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Distributional semantic models provide vector representations for words by gathering co-occurrence frequencies from corpora of text. Compositional distributional models extend these from words to phrases and sentences. In categorical compositional distributional semantics, phrase and sentence representations are functions of their grammatical structure and representations of the words therein. In this setting, grammatical structures are formalised by morphisms of a compact closed category and meanings of words are formalised by objects of the same category. These can be instantiated in the form of vectors or density matrices. This paper concerns the applications of this model to phrase and sentence level entailment. We argue that entropy-based distances of vectors and density matrices provide a good candidate to measure word-level entailment, show the advantage of density matrices over vectors for word level entailments, and prove that these distances extend compositionally from words to phrases and sentences. We exemplify our theoretical constructions on real data and a toy entailment dataset and provide preliminary experimental evidence. [ABSTRACT FROM AUTHOR]