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Operational methods in quantum information theory

量子信息论中的运算方法

基本信息

批准号：
11640115
负责人：
FUJIWARA Akio
金额：
$ 2.3万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640115/
关键词：
quantum information quantum estimation quantum channel random matrix entropy coding theory asymptotics entangled state 操作的容量直交性

项目摘要

The purpose of this project is to investigate operational methods in quantum information theory. In particular, we have studied (a) novel operational characterizations of quantum entropies, and (b) quantum channel identification problem. The results of these studies are summarized as follows.(a) Let p be a probability measure on a Hilbert space H the support of which being a countable set of mutually nonparallel unit vectors. Let p^<(n)> be the probability measure on H^<【cross product】n> defined by the nth i.i.d. extension of p, and consider L^<(n)> random vectors X (1) , ..., X (L^<(n)>) on H^<【cross product】n> which are subjected to p^<(n)>. We have introduced several definitions of "asymptotic orthogonality" for the random vectors and have studied the corresponding orthogonality capacity, i.e., the supremum of lim sup_n log L_n/n over all sequences {L_n}_n that satisfy each orthogonality criterion. Under the weak orthogonality condition that represents the situation in which the vec … More tor X (1) is almost orthogonal to the other vectors, the orthogonality capacity has turned out to be identical to the von Neumann entropy for the density operator ρ that corresponds to the measure p. Moreover we have clarified that the noiseless quantum channel coding theorem by Hausladen et al. is a direct consequence of this characterization. Under the strong orthogonality condition that represents the situation in which the vectors X (1), ..., X (L^<(n)>) are mutually almost orthogonal, on the other hand, the orthogonality capacity has turned out to be identical to half the quantum Renyi entropy of degree 2.(b) A quantum channel identification problem is this : given a parametric family {Γ_θ}_θ of quantum channels, find the best strategy of estimating the true value of the parameter θ. We have studied this problem from a noncommutative statistical point of view. In particular, we have demonstrated a nontrivial aspect of this problem as follows. Let Γ_θ be the isotropic depolarization channel acting on the two-level quantum system, in which the parameter θ represents the magnitude of depolarization. By using the Stokes' parametrization, it is represented as (x, y, z) → (θx, θy, θz). Due to the requirement of complete positivity of the map Γ_θ, the parameter θ must lie in the closed interval [-1/3, 1]. Up to the second extension H 【cross product】 H of the quantum system, the best strategy of estimating the isotropic depolarization parameter θ is the following. For 1/(√<3>)【less than or equal】θ【less than or equal】1, use Γ_θ 【cross product】 Γ_θ and input a maximally entangled state ; For 1/3【less than or equal】θ【less than or equal】1/(√<3>) use Γ_θ 【cross product】 Γ_θ and input a disentangled state ; For-1/3【less than or equal】θ【less than or equal】1/3, use Γ_θ 【cross product】 Id and input a maximally entangled state. It is surprising that the seemingly homogeneous family {Γ_θ}_θ of depolarization channels involves a transitionlike behavior. Less

本项目的目的是研究量子信息理论中的操作方法。特别地，我们研究了（a）量子熵的新的操作特征，和（B）量子信道识别问题。这些研究的结果总结如下。(a)设p是Hilbert空间H上的概率测度，其支撑是可数的互不平行的单位向量集。设p^<（n）>是H^<[叉积]n>上由第n个i.i.d.定义的概率测度。p的扩展，并且考虑L（n）>随机向量X（1），.，X（L^<（n）>）在H^<[叉积]n>上的映射，它们服从p^<（n）>。本文介绍了随机向量“渐近正交性”的几种定义，并研究了相应的正交性容量，即，证明了满足每个正交准则的序列{L_n}_n上的limsup_nlog L_n/n的上确界。在弱正交性条件下，该条件表示向量 ...更多信息由于X（1）几乎正交于其它向量，所以对于对应于测度p的密度算符ρ，正交性容量与von Neumann熵是相同的。此外，我们还阐明了Hausladen等人的无噪声量子信道编码定理是这一特征的直接结果。在表示向量X（1），...，X（L^<（n）>）相互几乎正交，另一方面，正交性容量已经证明等于2次量子Renyi熵的一半。(b)一个量子信道识别问题是：给定一个量子信道参数族{Γ_θ}_θ，找出估计参数θ真值的最佳策略。我们从非对易统计的观点研究了这个问题。特别是，我们已经证明了这个问题的一个重要方面如下。设Γ θ为作用于二能级量子系统的各向同性退偏振通道，其中参数θ表示退偏振的大小。用Stokes参数化表示为（x，y，z）→（θx，θy，θz）。由于映射Γ_θ的完全正性要求，参数θ必须位于闭区间[-1/3，1]内。直到量子系统的第二次扩展H [叉积] H，估计各向同性退偏振参数θ的最佳策略如下。对于1/（λ<3>）[小于或等于]θ[小于或等于]1，使用Γ_θ [叉积] Γ_θ，输入一个最大纠缠态;对于1/3[小于或等于]θ[小于或等于]1/（λ<3>），使用Γ_θ [叉积] Γ_θ，输入一个解纠缠态;对于-1/3 [小于或等于]θ[小于或等于]1/3，使用Γ_θ [叉积] Id，输入一个最大纠缠态。令人惊讶的是，看似同质的去极化通道族{Γ_θ}_θ包含类似于过渡的行为。少