Generic properties for random repeated quantum iterations

Abstract

We denote by $M^n$ the set of $n$ by $n$ complex matrices. Given a fixed density matrix $\beta:\mathbb{C}^n \to \mathbb{C}^n$ and a fixed unitary operator $U : \mathbb{C}^n \otimes \mathbb{C}^n \to \mathbb{C}^n \otimes \mathbb{C}^n$, the transformation $\Phi: M^n \to M^n$ $$ Q \to \Phi (Q) =\, \text{Tr}_2 (\,U \, ( Q \otimes \beta )\, U^*\,)$$ describes the interaction of $Q$ with the external source $\beta$. The result of this is $\Phi(Q)$. If $Q$ is a density operator then $\Phi(Q)$ is also a density operator. The main interest is to know what happen when we repeat several times the action of $\Phi$ in an initial fixed density operator $Q_0$. This procedure is known as random repeated quantum iterations and is of course related to the existence of one or more fixed points for $\Phi$. In \cite{NP}, among other things, the authors show that for a fixed $\beta$ there exists a set of full probability for the Haar measure such that the unitary operator $U$ satisfies the property that for the associated $\Phi$ there is a unique fixed point $ Q_\Phi$. Moreover, there exists convergence of the iterates $\Phi^n (Q_0) \to Q_\Phi$, when $n \to \infty$, for any given $Q_0$ We show here that there is an open and dense set of unitary operators $U: \mathbb{C}^n \otimes \mathbb{C}^n \to \mathbb{C}^n \otimes \mathbb{C}^n $ such that the associated $\Phi$ has a unique fixed point. We will also consider a detailed analysis of the case when $n=2$. We will be able to show explicit results. We consider the $C^0$ topology on the coefficients of $U$. In this case we will exhibit the explicit expression on the coefficients of $U$ which assures the existence of a unique fixed point for $\Phi$. Moreover, we present the explicit expression of the fixed point $Q_\Phi$