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Journal of Mathematical Physics

Quantum stochastic calculus associated with quadratic quantum noises

Un Cig Ji, Kalyan B. Sinha

Published Jan 25, 2016• 4 citations
Publisher

Abstract

We first study a class of fundamental quantum stochastic processes induced by the generators of a six dimensional non-solvable Lie †-algebra consisting of all linear combinations of the generalized Gross Laplacian and its adjoint, annihilation operator, creation operator, conservation, and time, and then we study the quantum stochastic integrals associated with the class of fundamental quantum stochastic processes, and the quantum Itô formula is revisited. The existence and uniqueness of solution of a quantum stochastic differential equation is proved. The unitarity conditions of solutions of quantum stochastic differential equations associated with the fundamental processes are examined. The quantum stochastic calculus extends the Hudson-Parthasarathy quantum stochastic calculus.

DOI
10.1063/1.4939919

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We first study a class of fundamental quantum stochastic processes induced by the generators of a six dimensional non-solvable Lie †-algebra consisting of all linear combinations of the generalized Gross Laplacian and its adjoint, annihilation operator, creation operator, conservation, and time, and then we study the quantum stochastic integrals associated with the class of fundamental quantum stochastic processes, and the quantum Itô formula is revisited. The existence and uniqueness of solution of a quantum stochastic differential equation is proved. The unitarity conditions of solutions of quantum stochastic differential equations associated with the fundamental processes are examined. The quantum stochastic calculus extends the Hudson-Parthasarathy quantum stochastic calculus.
In this paper we investigate a quantum stochastic calculus build of creation, annihilation and number of particles operators which fulfill some deformed commutation relations. Namely, we introduce a deformation of a number of particles operator which has simple commutation relations with well known q-deformed creation and annihilation operators. Since all operators considered in this theory are bounded we do not deal with some difficulties of a non deformed theory of Hudson and Parthasarathy. We define stochastic integrals and estimate them in the operator norm. We prove Ito's formula as well.
We establish the existence of viable solutions of quantum stochastic differential equations (QSDE) within the framework of Hudson and Parthasarathy formulation of quantum stochastic calculus.The main results are established for QSDE whose coefficients are Lipschitzian and quasicompact.This work extends the Nagumo viability results for classical differential equations on finite dimensional Euclidean spaces.Viable solutions of QSDE in the present formulation take values in some closed subsets of the infinite dimensional locally convex spaces of non-commutative stochastic processes.
Kurzweil or generalized differential equations associated with Lipschitzian quantum stochastic differential equations (QSDEs) are introduced and studied. This is accomplished within the framework of the Hudson-Parthasarathy formulations of quantum stochastic calculus. Results concerning the equivalence of these classes of equations satisfying the Caratheodory conditions are presented. It is further shown that the associated Kurzweil equation may be used to obtain a reasonably high accurate approximate solutions of the QSDEs. This generalize analogous results for classical initial value problems to the noncommutative quantum setting involving unbounded linear operators on a Hilbert space. Numerical examples are given.