  We establish a second anti-blocker theorem for non-commutative convex
corners, show that the anti-blocking operation is continuous on bounded sets of
convex corners, and define optimisation parameters for a given convex corner
that generalise well-known graph theoretic quantities. We define the entropy of
a state with respect to a convex corner, characterise its maximum value in
terms of a generalised fractional chromatic number and establish entropy
splitting results that demonstrate the entropic complementarity between a
convex corner and its anti-blocker. We identify two extremal tensor products of
convex corners and examine the behaviour of the introduced parameters with
respect to tensoring. Specialising to non-commutative graphs, we obtain quantum
versions of the fractional chromatic number and the clique covering number, as
well as a notion of non-commutative graph entropy of a state, which we show to
be continuous with respect to the state and the graph. We define the
Witsenhausen rate of a non-commutative graph and compute the values of our
parameters in some specific cases.
