We investigate the entanglement properties of the joint state of a distinguished quantum system and its environment in the quantum Brownian motion model. This model is a frequent starting point for investigations of environment-induced superselection. Using recent methods from quantum information theory, we show that there exists a large class of initial states for which no entanglement will be created at all times between the system of salient interest and the environment. If the distinguished

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... system has been initially prepared in a pure Gaussian state, then entanglement is created immediately, regardless of the temperature of the environment and the non-vanishing coupling.