On pure quasi-quantum quadratic operators of M₂(C)

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 τ, 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) ⊗ M ₂(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.