Random Quantum Maps and Their Associated Quantum Markov Chains

Abstract: The notion of quantum family of maps' (QFM) has been defined by Piotr Soltan as a noncommutative analogue ofparameterized family of continuous maps' between locally compact spaces. A QFM between C*-algebras $B,A$, is given by a pair $(C,\phi)$ where $C$ is a C*-algebra and $\phi:B\rightarrow A\check{\otimes}C$ is a $*$-morphism. The main goal of this note, is to introduce the notion of random quantum map' (RQM), which is a noncommutative analogue ofrandom continuous map' between compact spaces. We define a RQM between $B,A$, to be given by a triple $(C,\phi,\nu)$ where $(C,\phi)$ is a QFM and $\nu$ a state (normalized positive linear functional) on $C$. Our first application of RQMs takes place in theory of completely positive maps (CPM): RQMs give rise canonically to a class of CPMs which we call implemented CPMs. We consider some partial results about the natural and important problem of characterization of implemented CPMs. For instance, using Stinespring's Theorem, we show that any CPM from $B$ to $A$ is implemented if $A$ is finite-dimensional. Our second application of RQMs takes place in theory of quantum stochastic processes: We show that iterations of any RQM with $B=A$, gives rise to a quantum Markov chain in a sense introduced by Luigi Accardi.