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An extension of Cencov's categorical description of classical inferencetheory to the domain of quantum systems is presented. It provides a novelcategorical foundation to the theory of quantum information that embraces bothclassical and quantum information theory in a natural way, while also allowingto formalise the notion of quantum environment. A first application of theseideas is provided by extending the notion of statistical manifold toincorporate categories, and investigating a possible, uniparametric Cramer-Raoinequality in this setting.

On the categorical foundations of quantum information theory: Categories and the Cramer-Rao inequality