Trace monoids with idempotent generators and measure-only quantum automata

In this paper, we analyze a model of 1-way quantum automaton where only measurements are allowed (MON-1qfa). The automaton works on a compatibility alphabet $$(\Sigma, E)$$ of observables and its probabilistic behavior is a formal series on the free partially commutative monoid $$\hbox{FI}(\Sigma, E)$$ with idempotent generators. We prove some properties of this class of formal series and we apply the results to analyze the class $${\bf LMO}(\Sigma, E)$$ of languages recognized by MON-1qfa’s with isolated cut point. In particular, we prove that $${\bf LMO}(\Sigma, E)$$ is a boolean algebra of recognizable languages with finite variation, and that $${\bf LMO}(\Sigma, E)$$ is properly contained in the recognizable languages, with the exception of the trivial case of complete commutativity.

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