Refinement of quantum Markov states on trees

In the present paper, we propose a refinement for notion of quantum Markov states (QMS) on trees. A structure theorem QMS general trees is proved. We notice that any restriction in sense Ref. \cite{AccFid03} not necessarily to be QMS. It turns out localized has mentioned property which called \textit{sub-Markov states}, this allows us characterize translation invariant regular