On Existence of Quantum Trajectories for the Linear Deterministic Processes

The method of "quantum trajectories", i.e. transitions from a pure to a pure quantum state, is a useful tool in the open quantum systems theory and applications. This method relies on the nonlinear stochastic differential equations as a dynamical model. In contrast to this, we pose the question of existence of quantum trajectories for the pure states, each of which would be a solution to a linear, deterministic master equation. It turns out that this task is rather delicate. In its full generality, the task is practically intractable. On the other hand, we do not obtain a general answer even for certain well-established and widely used Markovian processes. Only for a few models for the case of the environment on the absolute zero temperature, we obtain existence of the desired quantum trajectories. In all other cases, there is not even a single such quantum trajectory. In conjunction with the standard method of quantum trajectories, our findings pose some nontrivial challenges for the foundations of the open systems theory and some interpretational and cosmological contexts.