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

Birkhäuser Basel eBooks

Stochastic Integration and Quantum Ito’s Formula

Kalyanapuram Rangachari Parthasarathy

Published Jan 1, 1992• 0 citations
Publisher

Abstract

In Section 21 we have already seen how the classical stochastic processes with independent increments can be realised as suitable linear combinations of the creation, conservation and annihilation operators in the boson Fock space Γ s (ℋ) over a Hilbert space ℋ. This includes, in particular, the Brownian motion and Poisson process. Since a well-developed theory of stochastic integration with respect to these classical processes exists, it is natural to wonder whether it would be possible to introduce quantum probabilistic analogues of stochastic integrals with respect to processes defined by the creation, conservation and annihilation operators. To this end one needs the notions of filtration and adapted processes. This, in turn, depends on the notion of time. We adopt the view that time is an ℝ+-valued observable in a Hilbert space ℋ which, as a spectral measure, has no jumps. The absence of jumps is assumed more as a matter of mathematical convenience than a philosophical or conceptual necessity. We shall now examine how such a notion of time leads to a filtration and the definition of adapted processes. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-0348-8641-3_3

Related papers

Quantum Stochastic Calculus

Kalyanapuram Rangachari Parthasarathy

The basic integrator processes of quantum stochastic calculus, namely, creation, conservation, and annihilation, are introduced in the Hilbert space of square integrable Brownian functionals. Stochastic integrals with respect to these processes and a quantum Itô’s formula are described. As an application two examples of quantum stochastic differential equations are discussed. A continuous time version of Stinespring’s theorem on completely positive maps in C *-algebras is exploited to formulate the notion of a quantum Markov process and indicate how classical Markov processes are woven into the fabric of the Schodinger-Heisenberg dynamics of quantum theory.

Birkhäuser Basel eBooks•Jan 1, 1995•6 citations

Quantum Stochastic Calculus

Mou-Hsiung Chang

This chapter begins with introductions of symmetric Fock space Γ(ℍ) and symmetric Guichardet space Φ(ℍ) of a generic complex Hilbert space ℍ. It is shown that these 2 spaces are actually isomorphic and can and will be used interchangeably throughout the chapter. In particular, the simple and yet useful integral-summation formula (see Theorem (3.2.1)) for the Guichardet space is often employed to establish the essential results in quantum stochastic calculus. It has been shown that the symmetric Fock space (and hence the symmetric Guichadet space) provides a plausible mathematical tool for modeling phenomena in quantum optics or quantum electrodynamics. Many results in the quantum physics and quantum probability literature are actually established based on this concrete model space. When the generic complex Hilbert space ℍ = L 2 (ℝ + ; K) (where K is another complex Hilbert space), the class of exponential vectors along with 3 different types of quantum noise processes, namely, the creation, annihilation, and neutral (or guage) processes, can be introduced. It is shown that the subspace generated by the class of exponential vectors is dense in the Fock space. Therefore, it is convenient to verify the properties that hold for Fock space by verifying the same hold for the class of exponential vectors. Parallel to those of Itô integrals with respect to classical Brownian motion and/or Poisson process, the concepts of a quantum stochastic integral of a operator-valued process (as a member of the Fock space) with respect to each of the above mentioned quantum noise processes are constructed. The quantum stochastic calculus, parallel to those of classical Itô calculus, is developed. The theory of quantum stochastic calculus on the symmetric Fock space due originally to Hudson and Parthasarathy [HP84a] has been the topic of investigation by many researchers in the last 20 years. Much of he material presented in this chapter can be found in the books by Meyer [Mey95], Parthasarathy [Par92] and Attal [Att03].

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

Quantum stochastic integrals as belated integrals

Chris Barnett, J. Martin Lindsay, Ivan F. Wilde

Quantum stochastic integrals have been constructed in various contexts [2, 3, 4, 5, 9] by adapting the construction of the classical L2-Itô-integral with respect to Brownian motion. Thus, the integral is first defined for simple integrands as a finite sum, then one establishes certain isometry relations or suitable bounds to allow the extension, by continuity, to more general integrands. The integrator is typically operator-valued, the integrand is vector-valued or operator-valued and the quantum stochastic integral is then given as a vector in a Hilbert space, or as an operator on the Hilbert space determined by its action on suitable vectors.

Glasgow Mathematical Journal•May 1, 1992•1 citation

Quantum stochastic calculus with maximal operator domains

Stéphane Attal, J. Martin Lindsay

Quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains. Operator adaptedness, conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of Fock space, together with sections of the adapted gradient operator. Free from exponential vector domains, our stochastic integrals may be satisfactorily composed yielding quantum Itô formulas for operator products as sums of stochastic integrals. The calculus has seen two reformulations since its discovery---one closely related to classical Itô calculus; the other to noncausal stochastic analysis and Malliavin calculus. Our theory extends both of these approaches and may be viewed as a synthesis of the two. The main application given here is existence and uniqueness for the Attal--Meyer equations for implicit definition of quantum stochastic integrals.

The Annals of Probability•Jan 1, 2004•24 citations

A matrix formulation of quantum stochastic calculus

Alexander C. R. Belton

We develop the theory of chaos spaces and chaos matrices. A chaos space is a Hilbert space with a fixed, countably-infinite, direct-sum decomposition. A chaos matrix between two chaos spaces is a doubly-infinite matrix of bounded operators which respects this decomposition. We study operators represented by such matrices, particularly with respect to self-adjointness. This theory is used to re-formulate the quantum stochastic calculus of Hudson and Parthasarathy. Integrals of chaos-matrix processes are defined using the Hitsuda-Skorokhod integral and Malliavin gradient,following Lindsay and Belavkin. A new way of defining adaptedness is developed and the consequent quantum product Ito formula is used to provide a genuine functional Ito formula for polynomials in a large class of unbounded processes, which include the Poisson process and Brownian motion. A new type of adaptedness, known as $\Omega$-adaptedness, is defined. We show that quantum stochastic integrals of $\Omega$-adapted processes are well-behaved; for instance, bounded processes have bounded integrals. We solve the appropriate modification of the evolution equation of Hudson and Parthasarathy: $U(t)=I+\int_{0}^{t}E(s)\mathrm{d}\Lambda(s)+F(s)\mathrm{d} A(s)+ G(s)U(s)\mathrm{d} A^{\dagger}(s)+H(s)U(s)\mathrm{d} s, $ where the coefficients are time-dependent, bounded, $\Omega$-adapted processes acting on the whole Fock space. We show that the usual conditions on the coefficients, viz. $(E,F,G,H)=(W-I,L,-WL^{*},iK+\mbox{$\frac{1}{2}$}LL^{*})$ where $W$ is unitary and $K$ self-adjoint, are necessary and sufficient conditions for the solution to be unitary. This is a very striking result when compared to the adapted case.

•Jan 1, 1998•7 citations

Quantum stochastic calculus on full Fock space

Michael Skeide

We present a new version of integration of time-adapted processes with respect to creation, annihilation and conservation processes on the full Fock space. Among the new features, in the first place, there is a new formulation of adaptedness which is both

Banach Center Publications•Jan 1, 1998•3 citations
In Section 21 we have already seen how the classical stochastic processes with independent increments can be realised as suitable linear combinations of the creation, conservation and annihilation operators in the boson Fock space Γ s (ℋ) over a Hilbert space ℋ. This includes, in particular, the Brownian motion and Poisson process. Since a well-developed theory of stochastic integration with respect to these classical processes exists, it is natural to wonder whether it would be possible to introduce quantum probabilistic analogues of stochastic integrals with respect to processes defined by the creation, conservation and annihilation operators. To this end one needs the notions of filtration and adapted processes. This, in turn, depends on the notion of time. We adopt the view that time is an ℝ+-valued observable in a Hilbert space ℋ which, as a spectral measure, has no jumps. The absence of jumps is assumed more as a matter of mathematical convenience than a philosophical or conceptual necessity. We shall now examine how such a notion of time leads to a filtration and the definition of adapted processes. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The basic integrator processes of quantum stochastic calculus, namely, creation, conservation, and annihilation, are introduced in the Hilbert space of square integrable Brownian functionals. Stochastic integrals with respect to these processes and a quantum Itô’s formula are described. As an application two examples of quantum stochastic differential equations are discussed. A continuous time version of Stinespring’s theorem on completely positive maps in C *-algebras is exploited to formulate the notion of a quantum Markov process and indicate how classical Markov processes are woven into the fabric of the Schodinger-Heisenberg dynamics of quantum theory.
This chapter begins with introductions of symmetric Fock space Γ(ℍ) and symmetric Guichardet space Φ(ℍ) of a generic complex Hilbert space ℍ. It is shown that these 2 spaces are actually isomorphic and can and will be used interchangeably throughout the chapter. In particular, the simple and yet useful integral-summation formula (see Theorem (3.2.1)) for the Guichardet space is often employed to establish the essential results in quantum stochastic calculus. It has been shown that the symmetric Fock space (and hence the symmetric Guichadet space) provides a plausible mathematical tool for modeling phenomena in quantum optics or quantum electrodynamics. Many results in the quantum physics and quantum probability literature are actually established based on this concrete model space. When the generic complex Hilbert space ℍ = L 2 (ℝ + ; K) (where K is another complex Hilbert space), the class of exponential vectors along with 3 different types of quantum noise processes, namely, the creation, annihilation, and neutral (or guage) processes, can be introduced. It is shown that the subspace generated by the class of exponential vectors is dense in the Fock space. Therefore, it is convenient to verify the properties that hold for Fock space by verifying the same hold for the class of exponential vectors. Parallel to those of Itô integrals with respect to classical Brownian motion and/or Poisson process, the concepts of a quantum stochastic integral of a operator-valued process (as a member of the Fock space) with respect to each of the above mentioned quantum noise processes are constructed. The quantum stochastic calculus, parallel to those of classical Itô calculus, is developed. The theory of quantum stochastic calculus on the symmetric Fock space due originally to Hudson and Parthasarathy [HP84a] has been the topic of investigation by many researchers in the last 20 years. Much of he material presented in this chapter can be found in the books by Meyer [Mey95], Parthasarathy [Par92] and Attal [Att03].
Quantum stochastic integrals have been constructed in various contexts [2, 3, 4, 5, 9] by adapting the construction of the classical L2-Itô-integral with respect to Brownian motion. Thus, the integral is first defined for simple integrands as a finite sum, then one establishes certain isometry relations or suitable bounds to allow the extension, by continuity, to more general integrands. The integrator is typically operator-valued, the integrand is vector-valued or operator-valued and the quantum stochastic integral is then given as a vector in a Hilbert space, or as an operator on the Hilbert space determined by its action on suitable vectors.
Quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains. Operator adaptedness, conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of Fock space, together with sections of the adapted gradient operator. Free from exponential vector domains, our stochastic integrals may be satisfactorily composed yielding quantum Itô formulas for operator products as sums of stochastic integrals. The calculus has seen two reformulations since its discovery---one closely related to classical Itô calculus; the other to noncausal stochastic analysis and Malliavin calculus. Our theory extends both of these approaches and may be viewed as a synthesis of the two. The main application given here is existence and uniqueness for the Attal--Meyer equations for implicit definition of quantum stochastic integrals.
We develop the theory of chaos spaces and chaos matrices. A chaos space is a Hilbert space with a fixed, countably-infinite, direct-sum decomposition. A chaos matrix between two chaos spaces is a doubly-infinite matrix of bounded operators which respects this decomposition. We study operators represented by such matrices, particularly with respect to self-adjointness. This theory is used to re-formulate the quantum stochastic calculus of Hudson and Parthasarathy. Integrals of chaos-matrix processes are defined using the Hitsuda-Skorokhod integral and Malliavin gradient,following Lindsay and Belavkin. A new way of defining adaptedness is developed and the consequent quantum product Ito formula is used to provide a genuine functional Ito formula for polynomials in a large class of unbounded processes, which include the Poisson process and Brownian motion. A new type of adaptedness, known as Ω\OmegaΩ-adaptedness, is defined. We show that quantum stochastic integrals of Ω\OmegaΩ-adapted processes are well-behaved; for instance, bounded processes have bounded integrals. We solve the appropriate modification of the evolution equation of Hudson and Parthasarathy: $U(t)=I+\int_{0}^{t}E(s)\mathrm{d}\Lambda(s)+F(s)\mathrm{d} A(s)+ G(s)U(s)\mathrm{d} A^{\dagger}(s)+H(s)U(s)\mathrm{d} s, $ where the coefficients are time-dependent, bounded, Ω\OmegaΩ-adapted processes acting on the whole Fock space. We show that the usual conditions on the coefficients, viz. $(E,F,G,H)=(W-I,L,-WL^{*},iK+\mbox{$\frac{1}{2}$}LL^{*})$ where WWW is unitary and KKK self-adjoint, are necessary and sufficient conditions for the solution to be unitary. This is a very striking result when compared to the adapted case.