  Density operators allow for representing ambiguity about a vector
representation, both in quantum theory and in distributional natural language
meaning. Formally equivalently, they allow for discarding part of the
description of a composite system, where we consider the discarded part to be
the context. We introduce dual density operators, which allow for two
independent notions of context. We demonstrate the use of dual density
operators within a grammatical-compositional distributional framework for
natural language meaning. We show that dual density operators can be used to
simultaneously represent: (i) ambiguity about word meanings (e.g. queen as a
person vs. queen as a band), and (ii) lexical entailment (e.g. tiger ->
mammal). We provide a proof-of-concept example.
