Quantum markov chains and related nonlinear dynamical systems

Abstract

In this thesis, we study forward quantum Markov chains associated with some quantum models on a Cayley tree. As it usually is, the quantum picture is different from classical one. We obtain some unexpected phenomena in quantum settings. More precisely, we give a construction of forward quantum Markov chains and calculate the compatibility condition of boundary conditions for certain interaction operators. We prove that boundary conditions of quantum XY and Ising models can be considered as a trajectory of certain nonlinear dynamical systems. We deeply study asymptotically behaviors of derived dynamical systems. Based on these investigations, we proved an existence of forward quantum Markov chains associated with quantum XY and Ising models. We prove the uniqueness of the quantum Markov chain associated with the quantum model on the Cayley tree of order two. However, we detect an existence of a quantum phase transition on the Cayley tree of order three for the quantum model. Unlike usual quantum phase transition, a phase transition which was observed on the quantum XY ?model does not exhibit at zero temperature. The phase transition surprisingly occurs between two nonzero temperatures, i.e., there exists a lower bound and an upper bound on the temperature for the existence of the phase transition. In order to show similarities between forward Markov chains and Gibbs measures we study the forward quantum Markov Chain associated with the Ising model on the Cayley tree of any order. Contrary to the quantum XY ?model, in the Ising model there always exists a phase transition for any order of the Cayley tree. Moreover, we study supplementary properties of this forward quantum Markov chain. Dynamical systems derived from quantum Markov chains have inspired a renewed interest in the theory of nonlinear operators. We study “advanced fixed point theorems” for an implicit and explicit form of nonlinear operators. All presented results generalize and unify many previous results for different class of nonlinear operators.