Interdisciplinary Information Sciences

Special Issue: New Trends in Infinite Dimensional Analysis and Quantum Probability

Survey of the Theories for Analogue of Wiener Measure Space

Kun Sik RYU, Man Kyu IM, Ki Seong CHOI

2009 年 15 巻 3 号 p. 319-337
発行日: 2009年
公開日: 2009/11/02

DOIhttps://doi.org/10.4036/iis.2009.319

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We consider the analogue of Wiener measure on the space of all real continuous functions, and then establish the measure-valued measure and the operator-valued measure on our space and we investigate their important properties, for example, the integration formula, the rotation theorem, the Bartle integral, the Bochner integral and the Dobrakov integral, the simple formula for conditional expectation, the measure-valued Feynman-Kac formula and Volterra integral equation and integral transform.

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On the Question of the Thompson Group

Jaeseong HEO

2009 年 15 巻 3 号 p. 339-344
発行日: 2009年
公開日: 2009/11/02

DOIhttps://doi.org/10.4036/iis.2009.339

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This survey is concerned with the Thompson group F and the von Neumann algebra L(F) associated with F. We first review some basic results on the Thompson groups and group von Neumann algebras. We investigate the structure of the von Neumann algebra W^*(F,P) generated by L(F) and a projection P on l²(F). We show that the algebra (not necessarily *) algebraically generated by two generating unitaries of L(F) and the commutant L(F)′ is strong-operator dense in {\\mathcal B}({\\mathcal H}). Furthermore, we will discuss actions of F and F′ on a set of dyadic rational numbers in [0,1] and some outer automorphism of L(F) for the amenability question of F.

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Quantum Stochastic Gradients

Un Cig JI, Nobuaki OBATA

2009 年 15 巻 3 号 p. 345-359
発行日: 2009年
公開日: 2009/11/02

DOIhttps://doi.org/10.4036/iis.2009.345

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We obtain regularity properties of the quantum stochastic gradients (creation, annihilation and conservation gradients) by systematic study of their various domains. Accordingly, regularity properties of the (non-adapted) quantum stochastic integrals of Hitsuda–Skorohod type follow by duality.

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Limit Theorems for the Average Distance and the Degree Distribution of the Threshold Network Model

Akihiro FUJIHARA, Yusuke IDE, Norio KONNO, Naoki MASUDA, Hiroyoshi MIW ...

2009 年 15 巻 3 号 p. 361-366
発行日: 2009年
公開日: 2009/11/02

DOIhttps://doi.org/10.4036/iis.2009.361

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The threshold network model is a type of finite random graph. For this model, we obtain an almost-sure convergence theorem for the average distance and derive a variant of Poisson approximation of the degree distribution by using extreme value theory.

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Orthogonal Polynomials Induced by Discrete-Time Quantum Walks in One Dimension

Masatoshi HAMADA, Norio KONNO, Wojciech MŁOTKOWSKI

2009 年 15 巻 3 号 p. 367-375
発行日: 2009年
公開日: 2009/11/02

DOIhttps://doi.org/10.4036/iis.2009.367

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In this paper we obtain some properties of orthogonal polynomials given by a weight function which is a limit density of a rescaled discrete-time quantum walk on the line.

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Glauber and Kawasaki Dynamics for Determinantal Point Processes in Discrete Spaces

Myeongju CHAE, Hyun Jae YOO

2009 年 15 巻 3 号 p. 377-388
発行日: 2009年
公開日: 2009/11/02

DOIhttps://doi.org/10.4036/iis.2009.377

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We construct the equilibrium Glauber and Kawasaki dynamics on discrete spaces which leave invariant certain determinantal point processes. We will construct Fellerian Markov processes with specified core for the generators. Further, we discuss the ergodicity of the processes.

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A Generalization of the Monotonic and Boolean Independence of Algebras and Applications to Probability

Janusz WYSOCZAŃSKI

2009 年 15 巻 3 号 p. 389-397
発行日: 2009年
公開日: 2009/11/02

DOIhttps://doi.org/10.4036/iis.2009.389

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We present a notion of bm-independence, which generalizes the boolean and monotonic independences. The generalization is defined for partially ordered index sets. For each independence there is an analogue of the classical central limit theorem. In the case of bm-independence this depends also on the index set. Examples of such partially ordered index sets are discrete lattices in symmetric positive cones.

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Infrared Divergence of a Scalar Quantum Field Model on a Pseudo Riemannian Manifold

Christian GÉRARD, Fumio HIROSHIMA, Annalisa PANATI, Akito SUZUK ...

2009 年 15 巻 3 号 p. 399-421
発行日: 2009年
公開日: 2009/11/02

DOIhttps://doi.org/10.4036/iis.2009.399

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A scalar quantum field model defined on a pseudo Riemannian manifold is considered. The model is unitarily transformed to the one with a variable mass. By means of a Feynman-Kac-type formula, it is shown that when the variable mass is short range, the Hamiltonian has no ground state. Moreover the infrared divergence of the expectation values of the number of bosons in the ground state is discussed.

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Limit Theorems for Discrete-Time Quantum Walks on Trees

Kota CHISAKI, Masatoshi HAMADA, Norio KONNO, Etsuo SEGAWA

2009 年 15 巻 3 号 p. 423-429
発行日: 2009年
公開日: 2009/11/02

DOIhttps://doi.org/10.4036/iis.2009.423

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We consider a discrete-time quantum walk W_t given by the Grover transformation on the homogeneous tree. We reduce W_t to a quantum walk X_t on a half line with a wall at the origin. This paper presents two types of limit theorems for X_t. The first one is X_t as t→∞, which corresponds to a localization in the case of an initial qubit state. The second one is X_t⁄t as t→∞, whose limit density is given by the Konno density function [1–4]. The density appears in various situations of discrete-time cases. The corresponding similar limit theorem was proved in [5] for a continuous-time case on the homogeneous tree.

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A Gauss-Poisson Correspondence and the Lévy Laplacian

Kimiaki SAITÔ

2009 年 15 巻 3 号 p. 431-440
発行日: 2009年
公開日: 2009/11/02

DOIhttps://doi.org/10.4036/iis.2009.431

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In this paper we present recent results on infinite dimensional Laplacians. In particular, by introducing the operator which transfers regular white noise functionals to functionals of exponential white noise, we give a relationship between an infinite dimensional Fourier-Mehler transform and the Lévy Laplacian. The operator implies a Gauss-Poisson correspondence if we consider the Lévy Laplacian acting on multiple Wiener integrals by some Lévy process. We also give an infinite dimensional random field associated with the Lévy Laplacian.

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Non-Equilibrium Thermo Field Dynamics and Its Application to Error-Correction for Spatially Correlated Quantum Errors

Toshihico ARIMITSU

2009 年 15 巻 3 号 p. 441-471
発行日: 2009年
公開日: 2009/11/02

DOIhttps://doi.org/10.4036/iis.2009.441

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The framework of the canonical operator formalism for dissipative and/or stochastic quantum systems named non-equilibrium thermo field dynamics (NETFD) is presented in a self-contained manner. As an application of the formalism, a model of noisy-channel which is under the influence of spatially correlated quantum Brownian motion is investigated within the framework of NETFD. It is shown that errors due to spatially correlated noises can be corrected by the quantum error-correction code and error-correction procedure prepared for those for independent noises.

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