Random Quantum Maps and Their Associated Quantum Markov Chains

TL;DR

This paper introduces 'random quantum maps' as a noncommutative analogue of random continuous maps, exploring their role in quantum Markov chains and the characterization of implemented completely positive maps.

Contribution

It defines the concept of random quantum maps (RQMs), linking them to quantum Markov chains and providing insights into the implementation of completely positive maps.

Findings

Any CPM from B to A is implemented if A is finite-dimensional.

RQMs generate quantum Markov chains through iterative processes.

Partial characterization results for implemented CPMs are provided.

Abstract

The notion of `quantum family of maps' (QFM) has been defined by Piotr Soltan as a noncommutative analogue of `parameterized 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 of `random 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.