In the present paper, we construct quantum Markov chains (QMC) over the Comb graphs. As an application of this construction, it is proved the existence of the disordered phase for the Ising type models (within QMC scheme) over the Comb graphs. Moreov
er, it is also established that the associated QMC has clustering property with respect to translations of the graph. We stress that this paper is the first one where a nontrivial example of QMC over non-regular graphs is given.
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 textit{sub-Markov states}, this allows us to characterize translation invariant QMS on regular trees.
In the present paper deals with asymptotical stability of Markov operators acting on abstract state spaces (i.e. an ordered Banach space, where the norm has an additivity property on the cone of positive elements). Basically, we are interested in the
rate of convergence when a Markov operator $T$ satisfies the uniform $P$-ergodicity, i.e. $|T^n-P|to 0$, here $P$ is a projection. We have showed that $T$ is uniformly $P$-ergodic if and only if $|T^n-P|leq Cbeta^n$, $0<beta<1$. In this paper, we prove that such a $beta$ is characterized by the spectral radius of $T-P$. Moreover, we give Deoblins kind of conditions for the uniform $P$-ergodicity of Markov operators.
We introduce quantum Markov states (QMS) in a general tree graph $G= (V, E)$, extending the Cayley trees case. We investigate the Markov property w.r.t. the finer structure of the considered tree. The main result of this paper concerns the diagonaliz
ability of a locally faithful QMS $varphi$ on a UHF-algebra $mathcal A_V$ over the considered tree by means of a suitable conditional expectation into a maximal abelian subalgebra. Namely, we prove the existence of a Umegaki conditional expectation $mathfrak E : mathcal A_V to mathcal D_V$ such that $$varphi = varphi_{lceil mathcal D_V}circ mathfrak E.$$ Moreover, we clarify the Markovian structure of the associated classical measure on the spectrum of the diagonal algebra $mathcal D_V$.
In the present paper, we are aiming to study limiting behavior of infinite dimensional Volterra operators. We introduce two classes $tilde {mathcal{V}}^+$ and $tilde{mathcal{V}}^-$of infinite dimensional Volterra operators. For operators taken from t
he introduced classes we study their omega limiting sets $omega_V$ and $omega_V^{(w)}$ with respect to $ell^1$-norm and pointwise convergence, respectively. To investigate the relations between these limiting sets, we study linear Lyapunov functions for such kind of Volterra operators. It is proven that if Volterra operator belongs to $tilde {mathcal{V}}^+$, then the sets and $omega_V^{(w)}(xb)$ coincide for every $xbin S$, and moreover, they are non empty. If Volterra operator belongs to $tilde {mathcal{V}}^-$, then $omega_V(xb)$ could be empty, and it implies the non-ergodicity (w.r.t $ell^1$-norm) of $V$, while it is weak ergodic.
We introduce a concept of Kadison-Schwarz divisible dynamical maps. It turns out that it is a natural generalization of the well known CP-divisibility which characterizes quantum Markovian evolution. It is proved that Kadison-Schwarz divisible maps a
re fully characterized in terms of time-local dissipative generators. Simple qubit evolution illustrates the concept.
In the present paper, we propose a new construction of quantum Markov fields on arbitrary connected, infinite, locally finite graphs. The construction is based on a specific tessellation on the considered graph, that allows us to express the Markov p
roperty for the local structure of the graph. Our main result concerns the existence and uniqueness of quantum Markov field over such graphs.
In this paper, we consider the classical Ising model on the Cayley tree of order k and show the existence of the phase transition in the following sense: there exists two quantum Markov states which are not quasi-equivalent. It turns out that the fou
nd critical temperature coincides with usual critical temperature.
