcomes with quantum memory. Motivated by this connection, we develop a coherence-based approach to quantum uncertainty relations. Within this approach, we introduce a coherence measure induced by the two-parameter α - z -Rényi relative entropy and derive corresponding uncertainty relations for both pure and mixed quantum states. The resulting state-independent bounds recover known results in appropriate limits and yield tighter bounds over a broad parameter regime. Our results provide a systematic connection between quantum coherence, generalized conditional entropies, and uncertainty relations, thereby offering a unified perspective on quantum uncertainty in terms of Rényi-type information measures.
Coherence of a quantum state intrinsically depends on the choice of the reference basis. A natural question to ask is the following: if we use two or more incompatible reference bases, can there be some trade-off relation between the coherence measures in different reference bases? We show that the quantum coherence of a state as quantified by the relative entropy of coherence in two or more noncommuting reference bases respects uncertainty like relations for a given state of single and bipartite quantum systems. In the case of bipartite systems, we find that the presence of entanglement may tighten the above relation. Further, we find an upper bound on the sum of the relative entropies of coherence of bipartite quantum states in two noncommuting reference bases. Moreover, we provide an upper bound on the absolute value of the difference of the relative entropies of coherence calculated with respect to two incompatible bases.