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

Banach Center Publications

Quantum Itô algebra and quantum martingale

V. P. Belavkin, Un Cig Ji

Published Jan 1, 2007• 0 citations
Publisher

Abstract

In this paper, we study a representation of the quantum Itô algebra in Fock space and then by using a noncommutative Radon-Nikodym type theorem we study the density operators of output states as quantum martingales, where the output states are absolutely
DOI
10.4064/bc78-0-3

Related papers

On Quantum Ito Algebras and their decompositions

V. P. Belavkin

A simple axiomatic characterization of the noncommutative Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every quotient Ito algebra has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise of a finite number of degrees of freedom is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin theorem in the classical case. Two basic examples of non-commutative Ito finite group algebras are considered.

arXiv:math-ph/0512071•Dec 21, 2005•2 citations

Chaotic states and stochastic integration in quantum systems

V. P. Belavkin

Quantum chaotic states over a noncommutative monoid, a unitalization of a noncommutative Ito algebra parametrizing a quantum stochastic Levy process, are described in terms of their infinitely divisible generating functionals over the monoid-valued processes on an atomless `space-time' set. A canonical decomposition of the logarithmic conditionally posive-definite generating functional is constructed in a pseudo-Euclidean space, given by a quadruple defining the monoid triangular operator representation and a cyclic zero pseudo-norm state in this space. It is shown that the exponential representation in the corresponding pseudo-Fock space yields the infinitely-divisible generating functional with respect to the exponential state vector, and its compression to the Fock space defines the cyclic infinitly-divisible representation associated with the Fock vacuum state. The structure of states on an arbitrary Ito algebra is studied with two canonical examples of quantum Wiener and Poisson states. A generalized quantum stochastic nonadapted multiple integral is explicitly defined in Fock scale, its continuity and quantum stochastic differentiability is proved. A unified non-adapted and functional quantum Ito formula is discovered and established both in weak and strong sense, and the multiplication formula on the exponential Ito algebra is found for the relatively bounded kernel-operators in Fock scale. The unitarity and projectivity properties of nonadapted quantum stochastic linear differential equations are studied, and their solution is constructed for the locally bounded nonadapted generators in terms of the chronological products in the underlying kernel algebra canonically represented by triangular operators in the pseudo-Fock space.

arXiv:math/0512265•Russian Mathematical Surveys•Feb 28, 1992•37 citations

Quantum Ito B*-algebras, their Classification and Decomposition

V. P. Belavkin

A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin Theorem in the classical case .

arXiv:math/0512508•Dec 21, 2005•0 citations

Infinite Dimensional Ito Algebras of Quantum White Noise

V. P. Belavkin

A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin theorem in the classical case.

arXiv:math/0512288•Dec 13, 2005•4 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

The Ito-Clifford Wong-Zakai Integrals and Martingale Representation

Joseph Spring

Classical stochastic integration is based upon a probability space involving a filtration of sigma‐algebras. This construction lends itself to non‐commutative quantum analogues based for example, on a Hilbert space, a filtration of von Neumann algebras and gage. We recall a non‐commutative construction for the two parameter case, these being integrals in the plane, resulting in type one and type two stochastic integrals which are orthogonal, centred L2 — martingales, obeying isometry properties and develop the construction to obtain an Ito‐Clifford Wong‐Zakai martingale representation.

AIP conference proceedings•Jan 1, 2007•1 citation
A simple axiomatic characterization of the noncommutative Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every quotient Ito algebra has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise of a finite number of degrees of freedom is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin theorem in the classical case. Two basic examples of non-commutative Ito finite group algebras are considered.
Quantum chaotic states over a noncommutative monoid, a unitalization of a noncommutative Ito algebra parametrizing a quantum stochastic Levy process, are described in terms of their infinitely divisible generating functionals over the monoid-valued processes on an atomless `space-time' set. A canonical decomposition of the logarithmic conditionally posive-definite generating functional is constructed in a pseudo-Euclidean space, given by a quadruple defining the monoid triangular operator representation and a cyclic zero pseudo-norm state in this space. It is shown that the exponential representation in the corresponding pseudo-Fock space yields the infinitely-divisible generating functional with respect to the exponential state vector, and its compression to the Fock space defines the cyclic infinitly-divisible representation associated with the Fock vacuum state. The structure of states on an arbitrary Ito algebra is studied with two canonical examples of quantum Wiener and Poisson states. A generalized quantum stochastic nonadapted multiple integral is explicitly defined in Fock scale, its continuity and quantum stochastic differentiability is proved. A unified non-adapted and functional quantum Ito formula is discovered and established both in weak and strong sense, and the multiplication formula on the exponential Ito algebra is found for the relatively bounded kernel-operators in Fock scale. The unitarity and projectivity properties of nonadapted quantum stochastic linear differential equations are studied, and their solution is constructed for the locally bounded nonadapted generators in terms of the chronological products in the underlying kernel algebra canonically represented by triangular operators in the pseudo-Fock space.
A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin Theorem in the classical case .
A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin theorem in the classical case.
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.
Classical stochastic integration is based upon a probability space involving a filtration of sigma‐algebras. This construction lends itself to non‐commutative quantum analogues based for example, on a Hilbert space, a filtration of von Neumann algebras and gage. We recall a non‐commutative construction for the two parameter case, these being integrals in the plane, resulting in type one and type two stochastic integrals which are orthogonal, centred L2 — martingales, obeying isometry properties and develop the construction to obtain an Ito‐Clifford Wong‐Zakai martingale representation.