标题： 量子力学可观测量不确定性的概率密度函数 显示英文标题

标题： Probability density functions of quantum mechanical observable uncertainties

摘要： 我们研究量子力学可观测量的不确定性，这些不确定性通过哈尔分布随机纯态中的标准差（方差的平方根）来量化。 我们解析地推导了任意比特可观测量的不确定性的概率密度函数（PDF）。 基于这些PDF，可观测量的不确定性区域由PDF的支持集来表征。 然后，与状态无关的不确定性关系被转化为不确定性区域上的优化问题，这为研究与状态无关的不确定性关系开辟了一个新视角。 我们的结果可以推广到高维空间中的多个可观测量情况。 显示英文摘要

摘要： We study the uncertainties of quantum mechanical observables, quantified by the standard deviation (square root of variance) in Haar-distributed random pure states. We derive analytically the probability density functions (PDFs) of the uncertainties of arbitrary qubit observables. Based on these PDFs, the uncertainty regions of the observables are characterized by the supports of the PDFs. The state-independent uncertainty relations are then transformed into the optimization problems over uncertainty regions, which opens a new vista for studying state independent uncertainty relations. Our results may be generalized to multiple observable case in higher dimensional spaces.