Operational methods in quantum information theory
Project/Area Number 
11640115

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Osaka University 
Principal Investigator 
FUJIWARA Akio 大阪大学, 大学院・理学研究科, 助教授 (30251359)

CoInvestigator(Kenkyūbuntansha) 
MANABE Shojiro 大阪大学, 大学院・理学研究科, 助教授 (20028260)
SUGIMOTO Mitsuru 大阪大学, 大学院・理学研究科, 助教授 (60196756)
ENOKI Ichiro 大阪大学, 大学院・理学研究科, 助教授 (20146806)
TAKAGOSHI Kensho 大阪大学, 大学院・理学研究科, 助教授 (20188171)
YAMAMOTO Yoshikiko 大阪大学, 大学院・理学研究科, 教授 (90028184)
高橋 智 大阪大学, 大学院・理学研究科, 講師 (70226835)
山崎 洋平 大阪大学, 大学院・理学研究科, 助教授 (00093477)

Project Period (FY) 
1999 – 2000

Project Status 
Completed (Fiscal Year 2000)

Budget Amount *help 
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2000: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 1999: ¥2,200,000 (Direct Cost: ¥2,200,000)

Keywords  quantum information / quantum estimation / quantum channel / random matrix / entropy / coding theory / asymptotics / entangled state / 操作的容量 / 直交性 
Research Abstract 
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
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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 twolevel 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 ; For1/3【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

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