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数理解析研究所講究録

QUANTIZATION OF INFORMATION THEORY (Mathematical Studies on Independence and Dependence Structure : Algebra meets Probability)

Ichiro Fujimoto, Hideo Miyata

Published Dec 1, 2012• 0 citations
OpenAlex

Abstract

In scope of $CP$ -convexity theory for $C^{*}$ -algebras (quantization of convexity, measure, entropy for completely positive maps), we investigate the operational stmcture of quantum interactions of entangled systems, and propose new information quantities which naturally generalize the classical information theory.

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OPERATIONAL STRUCTURE OF ENTANGLED QUANTUM SYSTEMS (Information and mathematics of non-additivity and non-extensivity : contacts with nonlinearity and non-commutativity)

Ichiro Fujimoto, 英男 宮田

We investigate the quantum interaction between two quantum sys- tems in terms of $C^{*}$ -algebras.We introduce the $co$ rrelation CP-map between the bipartite systems and study the operational and statistical structure of the quantum interaction.We then generalize the informational quantities into this quantum setting, and introduce a new entropy of CP-maps, which vanishes at extreme CP-maps in CP-convexity.Introduction.We prepare two quantum states described by density operators $\rho$ and $\sigma$ on Hilbert spaces $H_{1}$ and $K_{1}$ respectively, and so initially we have the compound state $\omega_{0}=\rho\otimes\sigma$ on the tensor product Hilbert space $H_{1}\otimes K_{1}$ .Suppose that there exists an interaction between the two subsystems for some time interval, and that the systems are in equilibrium with the exterior for this time period, then the compound state is changed to a state $\omega$ described by $\omega=U(\rho\otimes\sigma)U^{*}$ where $U$ is a unitary operator representing the quantum evolution of the systems.After the interaction, the states of the subsystems on $H_{2}$ and $K_{2}$ are described by $Tr_{K_{2}}w$ and $R_{H_{2}}\omega$ respectively.Then the channels $\varphi^{*}$ from $T(H_{1})$ to $T(H_{2})$ , and $\phi^{*}$ from $T(K_{1})$ to $T(K_{2})$ , are defined by $\varphi^{*}(\rho)=h_{K_{2}}\omega=r_{b_{K_{2}}U(\rho}\otimes\sigma)U^{*}$ and $\phi^{*}(\sigma)=b_{H_{2}}\omega=h_{H_{2}}U(\rho\otimes\sigma)U^{*}$ ,The detailed version of this paper will be submitted for publication elsewhere.Typaeet by $A_{A}\#?ffl$

数理解析研究所講究録•Feb 1, 2008•0 citations

Mathematical Foundations of Quantum Information Science

Alexander Semenovich Holevo

Abstract Quantum information science is a new and rapidly developing discipline that studies the general laws governing the transmission, storage, and processing of information in systems that obey the laws of quantum mechanics. It makes extensive use of matrix theory and operator theory on Hilbert spaces to investigate the capabilities of such systems and to develop principles for their rational design. An important consequence of the development of quantum information science is a substantial clarification of the logical structure and foundations of quantum mechanics and of its relation to reality. These lectures are devoted to the foundations of the mathematical theory of quantum systems: states, observables, communication channels, their entropic and information characteristics, and quantum information-processing protocols. The lectures provide a detailed introduction to the statistical structure of quantum theory and the necessary background from classical information theory; hence, knowledge of the basic mathematical subjects—analysis, linear algebra, and probability theory—is sufficient for understanding the material. In the first part of the course, we use the “standard” statistical model of a quantum system, but consider it in comparison with the classical statistical description. A consistent development of this approach leads to a generalized statistical model of quantum mechanics, in which observables (in general, “unsharp”) are described by probability operator-valued measures, while evolutions (in general, irreversible) are described by completely positive maps. Introductory material on these concepts, which underlie mathematical models of quantum information systems in the presence of noise and errors and have numerous applications in quantum information science, is presented in the second part of the course.

Proceedings of the Steklov Institute of Mathematics•Sep 1, 2026•0 citations

QUANTIZATION OF INFORMATION THEORY (Mathematical Studies on Independence and Dependence Structure : Algebra meets Probability : RIMS研究集会報告集)

Ichiro Fujimoto, 英男 宮田

数理解析研究所講究録•Dec 1, 2012•0 citations

Convexity of Information Theoretical Quantities in Tripartite Quantum Communication Systems (Information and mathematics of non-additivity and non-extensivity : non-additivity and convex analysis)

Hideo Miyata, Ichiro Fujimoto

数理解析研究所講究録•Feb 1, 2009•0 citations

APPLICATIONS OF FREE PROBABILITY TO QUANTUM INFORMATION THEORY (Mathematical Studies on Independence and Dependence Structure : Algebra meets Probability)

Benoı̂t Collins

We show how Free Probability Theory-and in particular, recent results regarding the norm convergence of random matrices-help to better understand im- portant problems in Quantum Information Theory, such as the problem of Minimum Output Entropy additivity.This is a report on joint papers and projects with Ion Nechita, Serban Belinschi, Camille Male and Motohisa Fukuda 1. ENTANGLEMENT AND ADDITIVITY 1.1.Entanglement and entropy.In Quantum Information Theory, a vector subspace $C$ of a tensor product $A\otimes B$ of two Hilbert spaces is said to be entangled iff $0$ is the only vector of $C$ that can be written as $a\otimes b$ with $a\in A,$ $b\in B$ .The problems of quantifying entanglement, and of finding 'highly' entangled spaces are very important in quantum information theory.For $x\in A\otimes B$ , its singular values -also known as Schmidt coefficients in Quantum Information Theory-are non-negative numbers $\lambda_{1}(x)\geq\ldots\geq\lambda_{k}(x)\geq 0$ such that $x= \sum_{i=1}^{k}\sqrt{\lambda_{i}}e_{i}(x)\otimes f_{i}(x)$ Here, $k=min($dim(A), $dim(B));e_{i}(x),$ $f_{i}(x)$ are orthonormal vectors.The sequence $\lambda(x)=(\lambda_{1}(x), \ldots)\in \mathbb{R}^{k}$ is uniquely defined.Let $K_{C}=\{\lambda(x), x\in C, ||x||=1\}\subset \mathbb{R}^{k}$ .This compact set is a subset of the set $\triangle_{k}^{+}=\{y\in \mathbb{R}_{+}^{k}:y_{1}\geq y_{2}\geq\cdots\geq y_{k}\geq 0, \sum_{i=1}^{k}y_{i}=1\}.$ Besides, we have $\Delta_{k}^{+}\subset\Delta_{k}$ , where $\Delta_{k}=\{x\in \mathbb{R}_{+}^{k}|\sum_{i=1}^{k}x_{i}=1\}$ is the $(k-1)-$ dimensional probability simplex.We will also make an abuse of notation and also denote by $K_{C}$ the subset of $\triangle_{k}$ that is obtained by symmetrizing $K_{C}$ under permuting the coordinates.For a positive real number $p>0$, we recall that the R\'enyi entropy of order $p$ of a probability vector $x\in\Delta_{k}$ to be $H^{p}(x):= \frac{1}{1-p}\log\sum_{i=1}^{k}x_{i}^{p}.$2010 Mathematics Subject Cassification.$46L65(46L37,46L54,.46L87)$ .

数理解析研究所講究録•Dec 1, 2012•0 citations

A representation of unital completely positive maps (Mathematical Studies on Independence and Dependence Structure : Algebra meets Probability)

Marie Choda

Let $M_{n}(\mathbb{C})$ be the algebra of $n\cross n$ complex mairices, and let $\Phi$ be a unital completely positive map of $M_{n}(\mathbb{C})$ to $M_{k}(\mathbb{C})$ .With the notion of the von Neumann entropy for a state in mind, we give a model ofThe $r$ is uniquely determined for $\Phi$ and the $r$ -tupple is also unique up to a $r\cross r$ unitary matrix.

数理解析研究所講究録•Dec 1, 2012•0 citations
In scope of CPCPCP -convexity theory for C∗C^{*}C∗ -algebras (quantization of convexity, measure, entropy for completely positive maps), we investigate the operational stmcture of quantum interactions of entangled systems, and propose new information quantities which naturally generalize the classical information theory.
We investigate the quantum interaction between two quantum sys- tems in terms of C∗C^{*}C∗ -algebras.We introduce the cococo rrelation CP-map between the bipartite systems and study the operational and statistical structure of the quantum interaction.We then generalize the informational quantities into this quantum setting, and introduce a new entropy of CP-maps, which vanishes at extreme CP-maps in CP-convexity.Introduction.We prepare two quantum states described by density operators ρ\rhoρ and σ\sigmaσ on Hilbert spaces H1H_{1}H1​ and K1K_{1}K1​ respectively, and so initially we have the compound state ω0=ρ⊗σ\omega_{0}=\rho\otimes\sigmaω0​=ρ⊗σ on the tensor product Hilbert space H1⊗K1H_{1}\otimes K_{1}H1​⊗K1​ .Suppose that there exists an interaction between the two subsystems for some time interval, and that the systems are in equilibrium with the exterior for this time period, then the compound state is changed to a state ω\omegaω described by ω=U(ρ⊗σ)U∗\omega=U(\rho\otimes\sigma)U^{*}ω=U(ρ⊗σ)U∗ where UUU is a unitary operator representing the quantum evolution of the systems.After the interaction, the states of the subsystems on H2H_{2}H2​ and K2K_{2}K2​ are described by TrK2wTr_{K_{2}}wTrK2​​w and RH2ωR_{H_{2}}\omegaRH2​​ω respectively.Then the channels φ∗\varphi^{*}φ∗ from T(H1)T(H_{1})T(H1​) to T(H2)T(H_{2})T(H2​) , and ϕ∗\phi^{*}ϕ∗ from T(K1)T(K_{1})T(K1​) to T(K2)T(K_{2})T(K2​) , are defined by φ∗(ρ)=hK2ω=rbK2U(ρ⊗σ)U∗\varphi^{*}(\rho)=h_{K_{2}}\omega=r_{b_{K_{2}}U(\rho}\otimes\sigma)U^{*}φ∗(ρ)=hK2​​ω=rbK2​​U(ρ​⊗σ)U∗ and ϕ∗(σ)=bH2ω=hH2U(ρ⊗σ)U∗\phi^{*}(\sigma)=b_{H_{2}}\omega=h_{H_{2}}U(\rho\otimes\sigma)U^{*}ϕ∗(σ)=bH2​​ω=hH2​​U(ρ⊗σ)U∗ ,The detailed version of this paper will be submitted for publication elsewhere.Typaeet by AA#?fflA_{A}\#?fflAA​#?ffl
Abstract Quantum information science is a new and rapidly developing discipline that studies the general laws governing the transmission, storage, and processing of information in systems that obey the laws of quantum mechanics. It makes extensive use of matrix theory and operator theory on Hilbert spaces to investigate the capabilities of such systems and to develop principles for their rational design. An important consequence of the development of quantum information science is a substantial clarification of the logical structure and foundations of quantum mechanics and of its relation to reality. These lectures are devoted to the foundations of the mathematical theory of quantum systems: states, observables, communication channels, their entropic and information characteristics, and quantum information-processing protocols. The lectures provide a detailed introduction to the statistical structure of quantum theory and the necessary background from classical information theory; hence, knowledge of the basic mathematical subjects—analysis, linear algebra, and probability theory—is sufficient for understanding the material. In the first part of the course, we use the “standard” statistical model of a quantum system, but consider it in comparison with the classical statistical description. A consistent development of this approach leads to a generalized statistical model of quantum mechanics, in which observables (in general, “unsharp”) are described by probability operator-valued measures, while evolutions (in general, irreversible) are described by completely positive maps. Introductory material on these concepts, which underlie mathematical models of quantum information systems in the presence of noise and errors and have numerous applications in quantum information science, is presented in the second part of the course.
We show how Free Probability Theory-and in particular, recent results regarding the norm convergence of random matrices-help to better understand im- portant problems in Quantum Information Theory, such as the problem of Minimum Output Entropy additivity.This is a report on joint papers and projects with Ion Nechita, Serban Belinschi, Camille Male and Motohisa Fukuda 1. ENTANGLEMENT AND ADDITIVITY 1.1.Entanglement and entropy.In Quantum Information Theory, a vector subspace CCC of a tensor product A⊗BA\otimes BA⊗B of two Hilbert spaces is said to be entangled iff 000 is the only vector of CCC that can be written as a⊗ba\otimes ba⊗b with a∈A,a\in A,a∈A, b∈Bb\in Bb∈B .The problems of quantifying entanglement, and of finding 'highly' entangled spaces are very important in quantum information theory.For x∈A⊗Bx\in A\otimes Bx∈A⊗B , its singular values -also known as Schmidt coefficients in Quantum Information Theory-are non-negative numbers λ1(x)≥…≥λk(x)≥0\lambda_{1}(x)\geq\ldots\geq\lambda_{k}(x)\geq 0λ1​(x)≥…≥λk​(x)≥0 such that x=∑i=1kλiei(x)⊗fi(x)x= \sum_{i=1}^{k}\sqrt{\lambda_{i}}e_{i}(x)\otimes f_{i}(x)x=∑i=1k​λi​​ei​(x)⊗fi​(x) Here, k=min(k=min(k=min(dim(A), dim(B));ei(x),dim(B));e_{i}(x),dim(B));ei​(x), fi(x)f_{i}(x)fi​(x) are orthonormal vectors.The sequence λ(x)=(λ1(x),…)∈Rk\lambda(x)=(\lambda_{1}(x), \ldots)\in \mathbb{R}^{k}λ(x)=(λ1​(x),…)∈Rk is uniquely defined.Let KC={λ(x),x∈C,∣∣x∣∣=1}⊂RkK_{C}=\{\lambda(x), x\in C, ||x||=1\}\subset \mathbb{R}^{k}KC​={λ(x),x∈C,∣∣x∣∣=1}⊂Rk .This compact set is a subset of the set △k+={y∈R+k:y1≥y2≥⋯≥yk≥0,∑i=1kyi=1}.\triangle_{k}^{+}=\{y\in \mathbb{R}_{+}^{k}:y_{1}\geq y_{2}\geq\cdots\geq y_{k}\geq 0, \sum_{i=1}^{k}y_{i}=1\}.△k+​={y∈R+k​:y1​≥y2​≥⋯≥yk​≥0,∑i=1k​yi​=1}. Besides, we have Δk+⊂Δk\Delta_{k}^{+}\subset\Delta_{k}Δk+​⊂Δk​ , where Δk={x∈R+k∣∑i=1kxi=1}\Delta_{k}=\{x\in \mathbb{R}_{+}^{k}|\sum_{i=1}^{k}x_{i}=1\}Δk​={x∈R+k​∣∑i=1k​xi​=1} is the (k−1)−(k-1)-(k−1)− dimensional probability simplex.We will also make an abuse of notation and also denote by KCK_{C}KC​ the subset of △k\triangle_{k}△k​ that is obtained by symmetrizing KCK_{C}KC​ under permuting the coordinates.For a positive real number p>0p>0p>0, we recall that the R\'enyi entropy of order ppp of a probability vector x∈Δkx\in\Delta_{k}x∈Δk​ to be $H^{p}(x):= \frac{1}{1-p}\log\sum_{i=1}^{k}x_{i}^{p}.$2010 Mathematics Subject Cassification.46L65(46L37,46L54,.46L87)46L65(46L37,46L54,.46L87)46L65(46L37,46L54,.46L87) .
Let Mn(C)M_{n}(\mathbb{C})Mn​(C) be the algebra of $n\cross n$ complex mairices, and let Φ\PhiΦ be a unital completely positive map of Mn(C)M_{n}(\mathbb{C})Mn​(C) to Mk(C)M_{k}(\mathbb{C})Mk​(C) .With the notion of the von Neumann entropy for a state in mind, we give a model ofThe rrr is uniquely determined for Φ\PhiΦ and the rrr -tupple is also unique up to a $r\cross r$ unitary matrix.