# Registered Data

## [02700] Recent developments on Infinite Dimensional Analysis, Stochastic Analysis and Quantum Probability

**Session Date & Time**:- 02700 (1/2) : 1C (Aug.21, 13:20-15:00)
- 02700 (2/2) : 1D (Aug.21, 15:30-17:10)

**Type**: Proposal of Minisymposium**Abstract**: There has been a great achievement of the theory in both white noise analysis $($as an infinite dimensional analysis$)$ and quantum probability theory starting from the seventies of the last century. These two theories have been developed and unified in order to get important directions on the quantum information theory, quantum computation and etc. In this session we would like to discuss connections between white noise analysis and quantum information theory through the experts’ talks in each fields including the mathematical finance.**Organizer(s)**: Isamu Doku, Kimiaki Saito**Classification**:__60H40__,__81P45__,__60H30__,__46L53__,__68Q11__,__White Noise Theory, Quantum Information, Applications of stochastic analysis, Noncommutative probability and statistics, Information complexity__**Speakers Info**:- Nobuaki Obata (Tohoku University)
- Noboru Watanabe (Tokyo University of Science)
- Un Cig Ji (Chungbuk National University)
- Shunsuke Kaji (Meijo University)

**Talks in Minisymposium**:**[02940] Note on complexities for the quantum compound systems****Author(s)**:**Noboru Watanabe**(Tokyo University of Science)

**Abstract**: In order to discuss the efficiency of information transmission of the quantum com- munication processes consistently, we consider the entropy type functional and the mutual entropy type functional with respect to the initial state and the quantum communication channel. In this study, the mutual entropy type measures are con- structed by the compound states between the initial and final systems. We modify the compound states and examine the entropy functional and the mutual entropy functional defined by the modified compound states by means of the initial state and the completely positive channel to study the efficiency of information transmission of the quantum communication processes.

**[02981] Multiplication Operators by White Noise Delta Functions and Associated Differential Equations****Author(s)**:**Un Cig Ji**(Chungbuk National University)

**Abstract**: We establish explicit forms of the multiplication operators induced by white noise delta functions, which are closely related to the Bogoliubov transformation and a quantum analogue of Girsanov transform. Then we study the differential equations for operators associated with the multiplication operators by the white noise delta functions. This talk is based on a joint work with L. Accardi and K. Saito.

**[03589] Asymptotics of densities of first passage times for spectrally negative Lévy processes****Author(s)**:**shunsuke kaji**(meijo university)

**Abstract**: We study a first passage time of a Lévy process over a positive constant level. In the spectrally negative case we give conditions for absolutely continuity of the distributions of the first passage times. The tail asymptotics of their densities are alsoclarified, where the asymptotics depend on tail behaviour of the corresponding Lévy measures.

**[03953] A combinatorial formula of the moments of a deformed filed operator****Author(s)**:**Nobuhiro ASAI**(Aichi University of Education)

**Abstract**: We shall construct the two parameterized deformed Fock space obtained from the weighted $q$-deformation technique. We shall explain a combinatorial moment formula of a Poisson type filed operator defined on this Fock space by using the set partitions with their statistics. In addition, we shall mention relationships with the recurrence formula for the orthogonal polynomials of the deformed Poisson distribution. This talk is based on the joint work with H. Yoshida (Ochanomizu Univ, Tokyo).

**[03986] Positivity of Q-matrices and quadratic embedding constants of graphs****Author(s)**:**Nobuaki Obata**(Tohoku University)

**Abstract**: Let $G=(V,E)$ be a graph. Positivity of the Q-matrix $Q=Q_q=[q^{d(x,y)}]$ is essential for q-deformed vacuum state of a graph and q-deformed CLT for a growing graph. Positivity of $Q_q$ is profoundly related to conditional negativity of the distance matrix $D=[d(x,y)]$. It is shown that the positivity region of $Q_q$ contains $[0,1]$ if and only if the quadratic embedding constant (QEC) of $G$ is non-positive. We report some results in this line and discuss open questions.