**Spontaneous wave localization as the solution of the Belavkin-Schrödinger nonlinear filtering equation
**

**Abstract:**

In this talk I shall discuss a nonlinear stochastic wave equation derived by
the author [1-4] to describe quantum jumps, state diffusion and
spontaneous localization of the wave function of a quantum particle under a
continuous observation. A dynamical model based on the quantum stochastic
endomorphic flow for such open system will be shown and the solution of the
reduced quantum filtering equation will be demonstrated on the example of a
single quantum particle with a continuous indirect position measurement. I
shall also show some open problems and discuss recent applications of the
corresponding quantum nonlinear filtering theory in quantum control and
communications.

References:

[1] V. P. Belavkin: A New Wave Equation for a Continuous Non-Demolition
Measurement. Phys Letters A 140(3) 355--358 (1989). quant-ph/0512136.

[2] V. P. Belavkin: A Posterior Schrödinger Equation for Continuous
Non-Demolition Measurement. J of Math Phys 31(12) 2930--2934 (1990).

[3] V. P. Belavkin: Continuous Non-Demolition Observation, Quantum Filtering
and Optimal Estimation. In Quantum Aspects of Optical Communication Lecture
notes in Physics 45 131-145, Springer, Berlin 1991.
quant-ph/0509205.

[4] V. P. Belavkin: Quantum Continual Measurements and a Posteriori Collapse
on CCR. Communications Mathematical Physics 146 611--635 (1992).
math-ph/0512070.