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

A Trotter Product Formula for quantum stochastic flows

B. Krishna Das, Debashish Goswami, Kalyan B. Sinha

Published Jan 1, 2010• 2 citations
OpenAlex

Abstract

We give a new method for proving the homomorphic property of a quantum stochastic ow satisfying a quantum stochastic differential equation with unbounded coefficients, under some further hypotheses. As an application, we prove a Trotter product formula for quantum stochastic ows and obtain quantum stochastic dilations of a class of quantum dynamical semigroups generalizing results of [5]

Related papers

A quantum stochastic Lie-Trotter product formula

J. Martin Lindsay, Kalyan B. Sinha

A Trotter product formula is established for unitary quantum stochastic processes governed by quantum stochastic differential equations with constant bounded coefficients.

arXiv:1011.4637•Nov 21, 2010•0 citations

A Homomorphism Theorem and a Trotter Product Formula for Quantum Stochastic Flows with Unbounded Coefficients

Biswarup Das, Debashish Goswami, Kalyan B. Sinha

arXiv:1001.0233•Communications in Mathematical Physics•Mar 18, 2014•5 citations

Quantum Stochastic Lie–Trotter Product Formula II

J. Martin Lindsay

A natural counterpart to the Lie–Trotter product formula for norm-continuous, one-parameter semigroups is proved, for the class of quasicontractive quantum stochastic (QS) operator cocycles whose expectation semigroup is norm continuous. Compared to previous such results, the assumption of a strong form of independence of the constituent cocycles is overcome. The analysis is facilitated by the development of some quantum Itô algebra. It is also shown how the maximal Gaussian component of a QS generator may be extracted—leading to a canonical decomposition of such generators, and the connection to perturbation theory is described. Finally, the quantum Itô algebra is extended to quadratic form generators, and a conjecture is formulated for the extension of the product formula to holomorphic QS cocycles.

arXiv:1707.05669•International Mathematics Research Notices•Jan 16, 2018•2 citations

Quantum Wiener chaos

Mateusz Jurczynski

In this thesis we develop the theory of quantum Wiener integrals on the bosonic Fock space. We study multiple quantum Wiener integrals as an algebra of unbounded operators, investigating its properties, including closedness, common domains and multiplication formulas. We show the applications of the new formalism by providing new proofs to the established theory of quantum stochastic calculus and new conditions for generating quantum stochastic cocycles and quantum stochastic evolutions. The corresponding quasifree case is also studied and the constructions extended to fit in that formalism. We construct the multiple quantum Wiener integral as one operator on a family of operators which we dub operator kernels. This in particular covers the case of quantum stochastic cocycles and evolutions. We show that the family of quantum Wiener integrals forms a WOT-dense algebra of unbounded operators on the bosonic Fock space. We provide more general conditions for an operator kernel to be multiple quantum Wiener integrable, which allows us to treat multiple quantum Wiener integrals as an algebra. We explore the influence of an initial space on the theory. Our setting gives natural conditions for a product of two cocycles (evolutions) to still be a cocycle (an evolution). We apply our theory by solving quantum stochastic differential equations (QSDEs) and by finding more elementary proofs of structure conditions on the generator of a quantum stochastic evolution and of the fundamental estimate in the proof of quantum stochastic Lie–Trotter formula. We also show how our theory unifies and generalises the theory of integral kernels and chaotic representation properties, proving in particular that every Hilbert–Schmidt operator is a quantum Wiener integral.

•Jan 1, 2017•0 citations

QUANTUM STOCHASTIC CALCULUS ON BOOLEAN FOCK SPACE

Anis Ben Ghorbal, Michael Schürmann

In this paper we establish a theory of stochastic integration with respect to the basic field operator processes in the Boolean case. This leads to a Boolean version of quantum Itô's product formula and has applications to the theory of dilations of quantum dynamical semigroups.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Nov 27, 2004•17 citations

A note on a formula of the Lévy-Khinchin type in quantum probability

Alberto Barchielli, Anna Maria Paganoni

In the past few years there has been an increasing interest in a certain class of stochastic differential equations (SDE’s) in Hilbert spaces for applications in quantum mechanics (measurements continuous in time [1-5]) and in quantum optics (photon-detection theory and numerical simulations of master equations [6-10]). Part of the mathematical theory of these equations has been developed in [11], where also “structural properties” of this class of equations have been studied. In the paper [11] it has been shown that such equations are connected with certain semigroups of linear operators and the form of the generator of semigroups related to such SDE’s has been established.

Nagoya Mathematical Journal•Mar 1, 1996•14 citations
We give a new method for proving the homomorphic property of a quantum stochastic ow satisfying a quantum stochastic differential equation with unbounded coefficients, under some further hypotheses. As an application, we prove a Trotter product formula for quantum stochastic ows and obtain quantum stochastic dilations of a class of quantum dynamical semigroups generalizing results of [5]
A natural counterpart to the Lie–Trotter product formula for norm-continuous, one-parameter semigroups is proved, for the class of quasicontractive quantum stochastic (QS) operator cocycles whose expectation semigroup is norm continuous. Compared to previous such results, the assumption of a strong form of independence of the constituent cocycles is overcome. The analysis is facilitated by the development of some quantum Itô algebra. It is also shown how the maximal Gaussian component of a QS generator may be extracted—leading to a canonical decomposition of such generators, and the connection to perturbation theory is described. Finally, the quantum Itô algebra is extended to quadratic form generators, and a conjecture is formulated for the extension of the product formula to holomorphic QS cocycles.
In this thesis we develop the theory of quantum Wiener integrals on the bosonic Fock space. We study multiple quantum Wiener integrals as an algebra of unbounded operators, investigating its properties, including closedness, common domains and multiplication formulas. We show the applications of the new formalism by providing new proofs to the established theory of quantum stochastic calculus and new conditions for generating quantum stochastic cocycles and quantum stochastic evolutions. The corresponding quasifree case is also studied and the constructions extended to fit in that formalism. We construct the multiple quantum Wiener integral as one operator on a family of operators which we dub operator kernels. This in particular covers the case of quantum stochastic cocycles and evolutions. We show that the family of quantum Wiener integrals forms a WOT-dense algebra of unbounded operators on the bosonic Fock space. We provide more general conditions for an operator kernel to be multiple quantum Wiener integrable, which allows us to treat multiple quantum Wiener integrals as an algebra. We explore the influence of an initial space on the theory. Our setting gives natural conditions for a product of two cocycles (evolutions) to still be a cocycle (an evolution). We apply our theory by solving quantum stochastic differential equations (QSDEs) and by finding more elementary proofs of structure conditions on the generator of a quantum stochastic evolution and of the fundamental estimate in the proof of quantum stochastic Lie–Trotter formula. We also show how our theory unifies and generalises the theory of integral kernels and chaotic representation properties, proving in particular that every Hilbert–Schmidt operator is a quantum Wiener integral.
In the past few years there has been an increasing interest in a certain class of stochastic differential equations (SDE’s) in Hilbert spaces for applications in quantum mechanics (measurements continuous in time [1-5]) and in quantum optics (photon-detection theory and numerical simulations of master equations [6-10]). Part of the mathematical theory of these equations has been developed in [11], where also “structural properties” of this class of equations have been studied. In the paper [11] it has been shown that such equations are connected with certain semigroups of linear operators and the form of the generator of semigroups related to such SDE’s has been established.