  In the present paper we study forward Quantum Markov Chains (QMC) defined on
a Cayley tree. Using the tree structure of graphs, we give a construction of
quantum Markov chains on a Cayley tree. By means of such constructions we prove
the existence of a phase transition for the XY-model on a Cayley tree of order
three in QMC scheme. By the phase transition we mean the existence of two now
quasi equivalent QMC for the given family of interaction operators
$\{K_{<x,y>}\}$.
