  We consider notions of freeness and ambiguity for the acceptance probability
of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study
the injectivity problem of determining if the acceptance probability function
of a MO-QFA is injective over all input words, i.e., giving a distinct
probability for each input word. We show that the injectivity problem is
undecidable for 8 state MO-QFA, even when all unitary matrices and the
projection matrix are rational and the initial state vector is real algebraic.
We also show undecidability of this problem when the initial vector is
rational, although with a huge increase in the number of states. We utilize
properties of quaternions, free rotation groups, representations of tuples of
rationals as linear sums of radicals and a reduction of the mixed modification
of Post's correspondence problem, as well as a new result on rational
polynomial packing functions which may be of independent interest.
