  Whenever a mathematical proposition to be proved requires more information
than it is contained in an axiomatic system, it can neither be proved nor
disproved, i.e. it is undecidable, or logically undetermined, within this
axiomatic system. I will show that certain mathematical propositions on a
d-valent function of a binary argument can be encoded in d-dimensional quantum
states of mutually unbiased basis (MUB) sets, and truth values of the
propositions can be tested in MUB measurements. I will then show that a
proposition is undecidable within the system of axioms encoded in the state, if
and only if the measurement associated with the proposition gives completely
random outcomes.
