Abstract. The paper contains a detailed computation about the algebra of canonical commutation relation, the representation of the Weyl unitaries, the quasi-free states and their von Neumann entropy. The Markov triplet is defined by constant entropy increase. The Markov property of a quasi-free state is described by the representing block matrix. The proof is based on results on the statistical sufficiency in the non-commutative case. The relation to classical Gaussian Markov triplets is also described.

AMS Subject Classification (1991): 46L53, 60J10, 40C05, 81R15

Keyword(s): Weyl unitaries, Fock representation, quasi-free state, von Neumann entropy, CCR algebra, Markov triplet

Received November 25, 2008, and in final form March 4, 2009. (Registered under 64/2008.)