Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. Radu Balescu, Equilibrium and nonequilibrium statistical mechanics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1975. MR 0408641
  • 2. A. Barchielli, A. M. Paganoni, and F. Zucca, On stochastic differential equations and semigroups of probability operators in quantum probability, Stochastic Process. Appl. 73 (1998), no. 1, 69–86. MR 1603834,
  • 3. H. Carmichael, An open systems approach to quantum optics. Lecture Notes in Physics. Springer-Verlag (1993).
  • 4. C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum mechanics. Volume 1. Hermann (1977).
  • 5. A. M. Chebotarev and F. Fagnola, Sufficient conditions for conservativity of minimal quantum dynamical semigroups, J. Funct. Anal. 153 (1998), no. 2, 382–404. MR 1614586,
  • 6. Franco Fagnola, Rolando Rebolledo, and Carlos Saavedra, Quantum flows associated to master equations in quantum optics, J. Math. Phys. 35 (1994), no. 1, 1–12. MR 1252096,
  • 7. Franco Fagnola, Quantum Markov semigroups and quantum flows, Proyecciones 18 (1999), no. 3, 144. MR 1814506
  • 8. F. Fagnola, S. J. Wills, Solving quantum stochastic differential equations with unbounded coefficients. J. Funct. Anal. 198 (2003), 279-310.
  • 9. Nicolas Gisin and Ian C. Percival, The quantum-state diffusion model applied to open systems, J. Phys. A 25 (1992), no. 21, 5677–5691. MR 1192024
  • 10. A. S. Holevo, On dissipative stochastic equations in a Hilbert space, Probab. Theory Related Fields 104 (1996), no. 4, 483–500. MR 1384042,
  • 11. A. S. Holevo, On dissipative stochastic equations in a Hilbert space, Probab. Theory Related Fields 104 (1996), no. 4, 483–500. MR 1384042,
  • 12. Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
  • 13. Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992. MR 1214374
  • 14. Tarso B.L. Kist, M. Orszag, T.A. Brun, L. Davidovich, Stochastic Schrödinger equations in cavity QDE: physical interpretation and localization. J. Opt. B: Quantum Semiclass. Opt. 1 (1999), 251-263.
  • 15. Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 5, Springer-Verlag, Berlin, 1992. Evolution problems. I; With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon; Translated from the French by Alan Craig. MR 1156075
  • 16. G.N. Milstein, The numerical integration of stochastic differentials equations. Ural University Press (1988).
  • 17. C. Mora, Solución numérica de ecuaciones diferenciales estocásticas mediante métodos exponenciales. Ph.D. Thesis, Pontificia Universidad Católica de Chile (2002).
  • 18. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
  • 19. Ian Percival, Quantum state diffusion, Cambridge University Press, Cambridge, 1998. MR 1666826
  • 20. Philip Protter, Stochastic integration and differential equations, Applications of Mathematics (New York), vol. 21, Springer-Verlag, Berlin, 1990. A new approach. MR 1037262
  • 21. M.O. Scully, M.S. Zubairy, Quantum optics. Cambridge University Press (1997).
  • 22. Walter T. Strunz, Lajos Diósi, Nicolas Gisin, and Ting Yu, Quantum trajectories for Brownian motion, Phys. Rev. Lett. 83 (1999), no. 24, 4909–4913. MR 1727574,
  • 23. Denis Talay, Efficient numerical schemes for the approximation of expectations of functionals of the solution of a SDE and applications, Filtering and control of random processes (Paris, 1983) Lect. Notes Control Inf. Sci., vol. 61, Springer, Berlin, 1984, pp. 294–313. MR 874837,
  • 24. Denis Talay, Discrétisation d’une équation différentielle stochastique et calcul approché d’espérances de fonctionnelles de la solution, RAIRO Modél. Math. Anal. Numér. 20 (1986), no. 1, 141–179 (French, with English summary). MR 844521
  • 25. Denis Talay and Luciano Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Anal. Appl. 8 (1990), no. 4, 483–509 (1991). MR 1091544,
  • 26. D. Talay, Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme, Markov Process. Related Fields 8 (2002), no. 2, 163–198. Inhomogeneous random systems (Cergy-Pontoise, 2001). MR 1924934

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 60H35, 60H10, 60H15, 60H30, 65C30, 65C05

Retrieve articles in all journals with MSC (2000): 60H35, 60H10, 60H15, 60H30, 65C30, 65C05

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