In this chapter, we introduce the theory of measurements in continuous time (diffusive case) starting from the particular but important case of complete observation. This allows to present the Hilbert space formulation of the theory, where the state of the observed quantum system is described by a vector in the Hilbert space H of the system. Even if this is a special case of the more general theory presented in Chap. 3, 4 and 5, it deserves a separate treatment for different reasons: it is instructive, it uses only the Hilbert space formulation of quantum mechanics, it is of interest on its own because the stochastic Scrödinger equation presented in this chapter has also been used in different contexts [1–6], some mathematical results of the following chapter will relay anyhow on the theory presented here, and Hilbert space SDEs are the key starting point for efficient numerical simulations of the dynamics of open quantum systems [1, 7].
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Barchielli, A., Gregoratti, M. (2009). The Stochastic Schrödinger Equation. In: Quantum Trajectories and Measurements in Continuous Time. Lecture Notes in Physics, vol 782. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01298-3_2
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