Algebraic Characterization of the Class of Languages Recognized by Measure Only Quantum Automata

We study a model of one-way quantum automaton where only measurement operations are allowed (MON-1QFA). We give an algebraic characterization of LMO(Sigma), showing that the syntactic monoids of the languages in LMO(Sigma) are exactly the J-trivial literally idempotent syntactic monoids, where J is the Green's relation determined by two-sided ideals. We also prove that LMO(Sigma) coincides with the literal variety of literally idempotent piecewise testable languages. This allows us to prove the existence of a polynomial-time algorithm for deciding whether a regular language belongs to LMO(Sigma).