摘要
不确定性是量子系统的一个基本特征.长期以来量子力学中一直采用可观测量的标准偏差来刻画这种不确定性.但近年来,研究者们通过分析一些具体例子发现,用可观测量的测量结果的Shannon熵来描述这种不确定性更为合适.形式上,Shannon熵也是一种更为一般的Rényi熵的极限形式.本文从对未知态的测量结果的可重复概率的角度,讨论了如何利用已有的测量结果预测新的测量结果,以及可观测量的不确定度的定量表示的问题.利用可观测量出现多次相同结果的概率定义了一种推广的Rényi熵,并用这种推广的Rényi熵给出了Maassen-Uffink型熵不确定关系的一种简单证明.
Uncertainty is a fundamental characteristic of quantum system.The degree of uncertainty of an observable has long been investigated by the standard deviation of the observable.In recent years,however,by analyzing some special examples,researchers have found that the Shannon entropy of the measurement outcomes of an observable is more suitable to quantify its uncertainty.Formally,Shannon entropy is a special limit of a more general Rényi entropy.In this paper,we discuss the problem of how to predict the measurement outcome of an observable by the existing measurement results of the observable,and how to quantitatively describe the uncertainty of the observable from the perspective of the repeatable probability of the measurement results of this observable in an unknown state.We will argue that if the same observable of different systems in the same state is repeatedly and independently measured many times,then the probability of obtaining an identical measurement result is a decaying function of the number of measurements of obtaining the same result,and the decay rate of the repeatable probability for obtaining the same measurement results and the repeatable number of measurements can represent the degree of uncertainty of the observable in this state.It means that the greater the uncertainty of an observable,the faster the repeatable probability decays with the number of repeatable measurements;conversely,the smaller the uncertainty,the slower the repeatable probability decays with the number of repeatable measurements.This observation enables us to give the Shannon entropy and the Rényi entropy of an observable uniformly by the functional relation between the repeatable probability and the number of repeatable measurements.We show that the Shannon entropy and the Rényi entropy can be formally regarded as the“decay index”of the repeatable probability with the number of repeatable measurements.In this way we also define a generalized Rényi entropy by the repeatable probability for consecutively observing
作者
张诗琪
杨化通
Zhang Shi-Qi;Yang Hua-Tong(School of Physics,Northeast Normal University,Changchun 130024,China)
出处
《物理学报》
SCIE
EI
CAS
CSCD
北大核心
2023年第11期141-146,共6页
Acta Physica Sinica
基金
国家自然科学基金(批准号:11875011)资助的课题.
关键词
不确定性
熵
不确定关系
uncertainty
entropy
uncertainty relation
