发表机构
Jianghan University; North Carolina State University; Central China Normal University; Agency for Science, Technology and Research (A*STAR)（江汉大学; 北卡罗来纳州立大学; 华中师范大学; 科学技术研究局（A*STAR））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于协方差的量子不确定性表述，从李代数及其表示推导出闭式最优界限，将不确定性转化为守恒律，并连接对称性与表示理论以揭示量子信息处理的固有极限。
AI 中文摘要
不确定性关系表达了量子可观测量之间的基本不相容性，然而最优的与状态无关的界限仅在少数情况下有解析解。这里我们引入一种基于协方差的量子不确定性表述，它为厄米可观测量和非厄米算子都产生了一个无限的不确定性关系族。对于构成李代数基的算子，我们从代数及其表示中以闭式形式推导出最优界限，将对量子态的优化替换为直接的代数计算。在每个权空间内，不确定性取固定值，将不等式转变为守恒律。这些结果展示了交换关系及其在量子态上的实现如何共同决定与不相容算子相关的最小量子不确定性。通过将不确定性与对称性和表示理论联系起来，该框架提供了一条通过其底层李代数揭示量子信息处理内在极限的途径。
英文摘要
Uncertainty relations impose fundamental constraints on quantum observables, yet optimal bounds are needed to determine their incompatibility through the minimum attainable uncertainty. Here we introduce a covariance-matrix formulation from which infinitely many uncertainty relations follow, encompassing Hermitian observables and non-Hermitian operators and recovering the variance sum as a special case. Within this family, a basis-independent uncertainty for simple and semisimple Lie algebras admits an exact solution: its optimal state-independent bound is an explicit pairing of root system with the highest weight of the representation. The minimum is attained by a highest-weight state, replacing a search over quantum states with a calculation in representation theory. A complementary result establishes that uncertainty is constant within each weight space, yielding a conservation law. The same commutation relations can therefore support distinct uncertainty limits, depending on their realization on the state space. By rendering these limits analytically accessible, the framework turns representation theory into a tool for quantifying quantum incompatibility and may help establish the boundary between attainable and impossible tasks in quantum theory.
Comments8 pages (main text) + 103 pages (supplemental material). Comments welcome!
