2007/02/23 by Matteo Gregoratti, Gregoratti, M.
Computer Science · Physics and Astronomy · #60J10 (Primary) #81S25 (Secondary) #FOS: Mathematics #Probability (math.PR) #Quantum Computing Algorithms and Architecture #Quantum Mechanics and Applications
paper · pdf · doi:10.48550/arxiv.math/0702690
openalex publication_date 2007/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the Classical Probability analogue of the 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 can be realized by a proper choice of the initial random state of the environment. We also compare this dilations with the 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.