Workshop on IDAQP and their Applications - IMS

Workshop on IDAQP and their Applications
(3 - 7 Mar 2014)

Dedicated to Professor Takeyuki Hida
Co-sponsored by RIST, Quantum Bio-informatics Research Division, Tokyo University of Science,
Aichi Prefectural University

~ Abstracts ~

Deduction of non--commutativity from commutativity: applications to quantum mechanics and classical probability theory
Luigi Accardi, University of Rome Tor Vergata, Italy

The talk explains how a simple and non trivial generalization of quantum mechanics emerges of from a well established topic of 19th century classical analysis: the theory of orthogonal polynomials.

In fact every classical random variable with all moments has a unique, intrinsic, quantum decomposition in terms of generalized creation, annihilation and preservation (CAP) operators.

The commutation relations among the CAP operators are uniquely determined by the principal Jacobi sequence of the probability distribution of the classical random variable. Standard quantum mechanics corresponds to the equivalence class (for the relation of having the same principal Jacobi sequence) of the standard Gaussian or Poisson measure on $\mathbb R$.

This shows that classical probability possesses a microscopic structure which is intrinsically non commutative. In this sense one can speak of emergence of non--commutativity from commutativity.

The recently obtained multi--dimensional generalization of these results suggests a new approach to non--commutative geometry.

Sensitive homology searching based on MTRAP alignment
Toshihide Hara, Tokyo University of Science, Japan

We develop a sensitive homology searching method by introducing the MTRAP alignment algorithm. Traditional method, such as BLAST, uses a measure determined on the assumption that there is no an intersite correlation on the sequences. On the other hand, our new method takes the correlation of consecutive residues.

Some of future directions of white noise theory
Takeyuki Hida, Nagoya University and Meijo University, Japan

The white noise theory has developed extensively, specifically by restricting the noise to be Gaussian. Non Gaussian cases are also interesting, in particular by observing non-similarities to the Gaussian case.

One of the significant future directions is concerned with the analysis of generalized functionals of a noise. Noting harmonic analysis applied to those generalized functionals, we can efficiently use the infinite dimensional rotation group. Successful results lead us to take the sympletic group and then to non-commutative analysis. By doing so, we can find intimate connections with quantum dynamics representations of some Lie groups and even with questions in economics.

We shall, therefore, be given many motivations to future directions of white noise theory.

Multiple Markov properties of Gaussian processes and their control
Win Win Htay, Yangon Technological University, Myanmar

We shall first make a short survey of multiple Markov properties of Gaussian processes, then come to the most general definition of this properties, where we use the white noise theory, in particular recent results on generalized white noise functionals.

Having established the analytic properties of those multiple Markov Gaussian processes, we can observe some basic properties of those processes then we shall come to some actual procedures to have the innovation as well as the best predictor of the future values based on the past observed data.

We also discuss the entropy loss which is one of the characteristics of multiple Markov Gaussian process expressing the rate of transmission of information.

Quantum white noise derivatives and series expansions of super operators
Un Cig Ji, Chungbuk National University, Korea

Within the framework of quantum white noise theory, we introduce the notion of quantum white noise derivatives of white noise (generalized) operators. Then as an application we study series expansions of super operators acting on white noise operators in terms of quantum white noise derivatives.

Local statistics for random self adjoint operators
Maddaly Krishna, The Institute of Mathematical Sciences, India

We discuss local statistics associated with random operators of the Schrodinger type on the lattice. To do this we look at finite dimensional approximations of the random operators and look at the random atomic measures associated with the eigen values in a neighbourhood of a point in the spectrum.

Weighted Fourier algebras on SUq(2): characters and finite dimensional representations
Hun Hee Lee, Seoul National University, Korea

In this talk we will consider weighted Fourier algebras on compact quantum groups mainly focusing on SUq(2) case. By assigning various weights we get different Banach algebras. We will consider characters and finite dimensional representations on this algebras in connection with the complexification SLq(2,C).

Two types of quantum correlation of quantum composite system
Takashi Matsuoka, Tokyo University of Science Suwa, Japan

We will show that the quantum mutual entropy can be divided into two parts of quantum correlation, one of them is called the T-correlation (Transmitted correlation) and another one is called the Q-correlation (non-classical-Quantum correlation). On the base of the decomposition we discuss the difference between classical and quantum correlations on the scheme of Shannon's information theory.

Quantum quadratic operators and their properties
Farrukh Mukhamedov, International Isalamic University, Malaysia

It is known that the linear positive maps play important role in the quantum information theory. In these studies one of the goal is to construct a map from the state space of a system to the state space of another system. In the literature on quantum information and communication systems, such a map is called a channel. Note that the concept of state in a physical system is a powerful weapon to study the dynamical behavior of that system. One of the important class of channels is so-called pure ones, which map pure states to pure ones. Therefore, itwould be interesting characterize such kind of maps (or channels). In the present work we are going to describepure quasi quantum quadratic operators. Note that such operators act on $B(H)$ to its tensor square. Moreover, certain properties of quantum quadratic operators will be discussed.

Dynamic behavior of some stochastic predator-prey models
Huu Du Nguyen, Vietnam Institute of Advanced Study in Mathematics, Vietnam

In this talk, we a sufficient and almost necessary condition for the permanence and ergodicity of a stochastic predator-prey model in both cases of white noise and real noise. We also investigate the ω-limit set and the support of a unique invariant probability measure and the convergence in total variation norm of transition probability to the i.p.m. Comparison to existing literatures and comments on other stochastic predator-prey are given.
Keywords. Extinction; permanence; stationary distribution; ergodicity.
Subject Classification. 34C12, 60H10, 92D25.

A mathematical realization of von Neumann's measurement scheme
Masanori Ohya, Tokyo University of Science, Japan

We solve the von Neumann reduction process in quantum measurement in terms of the lifting theory. We constructed the unitary dynamics giving the above reduction.

Evolution with gross Laplacian noise
Kalyan B. Sinha, JN Centre for Advanced Scientific Research, India

Quantum Stochastic Differential Equation, driven by the Gross Laplacian Process along with the three standard processes, viz. annihilation, creation and conservation processes , is considered and its solution leads to a new kind of unitary evolution in the initial Hilbert space and evolution semigroup on the observables as well .

Polymer measures - a progress report
Ludwig Streit, University Bielefeld, Germany
Campus Universitario da Penteada, Germany

The so-called polymer measures are informally given in white noise or Wiener space by density functions N exp(-gL) where L is the self-intersection local time of (fractional) Brownian motion (fBm). As a consequence self-intersections are suppressed, and paths will swell. Once the existence of this probability density is established the main issue is to determine the scaling behavior of paths. A very recent study extends these investigations to so-called k-tolerant measures where only higher order intersections are suppressed. For more and further references see W. Bock, J. Bornales, L. Streit: Self-Repelling Fractional Brownian Motion - The case of higher order self-intersections. Preprint, November 2013, http://arxiv.org/abs/1311.4375

A hysteresis effect on optical illusion and non-Kolmogorovian probability
Yoshiharu Tanaka, Tokyo University of Science, Japan

In this study, we discuss a non-Kolmogorovness of the optical illusion in the human visual perception. We show subjects the ambiguous figure of "Schroeder stair", which has two different meanings. We prepare 11 pictures which are inclined by different angles. The tendency to answer "left side is front" depends on the order of showing those pictures. For a mathematical treatment of such a context dependent phenomena, we propose a non-Kolmogorov probabilistic model which is based on adaptive dynamics.

Quantum holography and classical random fields
Igor Volovich, Steklov Mathematical Institute, Russia

The holographic principle in quantum gravity and string theory states that the description of properties of a volume of space can be thought of as encoded on a boundary of the region. Realizations of the holography are provided by the AdS/CFT duality between string or gravity theory in anti-de Sitter space and conformal field theory on its boundary and by the gravity/fluid duality. In the talk several examples of boundary value problems for stochastic equations of mathematical physics suitable for quantum holography with random boundary geometry will be discussed by using a universal representation of solutions of partial differential equations . They include the Hida white noise calculus, quantum entangled states, Brownian sheet, Levy's Brownian motion, and the black hole formation problem.

Quantum compressed sensing
Yazhen Wang, University of Wisconsin-Madison, USA

Quantum computation and quantum information are of great current interest in computer science, mathematics, physical sciences and engineering. They will likely lead to a new wave of technological innovations in communication, computation and cryptography.

As the theory of quantum physics is fundamentally stochastic, randomness and uncertainty are deeply rooted in quantum computation, quantum simulation and quantum information. Thus statistics can play an important role in quantum computation and quantum simulation, which in turn offer great potential to revolutionize computational statistics. This talk will give a brief review on quantum computation, introduce quantum tomography and compressed sensing and present the recent work on their statistical connection.

Note on entropy type complexity of communication processes
Noboru Watanabe, Tokyo University of Science, Japan

We briefly review the entropic complexities for classical and quantum systems. We introduce some complexities by means of entropy functionals in order to treat the transmission processes consistently. We apply the general frames of quantum communication to the Gaussian communication processes. Finally, we discuss about a construction of compound states including quantum correlations.

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