Language Modeling with Reduced Densities

We present a framework for modeling words, phrases, and longer expressions in a natural language using reduced density operators. We show these operators capture something of the meaning of these expressions and, under the Loewner order on positive semidefinite operators, preserve both a simple form of entailment and the relevant statistics therein. Pulling back the curtain, the assignment is shown to be a functor between categories enriched over probabilities.