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.

我们为 Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA) 的接受概率考虑了自由度和模糊度的概念。我们研究注入性确定 MO-QFA 的接受概率函数是否对所有输入词是单射的问题，即为每个输入词给出不同的概率。我们证明了 8 态 MO-QFA 的注入问题是不可判定的，即使所有酉矩阵和投影矩阵都是有理的并且初始状态向量是实代数的。当初始向量是有理数时，我们也展示了这个问题的不可判定性，尽管状态数量大幅增加。我们利用四元数的性质、自由旋转群、有理元组作为根的线性和的表示和 Post 对应问题的混合修改的简化，以及可能具有独立兴趣的有理多项式包装函数的新结果。