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Probability Theory and Related Fields

Stochastic integration with respect to q Brownian motion

Catherine Donati-Martin

Published Jan 1, 2003• 29 citations
Publisher
DOI
10.1007/s00440-002-0224-4

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Abstract In this paper, we investigate a linear Ito stochastic differential equation in the vector space from every conceivable perspective. In this case, the potential vector describes the magnitude of the classical noise acting on a quantum system. This vector potential can be expressed as a linear function of its parameters, with Hermitian operators serving as its coefficients since its parameters are assumed to be unknown. To the second order of perturbation, the unitary evolution operator can be determined with the aid of a potential perturbation parameter. As for the second term, it is written as a double-iterated stochastic integral with respect to Brownian motion, while the first term is written as an Ito stochastic integral. When controlling quantum systems, noise from the environment can be a major hindrance; this technique can help. Improve the dependability and practicality of quantum technologies like computers by learning to detect and regulate noise. If the potential’s parameters are affected by noise, then their reliability is thought to decrease. We focus on the special case where the potential is a linear function of these parameters with Hermitian operators as the coefficients. To find the unitary evolution operator up to O(ε), we can write the O(ε) term as an ito stochastic integral with respect to Brownian motion, and the O(ε2) term as a doubly iterated stochastic integral with respect to Brownian motion.
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.