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In the present paper we provide a construction of Quantum Markov chain on a Cayley tree. Moreover, we give a concrete example of such chains, which is shift invariant and has the clustering property.International Advanced of Technology Congress (ATCi ),PWTC, Malaysia. November 3-5, 2009 Quantum Markov Chains on a Caylay tree Farrukh Mukhamedov1 1De