On pure quasi-quantum quadratic operators of M_2(C) - Mukhamedov, Farrukh; Abduganiev, Abduaziz (World Scientific, 2013)
In this paper we study quasi-quantum quadratic operators (quasi-QQO) acting on the algebra of 2×2 matrices M2(C). It is known that a channel is called pure if it sends pure states to pure ones. In this paper, we introduce a weaker condition for the channel called q-purity. To study q-pure channels, we concentrate on quasi-QQO acting on M2(C). We describe all trace-preserving quasi-QQO on M2(C), which allows us to prove that if a trace-preserving symmetric quasi-QQO is such that the corresponding quadratic operator is linear, then its q-purity implies its positivity. If a symmetric quasi-QQO has a Haar state , then its corresponding quadratic operator is nonlinear, and it is proved that such q-pure symmetric quasi-QQO cannot be positive. We think that such a result will allow one to check whether a given mapping from M2(C) to M2(C)x M2(C) is pure or not. On the other hand, our study is related to the construction of pure quantum nonlinear channels. Moreover, we also indicate that nonlinear dynamics associated with pure quasi-QQO may have different kind of dynamics, i.e. it may behave chaotically or trivially.