  The zero-error capacity of a classical channel is a parameter of its
confusability graph, and is equal to the minimum of the values of graph
parameters that are additive under the disjoint union, multiplicative under the
strong product, monotone under homomorphisms between the complements, and
normalized. We show that any such function either has uncountably many
extensions to noncommutative graphs with similar properties, or no such
extensions at all. More precisely, we find that every extension has an exponent
that characterizes its values on the confusability graphs of identity quantum
channels, and the set of admissible exponents is either an unbounded
subinterval of $[1,\infty)$ or empty. In particular, the set of admissible
exponents for the Lov\'asz number, the projective rank, and the fractional
Haemers bound over the complex numbers are maximal, while the fractional clique
cover number does not have any extensions.
