On quantum quadratic operators of $\bm_2(\mathbb{C})$ and their dynamics

TL;DR

This paper investigates the nonlinear dynamics of quantum quadratic operators on 2x2 matrices, characterizing those with Haar states and Kadison-Schwarz property, and analyzing their stability and examples of operators lacking the property.

Contribution

It provides a detailed description of quantum quadratic operators with Haar states and Kadison-Schwarz property, including an example of an operator without this property, and studies their stability.

Findings

Characterization of q.q.o. with Haar state

Identification of q.q.o. with Kadison-Schwarz property

Example of q.q.o. not satisfying Kadison-Schwarz property

Abstract

In the present paper we study nonlinear dynamics of quantum quadratic operators (q.q.o) acting on the algebra of $2\times 2$ matrices $\bm_2(\bc)$. First, we describe q.q.o. with Haar state as well as quadratic operators with the Kadison-Schwarz property. By means of such a description we provide an example of q.q.o. which is not the Kadision-Schwarz operator. Then we study stability of dynamics of q.q.o.