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Statistical machine learning approaches have been at the epicenter of the ongoing research work in the field of robot learning by demonstration over the past few years. One of the most successful methodologies used for this purpose is a Gaussian mixture regression (GMR). In this paper, we propose an extension of GMR-based learning by demonstration models to incorporate concepts from the field of quantum mechanics. Indeed, conventional GMR models are formulated under the notion that all the observed data points can be assigned to a distinct number of model states (mixture components). In this paper, we reformulate GMR models, introducing some quantum states constructed by superposing conventional GMR states by means of linear combinations. The so-obtained quantum statistics-inspired mixture regression algorithm is subsequently applied to obtain a novel robot learning by demonstration methodology, offering a significantly increased quality of regenerated trajectories for computational costs comparable with currently state-of-the-art trajectory-based robot learning by demonstration approaches. We experimentally demonstrate the efficacy of the proposed approach.