We derive upper bounds for Hilbert–Schmidt’s quantum coherence of general states of a d-level quantum system, a qudit, in terms of its incoherent uncertainty, with the latter quantified using the linear and von Neumann’s entropies of the corresponding closest incoherent state. Similar bounds are obtained for Wigner–Yanase’s coherence. The reported inequalities are also given as coherence–populations trade-off relations. As an application example of these inequalities, we derive quantitative wave–particle duality relations for multi-slit interferometry. Our framework leads to the identification of predictability measures complementary to Hilbert–Schmidt’s, Wigner–Yanase’s, and \(l_{1}\)-norm quantum coherences. The quantifiers reported here for the wave and particle aspects of a quanton follow directly from the defining properties of the quantum density matrix (i.e., semi-positivity and unit trace), contrasting thus with most related results from the literature.

中文翻译：

---

密度矩阵性质的定量波粒对偶关系

我们根据非相干不确定性推导了d级量子系统qudit的一般态的希尔伯特-施密特量子相干的上限，后者使用相应最接近非相干态的线性熵和冯·诺依曼熵来量化。Wigner-Yanase的连贯性也获得了相似的界线。所报告的不平等现象也以连贯性与人口之间的权衡关系给出。作为这些不等式的应用示例，我们导出了用于多缝干涉测量的定量波粒对偶关系。我们的框架导致确定与Hilbert-Schmidt，Wigner-Yanase和\（l_ {1} \）互补的可预测性度量-规范量子相干性。此处报道的有关量子子的波和粒子方面的量词直接遵循量子密度矩阵的定义属性（即半正性和单位迹线），因此与文献中大多数相关结果形成对比。