AI 中文总结
研究多体自旋1/2系统上与态无关的不确定性关系，通过表示理论框架，利用克莱布施 - 戈尔丹分解和总自旋方差极值分析，推导出直至五体系统的精确界限，建立了首个统一代数框架。
AI 中文摘要
不确定性关系量化了量子可观测量同时测量的基本限制。传统表述依赖于状态，而与态无关的不确定性关系（SIURs）仅由算符的代数结构确定通用界限，应用于计量学、量子密码学和纠缠检测等领域。尽管已有广泛研究，但基于方差的精确分析SIURs主要在单体和二体系统中建立，熵SIURs通过信息论构造扩展到多体和记忆辅助设置。本文为集体自旋1/2系统中的多体SIURs开发了一个表示理论框架。利用克莱布施 - 戈尔丹分解和总自旋方差的极值分析，我们推导出了直至五体系统的精确与态无关界限。出现了明显的结构二分法：奇数n系统表现出严格正的通用界限，而偶数n在平凡扇区允许方差消失，但保留正的约化空间界限。这些结果为量子比特系综中基于方差的多体SIURs建立了第一个统一的代数框架。
英文摘要
Uncertainty relations quantify fundamental limits on simultaneous measurement of quantum observables. While conventional formulations are state-dependent, state-independent uncertainty relations (SIURs) impose universal bounds determined solely by the algebraic structure of the operators, with applications across metrology, quantum cryptography, and entanglement detection. Despite extensive study, exact analytical variance-based SIURs have so far been established primarily for one- and bi-partite systems, while entropic SIURs have been extended to multipartite and memory-assisted settings through information-theoretic constructions. In contrast, exact variance-based SIURs beyond the bipartite level have remained analytically unresolved.} Here we develop a representation-theoretic framework for multipartite SIURs in collective spin-$\tfrac{1}{2}$ systems. Using the Clebsch--Gordan decomposition and extremal analysis of total spin variance, we derive exact state-independent bounds up to quintipartite systems. A clear structural dichotomy emerges: odd $n$ systems exhibit strictly positive universal bounds (e.g., $Δ^2(\mathfrak{su}_2)\!\ge\!4/11$ for $n=3$), whereas even $n$ admit vanishing variance on trivial sectors but retain positive reduced-space bounds (e.g., $Δ^2(\mathfrak{su}_2)\!\ge\!1/8$ for $n=4$). These results establish the first unified, algebraic framework for multipartite variance-based SIURs in qubit ensembles.
Comments21pp
Journal refJ. Phys. A: Math. Theor. 59 (2026) 155302 (17pp)
