Markovian quasifree states on canonical anticommutation relation algebras

Abstract

The characterization of quasifree product states on CAR algebras is given. We also prove that the quasifree states on CAR algebra which saturate the strong additivity of von Neumann entropy with equality are product states.

Markov triplets on CCR-algebras

“…There are nontrivial Markovian quasi-free states which are not a product in the time localization. The existence of such state is interesting, because it is in contrast to the CAR case [20]. However, the first and the third subalgebras are always independent.…”

“…When ϕ 123 is quasi-free, it is given by a positive operator (corresponding to the 2-point function) and the main goal of the present paper is to describe the Markov property in terms of this operator. The paper [20] studies a similar question for the CAR algebra and [5] is about multivariate Gaussian distributions. Although the multivariate Gaussian case (in classical probability) is rather different from the present non-commutative setting, we use the same block matrix formalism (and the paper [5] was actually a preparation of this problem).…”