Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

数理解析研究所講究録

Analogue of Poisson Distribution in Monotone Fock Space

Naofumi Muraki

Published Jun 1, 1999• 1 citation
OpenAlex

Abstract

The monotone Fock space was introduced by the author to construct a new example of noncommutative “de Moivre Laplace theorem ” [1] and of noncommutative “Brow-nian motion ” [2] in quantum probability theory (see [3] [4] for general reference to quantum probability). We note that the essentially same structure was introduced

Related papers

Covariant Poisson Fields In Fock Space

David Applebaum

As I type these words, I have in front of me a collection of notes from a seminar in Quantum Probability run by Luigi Accardi in Roma in 1984-5 where I was at the time based as a research fellow. In those days I was rather meticulous in ordering my jottings and on turning to page 131 in my file I find twenty pages entitled “Representations of Current Groups” by K.R. Parthasarathy. I remember being fascinated by these lectures (I think there were two) in which Partha gave an account of how the Lévy-Khinchine formula arises, naturally from the construction of continuous tensor products of unitary group representations (see e.g. [St], [Ar], [PS]) and then revealed the beautiful Fock space decomposition of infinitely divisible random variables as sums of creation, conservation and annihilation operators. This is in fact one of the many perspectives through which the historian can see the “birth” of quantum probability — as a student of Robin Hudson, I had seen the subject evolve from a different point of view, through the search for quantum central limit theorems and non-commuting analogues of Brownian motion. The marriage of these two viewpoints turned out to be a wonderful thing, leading as it did to the creation of quantum stochastic calculus.

Texts and readings in mathematics•Jan 1, 1996•2 citations

On $(alpha, q)$-Poisson Operators and Their Distributions (Mathematical aspects of quantum fields and related topics)

Nobuhiro Asai

We propose an (α, q)-analogue of the Poisson operator on the Fock space of Bożejko-Ejsmont-Hasebe [6] and discuss a probability law of the operator. We show that the probability law is expressed by the q-Meixner distribution. Our results contain not only symmetric distributions as in [6], but also non-symmetric ones such as free Poisson, q and q²- deformations of Poisson, Pascal, Gamma, and Meixner distributions. This is the summary of the paper [4].

•Jun 1, 2021•0 citations

The poisson process in quantum stochastic calculus

Shayanthan Pathmanathan

Given a compensated Poisson process $(X_t)_{t \geq 0}$ based on $(\Omega, \mathcal{F}, \mathbb{P})$, the Wiener-Poisson isomorphism $\mathcal{W} : \mathfrak{F}_+(L^2 (\mathbb{R}_+)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$ is constructed. We restrict the isomorphism to $\mathfrak{F}_+(L^2 [0,1])$ and prove some novel properties of the Poisson exponentials $\mathcal{E}(f) := \mathcal{W}(e(f))$. A new proof of the result $\Lambda_t + A_t + A^{\dagger}_t = \mathcal{W}^{-1}\widehat{X_t} \mathcal{W}$ is also given. The analogous results for $\mathfrak{F}_+(L^2 (\mathbb{R}_+))$ are briefly mentioned. The concept of a compensated Poisson process over $\mathbb{R}_+$ is generalised to any measure space $(M, \mathcal{M}, \mu)$ as an isometry $I : L^2(M, \mathcal{M}, \mu) \to L^2 (\Omega,\mathcal{F}, \mathbb{P})$ satisfying certain properties. For such a generalised Poisson process we recall the construction of the generalised Wiener-Poisson isomorphism, $\mathcal{W}_I : \mathfrak{F}_+(L^2(M)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$, using Charlier polynomials. Two alternative constructions of $\mathcal{W}_I$ are also provided, the first using exponential vectors and then deducing the connection with Charlier polynomials, and the second using the theory of reproducing kernel Hilbert spaces. Given any measure space $(M, \mathcal{M}, \mu)$, we construct a canonical generalised Poisson process $I : L^2 (M, \mathcal{M}, \mu) \to L^2(\Delta, \mathcal{B}, \mathbb{P})$, where $\Delta$ is the maximal ideal space, with $\mathcal{B}$ the completion of its Borel $\sigma$-field with respect to $\mathbb{P}$, of a $C^*$-algebra $\mathcal{A} \subseteq \mathfrak{B}(\mathfrak{F}_+(L^2(M)))$. The Gelfand transform $\mathcal{A} \to \mathfrak{B}(L^2(\Delta))$ is unitarily implemented by the Wiener-Poisson isomorphism $\mathcal{W}_I: \mathfrak{F}_+(L^2(M)) \to L^2(\Delta)$. This construction only uses operator algebra theory and makes no a priori use of Poisson measures. A new Fock space proof of the quantum Ito formula for $(\Lambda_t + A_t + A^{\dagger}_t)_{0 \leq t \leq 1}$ is given. If $(F_{\ \! \! t})_{0 \leq t \leq 1}$ is a real, bounded, predictable process with respect to a compensated Poisson process $(X_t)_{0 \leq t \leq 1}$, we show that if $M_t = \int_0^t F_s dX_s$, then on $\mathsf{E}_{\mathrm{lb}} := \mathrm{linsp} \{ e(f) : f \in L^2_{\mathrm{lb}}[0,1] \}$, $\mathcal{W}^{-1} \widehat{M_t} \mathcal{W} = \int_0^t \mathcal{W}^{-1} \widehat{F_s} \mathcal{W} (d\Lambda_s + dA_s + dA^{\dagger}_s),$ and that $(\mathcal{W}^{-1} \widehat{M_t} \mathcal{W})_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale. We prove, using the classical Ito formula, that if $(J_t)_{0 \leq t \leq 1}$ is a regular self-adjoint quantum semimartingale, then $(\mathcal{W} \widehat{M_t} \mathcal{W}^{-1} + J_t)_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula, and hence the quantum Ito formula. The equivalent result for the sum of a Brownian and Poisson martingale, provided that the sum is essentially self-adjoint with core $\mathsf{E}_{\mathrm{lb}}$, is also proved.

•Jan 1, 2002•1 citation

Quantum filtering: a reference probability approach

Luc Bouten, Ramon van Handel

These notes are intended as an introduction to noncommutative (quantum) filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as the least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. Next we describe the Hudson-Parthasarathy quantum Ito calculus and its use in the modelling of physical systems. Finally, we use a reference probability method to obtain quantum filtering equations, in the Belavkin-Zakai (unnormalized) form, for several system-observation models from quantum optics. The normalized (Belavkin-Kushner-Stratonovich) form is obtained through a noncommutative analogue of the Kallianpur-Striebel formula.

arXiv:math-ph/0508006•Aug 1, 2005•22 citations

Generalized Quantum Stochastic Processes on Fock Space

Nobuaki Obata

As is highlighted in the excellent books by Meyer [21] and by Parthasarathy [26] quantum stochastic calculus on (Boson) Fock space has developed into a new field of mathematics keeping a profound contact with physical applications. Since Hudson and Parthasarathy [12] first formulated quantum stochastic integrals of Ito type in 1984 a crucial role has been played by three basic quantum stochastic processes:

Publications of the Research Institute for Mathematical Sciences•Aug 31, 1995•24 citations

Quantum Probability and Applications III: Proceedings of a Conference Held in Oberwolfach, Frg, January 25-31, 1987

Luigi Accardi, Wilhelm von Waldenfels

A note on Meyer's note.- Stochastic integration.- Quantum stochastic parallel transport on non-commutative vector bundles.- Input and output channels in quantum systems and quantum stochastic differential equations.- Some noncommutative Radon-Nikodym Theorems for von Neumann algebras.- Multidimensional quantum diffusions.- An application of de finetti's theorem.- The quantum Langevin equation from the independent-oscillator model.- Quantum poisson processes: Physical motivations and applications.- A noncommutative generalization of conditionally positive definite functions.- Contraction semigroups in L2 over a von neumann algebra.- Survey on a theory of non-commutative stationary markov processes.- Dynamical entropy for quantum systems.- An integral kernel approach to noise.- A note on shifts and cocycles.- Local measures in fock space stochastic calculus and a generalized ito-tanaka formula.- Representation of a class of quantum martingales II.- Conditional expectation in quantum probability.- Mutual quadratic variation and ito's table in quantum stochastic calculus.- On mixing properties of automorphisms of von neumann algebras related to measure space transformations.- First exit time: A theory of stopping times in quantum processes.- A central limit theorem on the free lie group.- Entropy, observability and the generalised second law of thermodynamics.- Remarks on asymptotic properties of groups of the bogoliubov transformations of car c*-algebras.- Linear and non-linear stochastic processes.- Detailed balance and critical slowing down.- Quantum martingales and stochastic integrals.

•May 1, 1988•1 citation
As I type these words, I have in front of me a collection of notes from a seminar in Quantum Probability run by Luigi Accardi in Roma in 1984-5 where I was at the time based as a research fellow. In those days I was rather meticulous in ordering my jottings and on turning to page 131 in my file I find twenty pages entitled “Representations of Current Groups” by K.R. Parthasarathy. I remember being fascinated by these lectures (I think there were two) in which Partha gave an account of how the Lévy-Khinchine formula arises, naturally from the construction of continuous tensor products of unitary group representations (see e.g. [St], [Ar], [PS]) and then revealed the beautiful Fock space decomposition of infinitely divisible random variables as sums of creation, conservation and annihilation operators. This is in fact one of the many perspectives through which the historian can see the “birth” of quantum probability — as a student of Robin Hudson, I had seen the subject evolve from a different point of view, through the search for quantum central limit theorems and non-commuting analogues of Brownian motion. The marriage of these two viewpoints turned out to be a wonderful thing, leading as it did to the creation of quantum stochastic calculus.
On (alpha,q)(alpha, q)(alpha,q)-Poisson Operators and Their Distributions (Mathematical aspects of quantum fields and related topics)
Given a compensated Poisson process (Xt)t≥0(X_t)_{t \geq 0}(Xt​)t≥0​ based on (Ω,F,P)(\Omega, \mathcal{F}, \mathbb{P})(Ω,F,P), the Wiener-Poisson isomorphism W:F+(L2(R+))→L2(Ω,F,P)\mathcal{W} : \mathfrak{F}_+(L^2 (\mathbb{R}_+)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})W:F+​(L2(R+​))→L2(Ω,F,P) is constructed. We restrict the isomorphism to F+(L2[0,1])\mathfrak{F}_+(L^2 [0,1])F+​(L2[0,1]) and prove some novel properties of the Poisson exponentials E(f):=W(e(f))\mathcal{E}(f) := \mathcal{W}(e(f))E(f):=W(e(f)). A new proof of the result Λt+At+At†=W−1Xt^W\Lambda_t + A_t + A^{\dagger}_t = \mathcal{W}^{-1}\widehat{X_t} \mathcal{W}Λt​+At​+At†​=W−1Xt​​W is also given. The analogous results for F+(L2(R+))\mathfrak{F}_+(L^2 (\mathbb{R}_+))F+​(L2(R+​)) are briefly mentioned. The concept of a compensated Poisson process over R+\mathbb{R}_+R+​ is generalised to any measure space (M,M,μ)(M, \mathcal{M}, \mu)(M,M,μ) as an isometry I:L2(M,M,μ)→L2(Ω,F,P)I : L^2(M, \mathcal{M}, \mu) \to L^2 (\Omega,\mathcal{F}, \mathbb{P})I:L2(M,M,μ)→L2(Ω,F,P) satisfying certain properties. For such a generalised Poisson process we recall the construction of the generalised Wiener-Poisson isomorphism, WI:F+(L2(M))→L2(Ω,F,P)\mathcal{W}_I : \mathfrak{F}_+(L^2(M)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})WI​:F+​(L2(M))→L2(Ω,F,P), using Charlier polynomials. Two alternative constructions of WI\mathcal{W}_IWI​ are also provided, the first using exponential vectors and then deducing the connection with Charlier polynomials, and the second using the theory of reproducing kernel Hilbert spaces. Given any measure space (M,M,μ)(M, \mathcal{M}, \mu)(M,M,μ), we construct a canonical generalised Poisson process I:L2(M,M,μ)→L2(Δ,B,P)I : L^2 (M, \mathcal{M}, \mu) \to L^2(\Delta, \mathcal{B}, \mathbb{P})I:L2(M,M,μ)→L2(Δ,B,P), where Δ\DeltaΔ is the maximal ideal space, with B\mathcal{B}B the completion of its Borel σ\sigmaσ-field with respect to P\mathbb{P}P, of a C∗C^*C∗-algebra A⊆B(F+(L2(M)))\mathcal{A} \subseteq \mathfrak{B}(\mathfrak{F}_+(L^2(M)))A⊆B(F+​(L2(M))). The Gelfand transform A→B(L2(Δ))\mathcal{A} \to \mathfrak{B}(L^2(\Delta))A→B(L2(Δ)) is unitarily implemented by the Wiener-Poisson isomorphism WI:F+(L2(M))→L2(Δ)\mathcal{W}_I: \mathfrak{F}_+(L^2(M)) \to L^2(\Delta)WI​:F+​(L2(M))→L2(Δ). This construction only uses operator algebra theory and makes no a priori use of Poisson measures. A new Fock space proof of the quantum Ito formula for (Λt+At+At†)0≤t≤1(\Lambda_t + A_t + A^{\dagger}_t)_{0 \leq t \leq 1}(Λt​+At​+At†​)0≤t≤1​ is given. If (F  ⁣ ⁣t)0≤t≤1(F_{\ \! \! t})_{0 \leq t \leq 1}(F t​)0≤t≤1​ is a real, bounded, predictable process with respect to a compensated Poisson process (Xt)0≤t≤1(X_t)_{0 \leq t \leq 1}(Xt​)0≤t≤1​, we show that if Mt=∫0tFsdXsM_t = \int_0^t F_s dX_sMt​=∫0t​Fs​dXs​, then on Elb:=linsp{e(f):f∈Llb2[0,1]}\mathsf{E}_{\mathrm{lb}} := \mathrm{linsp} \{ e(f) : f \in L^2_{\mathrm{lb}}[0,1] \}Elb​:=linsp{e(f):f∈Llb2​[0,1]}, W−1Mt^W=∫0tW−1Fs^W(dΛs+dAs+dAs†),\mathcal{W}^{-1} \widehat{M_t} \mathcal{W} = \int_0^t \mathcal{W}^{-1} \widehat{F_s} \mathcal{W} (d\Lambda_s + dA_s + dA^{\dagger}_s),W−1Mt​​W=∫0t​W−1Fs​​W(dΛs​+dAs​+dAs†​), and that (W−1Mt^W)0≤t≤1(\mathcal{W}^{-1} \widehat{M_t} \mathcal{W})_{0 \leq t \leq 1}(W−1Mt​​W)0≤t≤1​ is an essentially self-adjoint quantum semimartingale. We prove, using the classical Ito formula, that if (Jt)0≤t≤1(J_t)_{0 \leq t \leq 1}(Jt​)0≤t≤1​ is a regular self-adjoint quantum semimartingale, then (WMt^W−1+Jt)0≤t≤1(\mathcal{W} \widehat{M_t} \mathcal{W}^{-1} + J_t)_{0 \leq t \leq 1}(WMt​​W−1+Jt​)0≤t≤1​ is an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula, and hence the quantum Ito formula. The equivalent result for the sum of a Brownian and Poisson martingale, provided that the sum is essentially self-adjoint with core Elb\mathsf{E}_{\mathrm{lb}}Elb​, is also proved.
These notes are intended as an introduction to noncommutative (quantum) filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as the least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. Next we describe the Hudson-Parthasarathy quantum Ito calculus and its use in the modelling of physical systems. Finally, we use a reference probability method to obtain quantum filtering equations, in the Belavkin-Zakai (unnormalized) form, for several system-observation models from quantum optics. The normalized (Belavkin-Kushner-Stratonovich) form is obtained through a noncommutative analogue of the Kallianpur-Striebel formula.
A note on Meyer's note.- Stochastic integration.- Quantum stochastic parallel transport on non-commutative vector bundles.- Input and output channels in quantum systems and quantum stochastic differential equations.- Some noncommutative Radon-Nikodym Theorems for von Neumann algebras.- Multidimensional quantum diffusions.- An application of de finetti's theorem.- The quantum Langevin equation from the independent-oscillator model.- Quantum poisson processes: Physical motivations and applications.- A noncommutative generalization of conditionally positive definite functions.- Contraction semigroups in L2 over a von neumann algebra.- Survey on a theory of non-commutative stationary markov processes.- Dynamical entropy for quantum systems.- An integral kernel approach to noise.- A note on shifts and cocycles.- Local measures in fock space stochastic calculus and a generalized ito-tanaka formula.- Representation of a class of quantum martingales II.- Conditional expectation in quantum probability.- Mutual quadratic variation and ito's table in quantum stochastic calculus.- On mixing properties of automorphisms of von neumann algebras related to measure space transformations.- First exit time: A theory of stopping times in quantum processes.- A central limit theorem on the free lie group.- Entropy, observability and the generalised second law of thermodynamics.- Remarks on asymptotic properties of groups of the bogoliubov transformations of car c*-algebras.- Linear and non-linear stochastic processes.- Detailed balance and critical slowing down.- Quantum martingales and stochastic integrals.