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Mathematics of Computation

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Numerical simulation of stochastic evolution equations associated to quantum Markov semigroups

Author: Carlos M. Mora
Journal: Math. Comp. 73 (2004), 1393-1415
MSC (2000): Primary 60H35; Secondary 60H10, 60H15, 60H30, 65C30, 65C05
Published electronically: August 4, 2003
MathSciNet review: 2047093
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Abstract: We address the problem of approximating numerically the solutions $\left( X_{t}:t\in \left[ 0,T\right] \right) $ of stochastic evolution equations on Hilbert spaces $\left( \mathfrak{h},\left\langle \cdot ,\cdot \right\rangle \right) $, with respect to Brownian motions, arising in the unraveling of backward quantum master equations. In particular, we study the computation of mean values of $\left\langle X_{t},AX_{t}\right\rangle $, where $A$ is a linear operator. First, we introduce estimates on the behavior of $X_{t}$. Then we characterize the error induced by the substitution of $X_{t}$ with the solution $X_{t,n}$ of a convenient stochastic ordinary differential equation. It allows us to establish the rate of convergence of $\mathbf{E} \left\langle \tilde{X}_{t,n},A\tilde{X}_{t,n}\right\rangle $ to $\mathbf{E} \left\langle X_{t},AX_{t}\right\rangle $, where $\tilde{X}_{t,n}$ denotes the explicit Euler method. Finally, we consider an extrapolation method based on the Euler scheme. An application to the quantum harmonic oscillator system is included.

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Additional Information

Carlos M. Mora
Affiliation: Departamento de Ingeniería Matemática, Universidad de Concepción, Concepción, Chile
Address at time of publication: Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160 C, Concepción, Chile

Keywords: Stochastic evolution equation, numerical solution, rate of convergence, Euler scheme, Galerkin method, quantum dynamical semigroup, quantum master equation.
Received by editor(s): August 25, 2002
Received by editor(s) in revised form: January 7, 2003
Published electronically: August 4, 2003
Additional Notes: This research has been partially supported by FONDECYT grant 2000036, a DIPUC Ph.D. grant and the program “Cátedra Presidencial on Qualitative Analysis of Quantum Dynamical Systems”.
Article copyright: © Copyright 2003 American Mathematical Society