Exact separation for all positive-word differences

Establish whether for every pair of distinct positive words u and v over {a,b} with equal letter counts (so that w := u^{-1}v lies in the commutator subgroup F_2'), there exist matrices A,B in SU(2) such that tr(w(A,B)) = 0, i.e., determine universal exact separation for all positive-word differences in the two-state SU(2) measure-once quantum finite automaton model.

Background

The paper studies the exact separation problem for two-state measure-once quantum finite automata (MO-QFAs), which reduces to finding A,B in SU(2) such that tr(w(A,B))=0 for w=u^{-1}v, where u and v are distinct positive words with the same abelianization. The authors develop a slice-driven framework and prove exact separation under several certified criteria, leaving a sharply delimited residual class.

Despite these advances, the authors note that a complete solution covering every positive-word difference is not yet known. Hence the universal exact-separation question for this class remains unresolved.