Dilations à la quantum probability of Markov evolutions in discrete time. (English. Russian original) Zbl 1206.60070

Gregoratti, M.

Theory Probab. Appl. 54, No. 1, 140-150 (2010); translation from Teor. Veroyatn. Primen. 54, No. 1, 185-196 (2009).

Author’s abstract: We study the classical probability analogue of the unitary dilations of a quantum dynamical semigroup in quantum probability. Given a (not necessarily homogeneous) Markov chain in discrete time in a finite state space \(E\), we introduce a second system, an environment, and a deterministic invertible time-homogeneous global evolution of the system \(E\) with this environment such that the original Markov evolution of \(E\) is realized by a proper choice of the initial random state of the environment. We also compare these dilations with the unitary dilations of a quantum dynamical semigroup in quantum probability: given a classical Markov semigroup, we show that it can be extended to a quantum dynamical semigroup for which we can find a quantum dilation to a group of *-automorphisms admitting an invariant abelian subalgebra where this quantum dilation gives just our classical dilation.

Reviewer: Vitalii Gasanenko (Kyiv)

MSC:

60J35	Transition functions, generators and resolvents
60K37	Processes in random environments
60K40	Other physical applications of random processes

Keywords:

stochastic matrix; invertible bimeasurable map; dilation of a Markov evolution; dilation of the quantum dynamical semigroups