Mathematics of Computation

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ISSN 1088-6842(e) ISSN 0025-5718(p)

Numerical simulation of stochastic evolution equations associated to quantum Markov semigroups

Author(s): Carlos M. Mora.
Journal: Math. Comp. 73 (2004), 1393-1415.
MSC (2000): Primary 60H35; Secondary 60H10, 60H15, 60H30, 65C30, 65C05
Posted: 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 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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