On pure quasi quantum quadratic operators of M_2(C)

TL;DR

This paper investigates quasi quantum quadratic operators on 2x2 matrices, introducing a weaker form of purity called q-purity, and explores conditions under which these operators are positive or nonlinear, with implications for quantum channel construction.

Contribution

It characterizes all trace-preserving quasi q.q.o. on M_2(C) and links q-purity to positivity and nonlinearity, advancing understanding of quantum nonlinear channels.

Findings

Q-purity implies positivity for linear, symmetric quasi q.q.o.

Symmetric quasi q.q.o. with Haar state cannot be positive if q-pure.

Nonlinear quasi q.q.o. with Haar state exhibit non-positivity.

Abstract

In the present paper we study quasi quantum quadratic operators (q.q.o) acting on the algebra of $2\times 2$ matrices $M_2(C)$. It is known that a channel is called pure if it sends pure states to pure ones. In this papers, we introduce a weaker condition, called $q$-purity, than purity of the channel. To study $q$-pure channels, we concentrate ourselves to quasi q.q.o. acting on $M_2(C)$. We describe all trace-preserving quasi q.q.o. on $M_2(C)$, which allowed us to prove that if a trace-preserving symmetric quasi q.q.o. such that the corresponding quadratic operator is linear, then its $q$-purity implies its positivity. If a symmetric quasi q.q.o. has a Haar state $\tau$, then its corresponding quadratic operator is nonlinear, and it is proved that such $q$-pure symmetric quasi q.q.o. cannot be positive. We think that such a result will allow to check whether a given mapping from $M_2(C)$ to $M_2(C)\o M_2(C)$ is pure or not. On the other hand, our study is related to construction of pure quantum nonlinear channels. Moreover, it is also considered that nonlinear dynamics associated with quasi pure q.q.o. may have differen kind of dynamics, i.e. it may behave chaotically or trivially, respectively.