  The DisCoCat model of natural language meaning assigns meaning to a sentence
given: (i) the meanings of its words, and, (ii) its grammatical structure. The
recently introduced DisCoCirc model extends this to text consisting of multiple
sentences. While in DisCoCat all meanings are fixed, in DisCoCirc each sentence
updates meanings of words. In this paper we explore different update mechanisms
for DisCoCirc, in the case where meaning is encoded in density matrices---which
come with several advantages as compared to vectors.
  Our starting point are two non-commutative update mechanisms, borrowing one
from quantum foundations research, from Leifer and Spekkens. Unfortunately,
neither of these satisfies any desirable algebraic properties, nor are internal
to the meaning category. By passing to double density matrices we do get an
elegant internal diagrammatic update mechanism.
  We also show that (commutative) spiders can be cast as an instance of the
Leifer-Spekkens update mechanism. This result is of interest to quantum
foundations, as it bridges the work in Categorical Quantum Mechanics (CQM) with
that on conditional quantum states. Our work also underpins implementation of
text-level natural language processing on quantum hardware (a.k.a. QNLP), for
which exponential space-gain and quadratic speed-up have previously been
identified.
