Quantum extensions of dynamical systems and of Markov semigroups

Ivan Bardet (1)

1 EDPA - Équations aux dérivées partielles, analyse

We investigate some particular completely positive maps which admit a stable commutative Von Neumann subalgebra. The restriction of such maps to the stable algebra is then a Markov operator. In the first part of this article, we propose a recipe in order to find a quantum extension of a given Markov operator in the above sense. We show that the existence of such an extension is linked with the existence of a special form of dilation for the Markov operator studied by Attal in \cite{Att1}, reducing the problem to the extension of dynamical system. We then apply our method to the same problem in continuous time, proving the existence of a quantum extension for L\'evy processes. In the second part of this article, we focus on the case where the commutative algebra is isomorphic to $\mathcal{A}=l^\infty(1,...,N)$ with $N$ either finite or infinite. We propose a classification of the CP maps leaving $\mathcal{A}$ stable, producing physical examples of each classes.

Mots clés: Quantum Extension; Quantum Probability; Quantum Open System; Dynamical System

Domaines: Physique mathématique [math-ph]; Probabilités [math.PR]; Systèmes dynamiques [math.DS]; Physique Quantique [quant-ph]

Ivan Bardet. Quantum extensions of dynamical systems and of Markov semigroups. 2015. ⟨hal-01199552⟩