DILATIONS A LA QUANTUM PROBABILITY OF MARKOV EVOLUTIONS IN DISCRETE TIME

作者：Gregoratti M^*

来源：Theory of Probability and Its Applications, 2010, 54(1): 140-150.

DOI：10.1137/S0040585X97983997

• Markov chain
• quantum dynamical semigroup
• unitary dilation

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.

• 出版日期2010