Refinement of quantum Markov states on trees

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