Existence, uniqueness and approximation of a stochastic Schrödinger equation: The diffusive case. (English) Zbl 1167.60006

Stochastic Schödinger equations, or Belavkin equations, are perturbations of Schrödinger type equations. They describe the evolution of an open quantum system undergoing a continuous quantum measurement which is at the origin of the stochastic character of the evolution. Existence and uniqueness are proven for the solution of these equations in the diffusive case with a two-level system. Then the model is physically justified by proving that it is a continuous time limit of a discrete physical procedure with repeated quantum interactions.


60F05 Central limit and other weak theorems
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60J60 Diffusion processes
Full Text: DOI arXiv


[1] Attal, S. and Pautrat, Y. (2006). From repeated to continuous quantum interactions. Ann. Henri Poincaré 7 59-104. · Zbl 1099.81040
[2] Bouten, L., Guţă, M. and Maassen, H. (2004). Stochastic Schrödinger equations. J. Phys. A 37 3189-3209. · Zbl 1074.81040
[3] Dacunha-Castelle, D. and Duflo, M. (1986). Probability and Statistics . II . Springer, New York. · Zbl 0586.62004
[4] Davies, E. B. (1976). Quantum Theory of Open Systems . Academic Press, London. · Zbl 0388.46044
[5] Gough, J. and Sobolev, A. (2004). Stochastic Schrödinger equations as limit of discrete filtering. Open Syst. Inf. Dyn. 11 235-255. · Zbl 1086.81057
[6] Kümmerer, B. and Maassen, H. (2003). An ergodic theorem for quantum counting processes. J. Phys. A 36 2155-2161. · Zbl 1040.82042
[7] Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035-1070. · Zbl 0742.60053
[8] Kurtz, T. G. and Protter, P. (1991). Wong-Zakai corrections, random evolutions, and simulation schemes for SDEs. In Stochastic Analysis 331-346. Academic Press, Boston, MA. · Zbl 0762.60047
[9] Kurtz, T. G. and Protter, P. E. (1996). Weak convergence of stochastic integrals and differential equations. In Probabilistic Models for Nonlinear Partial Differential Equations ( Montecatini Terme , 1995). Lecture Notes in Math. 1627 1-41. Springer, Berlin. · Zbl 0862.60041
[10] Parthasarathy, K. R. (1992). An Introduction to Quantum Stochastic Calculus. Monographs in Mathematics 85 . Birkhäuser, Basel. · Zbl 0751.60046
[11] Pellegrini, C. (2007). Existence, uniqueness and approximation of stochastic Schrödinger equation: the Poissonian case. · Zbl 1207.17013
[12] Protter, P. E. (2004). Stochastic Integration and Differential Equations , 2nd ed. Applications of Mathematics ( New York ) 21 . Springer, Berlin. · Zbl 1041.60005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.