Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T18:33:00.199Z Has data issue: false hasContentIssue false
  • English
  • Français

The passage from random walk to diffusion in quantum probability

Published online by Cambridge University Press:  14 July 2016

K. R. Parthasarathy
Get access Rights & Permissions [Opens in a new window]

Abstract

The notion of a quantum random walk in discrete time is formulated and the passage to a continuous time diffusion limit is established. The limiting diffusion is described in terms of solutions of certain quantum stochastic differential equations.

Keywords

QUANTUM RANDOM WALKTOY COHERENT VECTORCOHERENT VECTORWEYL OPERATORCREATIONGAUGE AND ANNIHILATION PROCESSESQUANTUM STOCHASTIC DIFFERENTIAL EQUATIONQUANTUM DIFFUSIONQUANTUM MARKOV SEMIGROUPBOSON FOCK SPACE
Type
Part 4 - Applied Probability and Quantum Theory
Information
Journal of Applied Probability , Volume 25 , Issue A , 1988 , pp. 151 - 166
Copyright
Copyright © Applied Probability Trust 1988 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

References

[1] Cockroft, A. M. and Hudson, R. L. (1977) Quantum mechanical Wiener processes. J. Multivariate. Anal. 7, 107–124.Google Scholar
[2] Gorini, V., Kossakowski, A. and Sudarshan, E. C. G. (1976) Completely positive dynamical semigroups of n-level systems. J. Math. Phys. 17, 821–825.Google Scholar
[3] Hudson, R. L. and Parthasarathy, K. R. (1984) Quantum Ito's formula and stochastic evolutions. Commun. Math. Phys. 93, 301–323.Google Scholar
[4] Hudson, R. L. and Parthasarathy, K. R. (1984) Construction of quantum diffusions. In Quantum Probability and Applications to the Quantum Theory of Irreversible Evolutions, ed. Accardi, L., Frigerio, A. and Gorini, V., Lecture Notes in Mathematics 1055, Springer-Verlag, Berlin, 173–198.Google Scholar
[5] Hudson, R. L. and Parthasarathy, K. R. (1984) Stochastic dilations of uniformly continuous completely positive semigroups. Acta Appl. Math. 2, 353.CrossRefGoogle Scholar
[6] Lindblad, G. (1976) On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130.Google Scholar
[7] Meyer, P. A. (1986) Elements de probabilités quantiques. In Seminaire de Probabilités XX 1984/85, ed. Azéma, J. et Yor, M., Lecture Notes in Mathematics 1204, Springer-Verlag, Berlin, 186–312.Google Scholar
[8] Meyer, P. A. (1987) Elements de probabilités quantiques. In Seminaire de Probabilités XXI, ed. Azema, J. et Yor, M. CrossRefGoogle Scholar
[9] Parthasarathy, K. R. (1987) Infinitely divisible distributions Dizionario delle Scienze Fisiche Istituto della Enciclopedia Italiana.Google Scholar

Reference added in proof

[10] Lindsay, J. M. and Parthasarathy, K. R. (1988) The passage from random walk to diffusion in quantum probability. Sankhya. To appear.Google Scholar