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

Stochastics

Solution of the Hudson–Parthasarathy equation by ordered products

Wilhelm von Waldenfels

Published Jul 12, 2016• 0 citations
Publisher

Abstract

The solution of the quantum stochastic differential equation of Hudson and Partasarathy with constant coefficients can easily be formulated with the help of the normal ordered product. Introducing time ordering we obtain a new representation of the solution. This new representation gives an easy access to the Evans-Hudson flow and the unitary condition of Hudson and Parthasarathy. We use the measure theoretical white noise calculus.

DOI
10.1080/17442508.2016.1204300

Related papers

Quantum Stochastic Differential Equations

Mou-Hsiung Chang

Quite a few researchers have contributed to the development of quantum stochastic differential equations driven by quantum noises, initiated by Hudson and Parthasarathy [HP84a]. A nonexhaustive list of contributors includes Fagnola and Wills [FW00], [FW03], Obata [Oba97], and Linsay and Skalski [LS10]. Based on the results presented in Chapter 3, this chapter derives and considers a general form of linear (left) and (right) Hudson-Parthasarathy quantum stochastic differential equations driven by quantum noises in symmetric Fock space with operator-valued matrices as coefficients. Specifically, it studies the existence and uniqueness of the strong and mild solution processes for both the left and right quantum stochastic differential equations and conditions under which the solution processes are unitary, isometry, co-isometry, and regular. These results make extensive use of the properties of stochastic integral driven by quantum noise and its quantum stochastic calculus in the context of a symmetric Fock space, which was explored in Chapter 3. Various discrete approximation schemes, including Bouten et al. [BvH07], of the left Hudson-Parthasarathy QSDE are also explored for numerical computation. Specifically, it is shown that the solution of the Hudson- Parthasarathy QSDE can be approximated by a sequence of discrete interaction models with decreasing time step. In order to study this problem, discrete interaction models are embedded in a limiting space. This allows one to present strong convergence of the embedded discrete cocycles to the solution of the Hudson-Parthasarathy QSDE. It will be pointed out that the way in which the embedding is done does not affect the proof of the main result for discrete approximations. Examples of a QSDE that is analytically solvable and a QSDE that possesses a mild solution but not a strong solution are also given. QSDE of Hudson-Parthasarathy Type It has been mentioned earlier in Chapter 3 that an open quantum systems treated in this monograph are to be formulated as a quantum system in which the underlined (or the initial) quantum system is perturbed by quantum noises that represent the environment.

Cambridge University Press eBooks•Feb 9, 2015•0 citations

Mild Solutions of Quantum Stochastic Differential Equations

Franco Fagnola, Stephen J. Wills

We introduce the concept of a mild solution for the right Hudson-Parthasarathy quantum stochastic differential equation, prove existence and uniqueness results, and show the correspondence between our definition and similar ideas in the theory of classical stochastic differential equations. The conditions that a process must satisfy in order for it to be a mild solution are shown to be strictly weaker than those for it to be a strong solution by exhibiting a class of coefficient matrices for which a mild unitary solution can be found, but for which no strong solution exists.

Electronic Communications in Probability•Jan 1, 2000•6 citations

Existence and Uniqueness of Solution for the Kurzweil Equation Associated with Quantum Stochastic Differential Equation

Sheila Amina Bishop

We study the existence and uniqueness of solution of the Kurzweil equations ~ with the Non-Lipschitz quantum stochastic differential equation. This is accomplished within the framework of the Hudson and Parthasarathy formulations of quantum stochastic calculus and the Schwabic generalized ordinary differential equations.

•Jan 1, 2011•0 citations

SYMMETRIC DIFFERENTIATION AND HAMILTONIAN OF A QUANTUM STOCHASTIC PROCESS

Wilhelm von Waldenfels

The main object of this paper is to establish the Hamiltonian of the Hudson–Parthasarathy quantum stochastic differential equation [Formula: see text] As its solution forms a cocycle with respect to the time shift, its product with the time shift establishes a strongly continuous one-parameter unitary group W(t)= exp (-itH) and H is called its Hamiltonian. Our main result is that H is the closure of the restriction of the singular operator [Formula: see text] on an appropriate domain that shall be explicitly described. The symmetric differentiation [Formula: see text] is a generalization of the generator of the time shift, [Formula: see text] and [Formula: see text] are annihilation and creation operators and [Formula: see text] is the symmetric Dirac δ-function. In one dimension the symmetric differentiation might be used to establish a quantum stochastic calculus without Ito term.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Mar 1, 2005•10 citations

On the fermion quantum white noise calculus

Yu‐Wei Liao, Ke Liu

The fermion interpretation of the white noise framework is considered. As an application, Ito's product formula of Applebaum, Hudson and Parthasarathy (1987) is extended to the generalized case.

Journal of Physics A Mathematical and General•Nov 21, 1993•2 citations

Some quantum dynamical semi-groups with quantum stochastic dilation

Lingaraj Sahu, Preetinder Singh

We consider the GNS Hilbert space H of a uniformly hyper-finite C * -algebra and study a class of unbounded Lindbladian arises from commutators.Exploring the local structure of UHF algebra, we have shown that the associated Hudson-Parthasarathy type quantum stochastic differential equation admits a unitary solution.The vacuum expectation of homomorphic cocycle, implemented by the Hudson-Parthasarathy flow, is conservative and gives the minimal semi-group associated with the formal Lindbladian.We also associate conservative minimal semi-groups to another class of Lindbladian by solving the corresponding Evan-Hudson equation.

arXiv:1505.05296•Communications on Stochastic Analysis•Sep 1, 2015•0 citations
The solution of the quantum stochastic differential equation of Hudson and Partasarathy with constant coefficients can easily be formulated with the help of the normal ordered product. Introducing time ordering we obtain a new representation of the solution. This new representation gives an easy access to the Evans-Hudson flow and the unitary condition of Hudson and Parthasarathy. We use the measure theoretical white noise calculus.
Quite a few researchers have contributed to the development of quantum stochastic differential equations driven by quantum noises, initiated by Hudson and Parthasarathy [HP84a]. A nonexhaustive list of contributors includes Fagnola and Wills [FW00], [FW03], Obata [Oba97], and Linsay and Skalski [LS10]. Based on the results presented in Chapter 3, this chapter derives and considers a general form of linear (left) and (right) Hudson-Parthasarathy quantum stochastic differential equations driven by quantum noises in symmetric Fock space with operator-valued matrices as coefficients. Specifically, it studies the existence and uniqueness of the strong and mild solution processes for both the left and right quantum stochastic differential equations and conditions under which the solution processes are unitary, isometry, co-isometry, and regular. These results make extensive use of the properties of stochastic integral driven by quantum noise and its quantum stochastic calculus in the context of a symmetric Fock space, which was explored in Chapter 3. Various discrete approximation schemes, including Bouten et al. [BvH07], of the left Hudson-Parthasarathy QSDE are also explored for numerical computation. Specifically, it is shown that the solution of the Hudson- Parthasarathy QSDE can be approximated by a sequence of discrete interaction models with decreasing time step. In order to study this problem, discrete interaction models are embedded in a limiting space. This allows one to present strong convergence of the embedded discrete cocycles to the solution of the Hudson-Parthasarathy QSDE. It will be pointed out that the way in which the embedding is done does not affect the proof of the main result for discrete approximations. Examples of a QSDE that is analytically solvable and a QSDE that possesses a mild solution but not a strong solution are also given. QSDE of Hudson-Parthasarathy Type It has been mentioned earlier in Chapter 3 that an open quantum systems treated in this monograph are to be formulated as a quantum system in which the underlined (or the initial) quantum system is perturbed by quantum noises that represent the environment.
We introduce the concept of a mild solution for the right Hudson-Parthasarathy quantum stochastic differential equation, prove existence and uniqueness results, and show the correspondence between our definition and similar ideas in the theory of classical stochastic differential equations. The conditions that a process must satisfy in order for it to be a mild solution are shown to be strictly weaker than those for it to be a strong solution by exhibiting a class of coefficient matrices for which a mild unitary solution can be found, but for which no strong solution exists.
The main object of this paper is to establish the Hamiltonian of the Hudson–Parthasarathy quantum stochastic differential equation [Formula: see text] As its solution forms a cocycle with respect to the time shift, its product with the time shift establishes a strongly continuous one-parameter unitary group W(t)= exp (-itH) and H is called its Hamiltonian. Our main result is that H is the closure of the restriction of the singular operator [Formula: see text] on an appropriate domain that shall be explicitly described. The symmetric differentiation [Formula: see text] is a generalization of the generator of the time shift, [Formula: see text] and [Formula: see text] are annihilation and creation operators and [Formula: see text] is the symmetric Dirac δ-function. In one dimension the symmetric differentiation might be used to establish a quantum stochastic calculus without Ito term.
We consider the GNS Hilbert space H of a uniformly hyper-finite C * -algebra and study a class of unbounded Lindbladian arises from commutators.Exploring the local structure of UHF algebra, we have shown that the associated Hudson-Parthasarathy type quantum stochastic differential equation admits a unitary solution.The vacuum expectation of homomorphic cocycle, implemented by the Hudson-Parthasarathy flow, is conservative and gives the minimal semi-group associated with the formal Lindbladian.We also associate conservative minimal semi-groups to another class of Lindbladian by solving the corresponding Evan-Hudson equation.