On Kadison`s Schwarz inequality for quantum quadratic operators on M2(C) and their dynamics

Abstract

In this thesis we describe bistochastic Kadison-Schwarz operators on M2(C). Such a description allows us to find positive, but not Kadison-Schwarz operators. Moreover, by means of such a characterization we construct Kadison-Schwarz operators, which are not completely positive. Then we describe quantum quadratic operators on M2(C) with Haar state. Using such a description, we find a necessary condition for quantum quadratic operators that satisfy the Kadison-Schwarz property. This condition allows us to construct quantum quadratic operators which are not Kadison-Schwarz ones. Also we provide examples of quadratic operators for which corresponding linear mappings are not positive. Furthermore, we study nonlinear dynamics of quadratic operators acting the set of states of M2(C). Namely, we find some conditions for the stability of unique fixed point of such operators. Besides, dynamics of certain concrete quadratic operators are investigated.