Journal of Mathematical Physics
Feedback | Help | Exit
Search:
   
 
 
 
Previous Article
Nine theorems on the unification of quantum mechanics and relativity
A mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics is proposed. For this a Hilbert space H of functions of four variables x,t furnished with an a...
Next Article
Classical limit for semirelativistic Hartree systems
We consider the three-dimensional semirelativistic Hartree model for fast quantum mechanical particles moving in a self-consistent field. Under appropriate assumptions on the initial density matrix as...

Discrete approximation of quantum stochastic models

J. Math. Phys. 49, 102109 (2008); doi:10.1063/1.3001109

Published 20 October 2008

You are logged in to this journal.

Luc Bouten1 and Ramon Van Handel2
1Physical Measurement and Control 266-33, California Institute of Technology, Pasadena, California 91125, USA
2Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey 08544, USA
We develop a general technique for proving convergence of repeated quantum interactions to the solution of a quantum stochastic differential equation. The wide applicability of the method is illustrated in a variety of examples. Our main theorem, which is based on the Trotter–Kato theorem, is not restricted to a specific noise model and does not require boundedness of the limit coefficients. ©2008 American Institute of Physics
History: Received 16 July 2008; accepted 26 September 2008; published 20 October 2008
Permalink: http://link.aip.org/link/?JMAPAQ/49/102109/1
FULL TEXT OPTIONS   (FREE)
Download PDF (208 kB) View Cart

KEYWORDS and PACS

Keywords
PACS
  • 03.65.Ta
    Foundations of quantum mechanics; measurement theory
  • 02.50.Ey
    Stochastic processes
  • 02.30.-f
    Function theory, analysis
  • YEAR: 2008

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
AIP is a member of CrossRef AIP

REFERENCES (30)
View in Separate Window/Tab
  1. Accardi, L., Frigerio, A., and Lewis, J. T., “Quantum stochastic processes,” Publ. Res. Inst. Math. Sci. 18, 97 (1982).
  2. Accardi, L., Frigerio, A., and Lu, Y. G., “The weak coupling limit as a quantum functional central limit,” Commun. Math. Phys. 131, 537 (1990). [Inspec] [ISI]
  3. Attal, S. and Pautrat, Y., “From repeated to continuous quantum interactions,” Ann. Henri Poincare 7, 59 (2006).
  4. Barchielli, A., in Open Quantum Systems III: Recent Developments, edited by S. Attal, A. Joye, and C. -A. Pillet (Springer, New York, 2006), pp. 207–292.
  5. Bouten, L. and Silberfarb, A., “Adiabatic elimination in quantum stochastic models,” Commun. Math. Phys. 283, 491 (2008).
  6. Bouten, L., van Handel, R., and James, M. R., “A discrete invitation to quantum filtering and feedback control,” SIAM Rev. (in press).
  7. Bouten, L., van Handel, R., and Silberfarb, A., “Approximation and limit theorems for quantum stochastic models with unbounded coefficients,” J. Funct. Anal. 254, 3123 (2008).
  8. Chebotarev, A. M. and Ryzhakov, G. V., “Strong resolvent convergence of the Schrödinger evolution to quantum stochastics,” Math. Notes 74, 762 (2003).
  9. Derezinski, J. and De Roeck, W., “Extended weak coupling limit for Pauli-Fierz operators,” Commun. Math. Phys. 279, 1 (2008).
  10. Ethier, S. N. and Kurtz, T. G., Markov Processes: Characterization and Convergence (Wiley, New York, 1986).
  11. Fagnola, F., “On quantum stochastic differential equations with unbounded coefficients,” Probab. Theory Relat. Fields 86, 501 (1990).
  12. Fagnola, F., in Quantum Probability and Related Topics, edited by L. Accardi (World Scientific, Singapore, 1993), Vol. VIII, pp. 143–164.
  13. Geremia, J. M., Stockton, J. K., Doherty, A. C., and Mabuchi, H., “Quantum Kalman filtering and the Heisenberg limit in atomic magnetometry,” Phys. Rev. Lett. 91, 250801 (2003). [MEDLINE]
  14. Gough, J., “Holevo-ordering and the continuous-time limit for open floquet dynamics,” Lett. Math. Phys. 67, 207 (2004).
  15. Gough, J., “Quantum flows as Markovian limit of emission, absorption and scattering interactions,” Commun. Math. Phys. 254, 489 (2005). [ISI]
  16. Holevo, A. S., Quantum Probability & Related Topics, QP-PQ Vol. VII (World Scientific, River Edge, NJ, 1992), pp. 175–202.
  17. Holevo, A. S., “Exponential formulae in quantum stochastic calculus,” Proc. R. Soc. Edinb [Biol] 126A, 375 (1996).
  18. Hudson, R. L. and Parthasarathy, K. R., “Quantum Itô's formula and stochastic evolutions,” Commun. Math. Phys. 93, 301 (1984). [Inspec] [ISI]
  19. James, M. R., Nurdin, H. I., and Petersen, I. R., “H[infinity] control of linear quantum stochastic systems,” IEEE Trans. Autom. Control 53, 1787 (2008). [Inspec]
  20. Kushner, H. J., Approximation and Weak Convergence Methods for Random Processes, With Applications to Stochastic Systems Theory (MIT, Cambridge, MA, 1984).
  21. Lindsay, J. M. and Parthasarathy, K. R., “The passage from random walk to diffusion in quantum probability ii,” Sankhya, Ser. A 50, 151 (1988). [ISI]
  22. Lindsay, J. M. and Wills, S. J., “Construction of some quantum stochastic operator cocycles by the semigroup method,” Proc. Indian Acad. Sci., Math. Sci. 116, 519 (2006).
  23. Lindsay, J. M. and Wills, S. J., “Quantum stochastic operator cocycles via associated semigroups,” Math. Proc. Cambridge Philos. Soc. 142, 535 (2007).
  24. Mabuchi, H., “Coherent-feedback quantum control with a dynamic compensator,” Phys. Rev. A 78, 032323 (2008).
  25. Nurdin, H. I., James, M. R., and Petersen, I. R., e-print arXiv:0711.2551.
  26. Parthasarathy, K. R., “The passage from random walk to diffusion in quantum probability,” J. Appl. Probab. 25A, 151 (1988).
  27. Parthasarathy, K. R., An Introduction to Quantum Stochastic Calculus (Birkhäuser, Basel, 1992).
  28. Reed, M. and Simon, B., Fourier Analysis, Self-Adjointness, Methods of Modern Mathematical Physics Vol. 2 (Academic, New York, 1975).
  29. Reed, M. and Simon, B., Functional Analysis, Methods of Modern Mathematical Physics Vol. 1 (Academic, New York, 1980).
  30. Stroock, D. W. and Varadhan, S. R. S., Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften Vol. 233 (Springer-Verlag, Berlin, 1979).






Copyright © American Institute of Physics