量子信息中的不确定性原理及其应用

发布时间：2018-01-20 03:33

  本文关键词： 不确定性关系 斜信息 WYD斜信息 广义度量调整斜信息 关联测度 方差 协方差 协方差矩阵 马尔科夫性 统一熵　出处：《陕西师范大学》2016年博士论文　论文类型：学位论文

【摘要】：本文利用算子理论与算子代数的知识研究了量子信息中的不确定性理论及其应用.建立了广义度量调整斜信息的不确定性关系;给出了非自伴算子对应的Wigner-Yanase-Dyson(WYD)斜信息以及相关的一些物理量的定义,并建立了相关的不确定性关系;证明了关于混合态的方差的和式以及乘积式的不确定性关系;从不确定性理论的角度,利用协方差矩阵刻画了非马尔科夫性;证明了统一熵关于参数的单调性以及凹凸性.本文共分五章,主要内容如下:第一章介绍了本文的研究背景及现状,并列出了本文要用到的符号,定义以及相关已有成果.第二章引入了量化关联的度量函数Fa,α(ρab),得到了如下结论:对于经典-量子态Pab,Fa,α(Pab)= 0当且仅当ρab是乘积态;Fa,α(ρab)是局部酉不变的且在具有相同的边际态ρa之集上是凸的;Fa,α(ρab)在B(Hb)上的局部随机酉运算作用下是不增的;对于一个量子-经典态ρab,Fa,α(ρab)在B(Hb)上的局部量子运算作用下是不增的;最后,我们分别计算了纯态以及Bell-对角态的关联度量Fa,α(ρab).第三章建立了关于广义度量调整斜信息以及广义度量调整关联测度的不确定性关系,并由此利用几种典型的算子单调函数得到一些关于斜信息和WYD斜信息的不确定性关系.引入了非自伴算子对应的广义WYD关联测度,广义WYD斜信息以及一些相关的物理量,讨论了它们的性质,建立了关于广义WYD斜信息的一些不确定性关系.此外,给出了两个可观测量关于混合态的方差的和的下界,保证了当两个可观测量在系统态上是不相容时,结果是不平凡的.我们还建立了关于两个可观测量方差的乘积的更强的不确定性关系.同时,我们得到三个可观测量关于混合态的几种强不确定性关系.第四章提出了量子演化的非马尔科夫性的新刻画,从不确定性角度利用协方差矩阵来刻画非马尔科夫性,得到了关于协方差矩阵的基本性质.考虑了几种典型的例子,并将我们的度量与Fisher信息矩阵刻画,可除性刻画以及Breuer-Laine-Piilo(BLP)刻画进行了比较研究.第五章刻画了统一量子(r,s)-熵关于参数的单调性以及凹凸性,主要结论如下:(ⅰ)对于任意给定的0rl,Ers(ρ)关于s ∈(-∞,+∞)是单调增加的,以及对于任意的r ≥ 1,Ers(ρ)关于s ∈(-∞,+∞)是单调减少的;(ⅱ)对于任意的s0,Ers(ρ)关于r ∈(0,+∞)是单调减少的;(ⅲ)对于乘积态ρab,存在实数a和b使得当r ≥ 1时,Irs(ρab)关于∈[0,a]是单调增加的,以及当0r1时,它关于s ∈[b,0]是单调减少的;(ⅳ)对于乘积态ρab,m2且m-21nm = 1,对于每一个 s0,Irs(ρab)关于 r ∈[rs,+∞)是减少的,其中 = max{as,bs},且a,ss满足trρaas= trρbbs=m =-1/s(ⅴ)对于任意的r0,Ers(ρ)关于s∈(-∞,+∞)是凸函数.
[Abstract]:In this paper, the uncertainty theory and its application in quantum information are studied by means of operator theory and operator algebra, and the uncertainty relation of generalized metric adjusted skew information is established. In this paper, the skew information of Wigner-Yanase-Dysonn WYDcorresponding to non-self-adjoint operators and the definition of some related physical quantities are given, and the related uncertainty relations are established. The sum of variances of mixed states and the uncertain relation of product are proved. The non-Markov property is characterized by covariance matrix from the perspective of uncertainty theory. This paper is divided into five chapters. The main contents are as follows: the first chapter introduces the research background and current situation of this paper, and lists the symbols to be used in this paper. In chapter 2, we introduce the quantized correlation metric function Fa, 伪 (蟻 abg), and obtain the following conclusion: for the classical quantum state Paban Fa. A Pabu = 0 if and only if 蟻 ab is a product state; Fa, 伪 (蟻 ab) is locally unitary invariant and convex on the set with the same marginal state 蟻 a; Faa, 伪 (蟻 ab) is not increased under the action of local random unitary operation on BHb. For a quantum-classical state 蟻 abn Faa, 伪 (蟻 abs) is not increased under the action of local quantum operations on BHb. Finally, we calculate the correlation metric Fa of pure states and Bell-diagonal states, respectively. In Chapter 3, the uncertainty relation of generalized metric adjusted skew information and generalized metric adjusted correlation measure is established. By using some typical operator monotone functions, some uncertain relations about skew information and WYD skew information are obtained, and the generalized WYD correlation measure corresponding to non-self-adjoint operators is introduced. The properties of generalized WYD oblique information and some related physical quantities are discussed, and some uncertain relations about generalized WYD oblique information are established. The lower bound of the sum of variance of two observable measurements for mixed states is given, which ensures that the two observable measurements are incompatible in the system state. The result is not trivial. We also establish a stronger uncertainty relation about the product of two observable measurements of variance. We obtain three strong uncertainty relations of observable measurements on mixed states. Chapter 4th presents a new characterization of non-Markov properties of quantum evolution. In this paper, we use the covariance matrix to characterize the non-Markov property from the uncertainty point of view, and obtain the basic properties of the covariance matrix. Some typical examples are considered. And we depict our metrics and Fisher information matrix. The divisibility characterization and Breuer-Laine-Piilo BLP characterization are compared. Chapter 5th characterizes the unified quantum r. For the monotonicity and concave convexity of parameters, the main conclusions are as follows: (I) is monotone increasing for any given 0 rln Ers (蟻) in relation to s 鈭，

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