  Categorical compositional distributional model of Coecke et al. (2010)
suggests a way to combine grammatical composition of the formal, type logical
models with the corpus based, empirical word representations of distributional
semantics. This paper contributes to the project by expanding the model to also
capture entailment relations. This is achieved by extending the representations
of words from points in meaning space to density operators, which are
probability distributions on the subspaces of the space. A symmetric measure of
similarity and an asymmetric measure of entailment is defined, where lexical
entailment is measured using von Neumann entropy, the quantum variant of
Kullback-Leibler divergence. Lexical entailment, combined with the composition
map on word representations, provides a method to obtain entailment relations
on the level of sentences. Truth theoretic and corpus-based examples are
provided.
